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Now is the right time to become an American Federation of Musicians member. From ragtime to rap, from the early phonograph to today's digital recordings, the AFM has been there for its members. And now there are more benefits available to AFM members than ever before, including a multi-million dollar pension fund, excellent contract protection, instrument and travelers insurance, work referral programs and access to licensed booking agents to keep you working.
As an AFM member, you are part of a membership of more than 80,000 musicians. Experience has proven that collective activity on behalf of individuals with similar interests is the most effective way to achieve a goal. The AFM can negotiate agreements and administer contracts, procure valuable benefits and achieve legislative goals. A single musician has no such power.
The AFM has a proud history of managing change rather than being victimized by it. We find strength in adversity, and when the going gets tough, we get creative - all on your behalf.
Like the industry, the AFM is also changing and evolving, and its policies and programs will move in new directions dictated by its members. As a member, you will determine these directions through your interest and involvement. Your membership card will be your key to participation in governing your union, keeping it responsive to your needs and enabling it to serve you better. To become a member now, visit www.afm.org/join.
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International Musician, Recent News
MTT to Step Down as San Francisco Symphony MD
November 9, 2017 IM
San Francisco Symphony Music Director and Conductor Michael Tilson Thomas announced he will step down as music director at the end of the 2019-2020 season. Thomas, who has been credited with propelling the symphony to international recognition, will celebrate his 25th season at the helm and his 75th birthday in December 2019.
As a member of Locals 9-535 (Boston, MA) and 47 (Los Angeles, CA), Thomas has promised to continue in the post of music director laureate, conducting at least four weeks per year and working on special programming projects. He is credited with altering San Francisco Symphony's performance style, while expanding repertoire to include emphasis on American and contemporary composers, as well as theatrical music. He also promoted music education and outreach, including his multimedia project Keeping Score.
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MICHAEL TREMBLAY: PANEL #10 & CLOSING REMARKS //
Michael Tremblay
Invest Ottawa and Bayview Yards
Michael brings more than 30 years of executive leadership experience with multinational corporations to Invest Ottawa and its stakeholders, most recently from Microsoft Canada where he served as Vice President, Public Services for over a decade. As part of this Fortune 100 company, he led and drove the growth of public sector business; collaborated extensively with employees, partners and customers; and contributed to Microsoft's Canadian leadership team.
Prior to joining Microsoft, he served as a Senior Vice President with SAP Canada and Fujitsu Consulting, and as a Vice President with JDS Uniphase, EDS Systemhouse and Digital Equipment Corporation. An Ottawa native, Michael is also deeply engaged in the community, contributing to the Algonquin College Board of Governors and Foundation Board for more than 13 years. Michael is an active member of the Audit and Corporate Services Committee and has previously served on the Advocacy Committee with the Children's Hospital of Eastern Ontario.

redpajama_set_name: RedPajamaCommonCrawl


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Q: Find the value of $\int_0^1 \frac{\ln(1+x)}{1+x^2}dx$ Find the value of following function's integral  with respect to x from 0 to 1
$$F(x)=\frac{\ln(1+x)}{1+x^2}$$

A: The hint: Let $x=\tan \theta$ and the rest is smooth:
$$\int\limits_0^1\frac{\ln(1+x)}{1+x^2}dx-\frac{\pi}{8}\ln2=\int\limits_{0}^{\frac{\pi}{4}}\ln(1+\tan\theta)d\theta-\frac{\pi}{8}\ln2=$$
$$=\int\limits_{0}^{\frac{\pi}{4}}\ln\frac{\sqrt2\sin\left(\frac{\pi}{4}+\theta\right)}{\cos\theta}d\theta-\frac{\pi}{8}\ln2=$$
$$=\int\limits_{0}^{\frac{\pi}{4}}\ln\sin\left(\frac{\pi}{4}+\theta\right)d\theta-\int\limits_{0}^{\frac{\pi}{4}}\ln\cos\theta d\theta=$$
$$=\int\limits_{0}^{\frac{\pi}{4}}\ln\sin\left(\frac{\pi}{4}+\theta\right)d\theta-\int\limits_{\frac{\pi}{4}}^{0}\ln\sin\left(\frac{\pi}{4}+\frac{\pi}{4}-\theta\right) d\left(\frac{\pi}{4}-\theta\right)=0.$$
Id est, $$\int\limits_0^1\frac{\ln(1+x)}{1+x^2}dx=\frac{\pi}{8}\ln2$$

A: Let us consider the double integral
$$I=\int_{0}^{1} \int_{0}^{1} \frac{x}{(1+xy)(1+x^2)} \ dy \ dx.$$ Do the inner integral to get
$$I= \int_{0}^{1} \frac{\ln(1+x)}{x^2+1} \ dx.$$
Now reverse the order of integration, justified by Fubini's Theorem.
$$I=\int_{0}^{1} \int_{0}^{1} \frac{x}{(1+xy)(1+x^2)} \ dx \ dy.$$ Now we use partial fractions on the integrand
$$\frac{x}{(1+xy)(1+x^2)}= \frac{x}{(x^2+1)(y^2+1)}+\frac{y}{(x^2+1)(y^2+1)} - \frac{y}{(1+xy)(y^2+1)},$$ and evaluate the double integral, which simplifies to
$$I= \int_{0}^{1} \frac{\ln(2)}{2(y^2+1)} + \frac{\pi y}{4(y^2+1)} - \frac{\ln(1+y)}{y^2+1} \ dy = \frac{\pi \ln(2)}{4}- I.$$
which gives us $$I=\frac{\pi \ln(2)}{8}.$$

redpajama_set_name: RedPajamaStackExchange


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Animal Arms Race
A type of arms race appears to be developing among many species of animal life, particularly the insects. Such a race demands a rapid evolution of improved hunting—defense tactics for all groups of organisms involved.
The monarch butterfly has developed a slick evolutionary trick that shields it from voracious birds such as the blue jay. In the past monarchs were memorable morsels for the blue jay, but in recent years, probably stemming from the butterflies' diet of milkweed, they sequester in their body packets of poison capable of sending a naïve predator into violent fits of retching. Any blue jay swallowing a monarch butterfly in this day and age is almost immediately taken ill. When recovered, the bird will avoid the monarch like the plague. Of course this doesn't do the original swallowed butterfly any good, but its sacrifice helps preserve the race. And because the blue jay will subsequently avoid any bug that even remotely resembles the butterfly, other insects are protected as well.
The evolving traits of protective body poisons apparently extend to other lowly forms of life. For years scientists assumed that invertebrates such as insects were of too limited intelligence to learn from feeding experiences. Recent experimentation indicates that this is definitely not so, at least for the praying mantis. One research entomologist, May Berenbaum, fed snacks of milkweed bugs to a type of mantis and watched as the predator regurgitated its meal. The mantis very plainly refused a second helping and went so far as to haul away the foul—tasting bug offered as dessert. The conclusion was definite, as the scientist observed: "Obviously they're not so stupid. They will reject noxious prey. They not only perceive differences between edible and inedible bugs, but they learn to generalize."
To test her theory further, Berenbaum and a graduate student raised milkweed bugs on two different diets. One group was fed a diet of milkweed, a plant containing a high concentration of cardenolides, a chemical known to be toxic to many animals. A second group of identical bugs was fed a special diet of sunflower seeds. The results were illuminating.
A mantis encountering milkweed bugs for the first time showed no reluctance in pursuing, capturing, and devouring the unfortunate insects. Those bugs fed sunflower seeds received the same treatment and were quickly converted into dinner. However, as was expected, the mantises that had eaten milkweed—fed bugs became violently sick and quickly gave up their meal. It was then very noticeable that the sadder but wiser predators learned to avoid and reject even harmless bugs that had been painted by the scientists to resemble milkweed—fed bugs.
The evolution of animal protective techniques has expanded in such a manner that insects can even compete for survival with higher forms of life. An excellent example in the growing competition for survival is one involving the bat.
The bat emits sounds at frequencies of up to 230,000 vibrations per second, well above the human range of hearing. The bat's built—in sonar system permits it to read echoes of these sounds, which bounce off intended victims, and the hunter can thereby judge the location of the prey as well as the speed and direction in which it is moving.
Certain species of moths have developed methods of countering the bat's sonar system. Their own hearing organs are, depending on the species, located on various parts of their bodies and are used as a built—in early warning system tuned to the bat's high—pitched cries. The element of surprise is thereby defeated, and the forewarned moth is often able to escape.
Extreme refinement seems to be developing in the tiger moth. This species has developed a way to jam the bat's sonar system by producing a clicking sound that actually mimics the bat's echo location cry. Thus the information relayed to the bat is false and causes it to swerve off course and miss its prey.
Admittedly research into the learning process of lower forms of life is only in the infant stage. But clearly such learning processes may be an important selective force that has permitted some insect species to survive through the ages. Most recent research has shown that creatures ranging from scorpions to spiders can also be trained to avoid distasteful prey, much to the benefit of the prey.


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Veniamin Yevgenyevich But (; born 1 August 1961) President of Russian rowing Federation (November 2012 – November 2016) is a retired Soviet rower. Competing for the Soviet Union in the men's eight, he won a world title in 1985 and an Olympic silver medal at the 1988 Summer Olympics.

References

1961 births
Living people
Russian male rowers
Soviet male rowers
Rowers at the 1988 Summer Olympics
Rowers at the 1992 Summer Olympics
Olympic silver medalists for the Soviet Union
Olympic rowers of the Soviet Union
Olympic rowers of the Unified Team
Olympic medalists in rowing
World Rowing Championships medalists for the Soviet Union
Medalists at the 1988 Summer Olympics
Honoured Masters of Sport of the USSR

redpajama_set_name: RedPajamaWikipedia


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Submission of the report by the Praesidium of the Convention on the Future of Europe (Rome, 18 July 2003)
On 18 July 2003, at an official ceremony in Rome, Valéry Giscard d'Estaing (on the left), President of the European Convention, submits the final report of the Draft Treaty establishing a Constitution for Europe to Silvio Berlusconi (on the right), Italian Prime Minister and President-in-Office of the European Council.
Source and copyright
Source: Remise du rapport de la présidence de la Convention sur l'avenir de l'Europe au président du Conseil européen. Rome: Commission européenne, 18.07.2003. Couleur.
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Historical events in the European integration process (1945–2014): 1998–2009 The unification of Europe
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Join Solar Oregon and Oregon Chapter of the Sierra Club to learn about the Portland Clean Energy Fund (PCEF) and how our community can benefit from clean energy technologies.
Solar Drinks is an opportunity for sustainable energy advocates to meet other community members who support solar renewable energy and green building design. Come join us for a relaxed social and networking event where you can speak with Solar Oregon staff, solar industry members, and your neighbors about any topic under the sun!
Solar Drink's is a free event, but a $5-$10 donation is suggested for non-Solar Oregon and Sierra Club Members to help us cover the costs of providing this event.


Cabárceno Nature Park brings together education, culture and science and recreation. One hundred and fifty species of animals from five continents roam semi-freely around the park, which has become one of the biggest tourist attractions in northern Spain.
More than 20km of roads criss-cross the Park, leading you to the different enclosures via fantastic gorges, placid lakes and intriguing rock formations. There are also numerous parking areas and paths that let you discover stunning hidden corners.
Cabárceno Nature Park is the perfect place to spend the day as a family and to enjoy the wonders of nature. What's more, you can also enjoy personalised experiences, like the wild tour or the explorer tour, which will make your visit unforgettable.


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You are here: Home > Canals & Rivers > Canals & Rivers > Rochdale Canal > Historical Information
Boaters Services
The Rochdale Canal was the first of three canals to be built across the Pennines and was opened in 1804. It was a busy commercial route during the nineteenth century but traffic declined during the First World War. In 1952, the majority of the canal was closed to navigation. Restoration was promoted by IWA from the early 1970s and with the founding of Rochdale Canal Society. After many years of long campaigning, the Rochdale Canal was the first major waterway restoration to receive a Millennium Lottery grant and this was matched by funding from English Partnerships (the government's then economic development division) in recognition of the social and economic benefits the restored canal would offer. The devolution of English Partnerships' responsibilities to the Regional Development Agencies presented funding difficulties but The Waterways Trust took over ownership of the Rochdale Canal Company (which owned the canal) and brought funding arrangements to fruition. British Waterways was appointed to carry out the work and operate the waterway.


Check out our Aqualane Shores Market Report to see market activity including Active and Sold Listings!
Aqualane Shore is comprised of approximately 350 home sites, most of which are waterfront with quick access to the Gulf. The unique, palm-lined streets of Aqualane Shores is at the south end of town situated between the elegance of Port Royal the ambiance of Old Naples. The area offers a quiet setting where deep water canals entice large boat owners. Streets are dotted with a variety of home styles, some of which were originally built in the '60s and '70s. Most of these homes have been expanded and renovated, or in many cases, razed to make room for a new grand residence.
Aqualane Shores borders on the Third Street shopping area with restaurants and wonderful shopping opportunities. Another great attraction for residents is that the beach is within walking or biking distance. The community encompasses nearly 300 acres and is located within the city limits of Naples.


Which Disney princess(or heroine) does Fluttershy remind anda of?
Snow White because they can both talk to animals, and Snow White's dress bottom is yellow which reminds me of Fluttershy. And also both are shy and kind.


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Inflyte Radar: Selective Response
Posted on 22nd August 2019
California's increasingly growing presence in the world of techno takes another step in the right direction as west coast studio talent Selective Response launches his very own imprint, Crisis Of Man.
The label announcement has found the producer appear on our Inflyte Radar this month, where we discussed everything from the reason for wanting his own point of distribution to what he thinks of the California and American techno scenes, what he has coming in the near future, and more. Find out more about his story below.
Selective Response! How are things?
Hey! Honestly, things are rather surreal at the moment.
You're on our Inflyte Radar this month, what do you have going on right now?
Where to begin? Currently trying to keep my head on straight and focus on getting music done without getting distracted by all the amazing opportunities that are forthcoming. It's a challenge, but a much welcomed and appreciated one.
We saw that you're about to launch your own imprint, Crisis Of Man, what's the concept and why is it the right time to launch your own label?
I've been producing for nearly a decade, and since day one, my music has never quite fit in with others. And, also since day one, I've never been willing to compromise my sound and vision to make something work. This has definitely made things much more challenging, but that's ok, because I'm proud and confident in who I am today, and my music has never been a better reflection of that.
That being said, I got tired of being told no, which is also ok and actually beneficial, so I decided to once again take matters into my own hands and give myself a platform to release my work. I'd always planned on starting the label anyways but decided I might as well do it now. Label hasn't even launched, and things are already looking extremely promising.
As for the name, it actually came from a track I've done, which is exactly how Selective Response came about. It's a take on today's society and where mankind is in our evolution. There's a quote that says, "Man's greatest burden is unfulfilled potential", which perfectly fits in with the theme the label. Preventing yourself from achieving anything at the risk of failure is a sure-fire way to live a life of regret. Fuck that.
And your most recent single, Malicious Intent, tell us about that? Was it a typical studio session that sparked the idea?
Yes, it was a typical studio session, but it's a funny story with that one. I'm involved in other areas of the industry, and constantly have new gear coming in and out of my studio. With that track, I was jamming on either the Korg Monologue or Prologue and came up with this blistering lead that right away struck me as something special. I'm a very visual creator with a vivid imagination, and in my head, I saw a peak time moment inside of a dark and sweaty bunker, people going absolutely apeshit, and that was the foundation of the track.
I actually made the track almost a year and a half ago, and as is the case with much of my earlier music, the track was greater than my ability to produce it. So, instead of forcing it, I sat on it. I sent it to a couple people and never got any feedback, but I knew that in time it would come together. Fast forward to a few months ago and the confidence in my skills finally caught up, so here we are.
You're based over in California, how do you compare the techno scene there to the rest of the world?
Obviously, the scene and market in the US in general is much smaller, but California specifically is really beginning to come into its own. The LA rave and warehouse scene back in the 90s and 00s played a crucial role in electronic music in America, so the roots are there. But trends come and go, and bigger clubs began to take over. However, thanks to people like Nik Wilson, the Synthetik Minds, Lights Down Low, Incognito, Omen Recordings and others, the underground is alive and well. Nearly every weekend, the best artists are playing here and having a great time. It warms my cockles!
And do you think you'll be moving to Europe to have easier access to a bigger range of techno clubs?
I'm never leaving! But in all seriousness, I've had this chat with many people, and while at first glance it would be an obvious decision, but the sheer amount of people who are doing that is a major deterrent. I actually was planning on doing that a few years ago, but then thought, why move into an oversaturated market when there is an opportunity to help build the scene right here? That's not to say I won't go over there for an extended period of time, but I feel like, with time, Los Angeles will become a global electronic music destination.
Also, there are quite a few artists who live here that seem to be doing great including Developer, Drumcell, Silent Servant (who actually lives probably 20 minutes from me), and Truncate, who all play places all over the world, including the legendary Berghain. Plus, I mean have you ever been to the beach in the middle of winter when it's 85 (or 25c) degrees? Two of my biggest inspirations and energy sources are sunshine and the ocean. Moving away from that would massively restrict my creativity.
Lastly, before we go, we heard you're working on something with Ansome?
That project isn't so much a collaboration, but more of me being a middle man. That being said, it's going to be a cool experience for producers and such – more to come on that! However, I have something similar in the works with an artist who has been a huge inspiration for me for so many years, but that's still in it's early development – so for now, I can't really say anything but watch this space!
inflyte radar
selective response


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Be Highly Visible - Part III: Programming
Dr. Geoffrey Zoeller, Patriots' Path Council vice president for membership and relations, continues his series on why your unit needs to be highly visible! This blog post deals with having engaging programming that makes your unit more visible to the community. It is also a great way to market Scouting to community and business leaders in your area.
Here are some concrete examples:
Local media outlets always welcome Scouts. Not only is this a great way for the boys to learn about making connections with the community, but some organization allow the boys to participate in making commercials, recording songs, etc. A great experience for the boys and an effective means of free advertising for Scouting.
Many regional learning centers and natural sites of interest have Scout-specific programs that are well thought-out and highly structured. Pictures from visits can be submitted to local papers, websites, and school newsletters. Having groups of Scouts visit these sites also makes Scouting highly visible to other patrons and visitors.
Your nearby historical society will be happy to provide programs to your Scouts that meet achievement and elective requirements. At the same time, your Scouts will be visible at local sites of interest, which can often lead to relationship building and even community service projects. Plus ... it's fun!
Be Highly Visible - Part II: Be seen
Dr. Geoffrey Zoeller, Patriots' Path Council vice president for membership and relations, continues his series on why your unit needs to be highly visible! This blog post deals with making connections within the community and interfacing with community and religious organizations.
BSA's Adopt-A-School program is a great way to build a relationship with your local school and also show the school community that your unit is active and doing positive things with youth. Go to http://www.bsaadoptaschool.org/ for more information about how to get your unit active.
Be sure to participate in community events and celebrations. This fulfills the Duty to Country aspect of our program, it shows off your unit to the public in a positive light, and most importantly, the boys love it. Event organizers love having Cub Scouts participate and this is another opportunity to build relationships with important community leaders.
Having Scouts earn their youth religious award not only fulfills our programs Duty to God beliefs, but is also a great way to develop relationships with the religious organizations themselves. Congregations love seeing Scouts earn these awards and non-Scouts who are in the church youth group might be interested in finding out more about Scouting. Go to http://www.praypub.org for more information about youth religious award programs.
Be Highly Visible - Part I: Make it easy for families to find you
Dr. Geoffrey Zoeller, Patriots' Path Council vice president for membership and relations, gives some solid advice for local units: "I've heard it said more than once that the Cub Scout Pack in a town was the community's 'best kept secret.' The last thing we want is for Cub Scouting to be any sort of secret. Your unit needs to be highly visible! Show the community the value of scouting for their youth, do community service, be out there and get noticed in meaningful and positive ways. That is what is going to attract new families and create excitement and anticipation with new boys."
The next few blog posts will provide units with some concrete examples:
Get your unit's outings published in the newspaper when they
are doing 'cool' things that other boys would want to do.
Recognize community leaders often and publically for their support of Scouting.
Handsome certificates are available at your local Scout Shop.
Be sure to have boys wear their uniforms to local athletic and community events. What's a better
recruiter than an excited little Cub Scout happy and looking sharp in their uniform?
There are many, many more ideas on how to be visible. The next few blog posts in this series will give you even more ideas.
Potential Cub Scouts come right to your door on Halloween
As we approach Halloween, Dr. Geoffrey Zoeller, Patriots' Path Council vice president for membership and relations, reminds us about engaging with the youth of our community. "Who is going to be coming to your door for candy in a few days? Potential Cub Scouts! When giving them treats, why not also invite them to join Scouting? Pass out information/invite cards, tie slips of paper with invites to bags, use labels and stick them to candy wrappers. You can even do a twist on front yard camping displays by creating a scary tent set-up. Most importantly, briefly engage youth about what Scouting has to offer and give them some means of getting more information."
The below cards were created by the Westchester-Putnam Council for this purpose and can be obtained by clicking here.
Remember, our youth deserve a safe and fun-filled Halloween, so make your approach sincere, informative, and quick!! -- there are many more houses for them to visit.
Use Formal Invitations to Recruit and Avoid School Restrictions on Flyers
Dr. Geoffrey Zoeller, Patriots' Path Council vice president of membership/relations, makes the observation that "some schools will not allow units to use tradition, council-generated flyers to be distributed for recruitment purposes. Others will only let such flyers be sent home in electronic form via a virtual backpack."
Many school policies allow for the distribution of written invitations, however, so that students can invite one another to birthday parties, special events, etc. Dr. Zoeller notes that "some crafty units are taking a different approach to recruitment and while still adhering to Board policies are instead creating formal 'invitations' as an alternative to prohibited flyers." It may seem like splitting hairs, but there is a definite difference between flyers and invitations in most Board Policy Manuals.
Why not create a formal looking invitation from a template many units have used annually for the Blue and Gold Dinner?
A neat design that recalls the traditional Cub Scout neckerchief.
Inside you can customize an "invite" message in any way you wish.
By creating a formal invitation, which many school district policies allow, you can circumvent the restriction on flyers and obtain the desired access to new families and youth -- all while putting your best foot forward and coming across as professional and creative.
To download the two customizable templates in Word format, click here and also here.
Good luck and happy recruiting!
Author's Note: A special "shout out" to Cub Scout Pack 182 in Warren, Watchung Mountain District, for their creative use of these invitations and for persevering in getting them distributed as invitations in their schools. Well done!!
Webelos Scouts and Boy Scout Camporees
Patriots' Path Council's vice president for membership and relations, Dr. Geoffrey Zoeller, discusses further ideas for effective Cub Scout Pack Planning . "The June/July 2013 edition of Advancement News offers some great insight into creating additional opportunities for Webelos that also supports quality Webelos to Scout Transition practices. The below article come directly from page 2 of the Advancement News." You can download the full edition at: http://library.constantcontact.com/download/get/file/1109945141041-14/06-07+2013+Advancement+News.pdf
Webelos Scouts and Boy Scout Camporees This time of year during annual program planning, Webelos leaders often have questions related to Webelos Scouts attending Boy Scout camporees. Most frequently asked: Can they attend or not? If they can, why should they, and what can they do while there?
Webelos dens might decide it would be a good idea to attend camporees to complete Arrow of Light Requirement 4, "With your Webelos den, visit at least one Boy Scout–oriented outdoor activity," or even Requirement 5, "Participate in a Webelos overnight campout or day hike." From an advancement view, camporee attendance seems to fulfill the "letter of the law" for the former, but probably not for the latter—unless the Webelos Scouts hike through the camporee on the way to their own separate campsite!
According to The Cub Scout Leader Book, a well planned Webelos-to-Scout transition program helps Webelos Scouts in many ways:
By introducing them to Boy Scouting skills and future advancement opportunities.
By giving them the opportunity to see youth leadership at work in the troop and sense their own potential as youth leaders.
By allowing them to become confident and enthusiastic about the patrol method.
By generating the desire for troop membership as the result of this gradual exposure to troop-oriented activities.
By letting them make friends in the troop before becoming Boy Scouts.
There is little doubt these benefits would be experienced at a camporee, but the multi-troop setting may not go far enough in sup-porting them. This is primarily because a Scout troop involved in camporee events may not be focused on the Webelos visitors. Consider a similar scenario: When high school seniors visit a prospective college, the college will probably not want those visits to be during homecoming when students are caught up in their own activities. Instead, the college may plan special visit weekends where the focus can be on the potential freshmen.
A Webelos Scout daytime visit to a camporee is encouraged, as mentioned in the Cub Scout Outdoor Program Guidelines (www.scouting.org/scoutsource/CubScouts/resources/CubScoutOutdoorProgramGuidelines.aspx). However, the boys should not compete in the activities designed for Boy Scouts. To return to our previous analogy, those visiting high school seniors would not be able to take classes for credit during their college visitations. Help in determining whether an activity is oriented to Boy Scouts or Cub Scouts, can be found at www.scouting.org/scoutsource/BoyScouts/Age-AppropriateGuidelines.aspx. Most importantly, though, Webelos Scouts cannot spend the night at a Boy Scout camporee.
So much for "letter of the law." What about intent? As noted above in the Cub Scout Leader Book excerpt, the point of the Boy Scout visit and joint activity is to help with the Webelos transition into a troop. This can best be accomplished at a special troop Webelos Scout welcoming event, rather than at a camporee.
Annual Summertime Program Planning - Revisited
Back in April Patriots' Path Council's vice president for membership and relations, Dr. Geoffrey Zoeller, discussed the newly updated national resources for Cub Scout Pack Planning . "Research conducted by Eli Lilly of Indianapolis, Indiana, illustrated that strong packs all have in common good annual pack programs that are developed a year in advance. These plans are then placed in a calendar and shared with all families right in the beginning of the year. This shows good organization, avoids conflicts later, attracts more families, and you'll retain Cub Scouts for longer. Why wouldn't we all want to conduct strong and thoughtful program planning??"
There is an overview of the six steps of the Annual Program Planning Process and additional resources that include:
Pack calendar template that allows you to build and share your calendar with your families electronically or printed
Pack newsletter template so you can share regular information with your pack families
Posters, both letter size and tabloid size, which you can customize for your needs
Family Talent Survey form
Cub Scout Den Meeting Program form
Den and Pack Meeting Resource Guide
Boys' Life Planning Calendar
Boys' Life resources
Pack Budget Planning Guide/worksheets
The national BSA resources on Pack Planning can be accessed at: http://www.scouting.org/scoutsource/CubScouts/Leaders/PackCommitteeResources/UnitProgramPlanningTools.aspx
Additionally, Dr. Zoeller notes that the Patriots' Path Council's own Cub Opportunities page allows for "one-stop-shopping" when it comes time for dropping in great council and district events to round out the Cub Scout Pack calendar. Events like district family camping, the Belt Loop Midway and Bonanza events, fishing tournaments, and a whole host of days trips, museums, sporting events, and so much more!!
Be sure to visit this hugely important resource when planning for the coming year at: http://cubopportunities.ppbsa.org/
The Religious Emblems Program Can Create Stonger Units
Today, Dr. Geoffrey Zoeller, Patriots' Path Council vice president for membership and relations, reminds us about the importance of the Duty to God component of Scouting. "So many of our chartered organizations are religiously-based, and yet so few of our Scouts earn the religious awards available to them. Participation in these programs could create an even stronger bond between the unit, families, and their chartered organization."
Youth members and families must obtain the specific set of materials for their religion. These materials will contain all of the information, lessons, and/or service projects that scouts will need to complete. Some religions also provide adult manuals for counselors and mentors. Check with your local council Scout Shop to see if your booklets are in stock or contact your religious organization directly.
Below are links to a series of webinars to help instruct unit leaders, clergy, and families about the various offerings available to our youth. Others can assist district and council leaders in creating large-scale programming. In addition, you may wish to visit the main P.R.A.Y. website for further information at http://www.praypub.org/main_frameset.htm
Past Webinar Recordings and Other Resources
THE P.R.A.Y. PROGRAM for Protestant and Independent Christian Churches
Anyone interested in the Protestant Christian program (i.e. clergy, lay people, Scout leaders, parents!)
Deb Hazlewood (P.R.A.Y. curriculum writer)
Thinking about offering a P.R.A.Y. class? Find out how easy it is! More importantly, discover how the P.R.A.Y. program can grow your church membership!
An Overview of the P.R.A.Y. Program
View Recorded Webinar (Requires name and email address to view)
PowerPoint:
Download PowerPoint Presentation (Right Click, Save As)
Religious Emblems Coordinator – Overview of the R.E.C. Position (Boy Scouts of America)
For members of BSA (staff and volunteers)
R. Chip Turner, P.R.A.Y. Director of Communication and Training
Learn how the new Religious Emblems Coordinator position can be a tool for Council growth and retention! An overview of the rationale, duties, and resources available to Council and District R.E.C.'s Click here for more R.E.C. Resources.
PowerPoint of REC Presentation (Right Click, Save As)
Religious Emblems Coordinator – Best Practices (Boy Scouts of America)
Steve Bays,
John Lyon, Atlanta Area Council, BSA
Updates and presentation of best practices by guest panelists
View PDF of PowerPoint Slides (PowerPoint coming soon.)
So You're A Unit REC! Now What?
For members of Boy Scouts of America (staff and volunteers)
Basic Orientation for Religious Emblems Coordinators at the Unit Level
Coordinating Council / District / Unit REC's
How All REC's Can Work Together to Achieve the Ultimate Goal of Greater Emblems Usage
FAITH-BASED INITIATIVES (F.B.I.) for Boy Scouts of America
For members of BSA, clergy, church lay leaders
Mark Hazlewood, P.R.A.Y. CEO
Learn what BSA and faith communities have in common and how the religious emblems programs can be the LINCHPIN for membership growth. Learn the rationale and benefits for collaborating with the faith community.
This "mini session" is a teaser for the weeklong workshops offered at Philmont and the Seabase.
Click here for more FBI Resources.
Teaching God and Me
For people who want to teach a God and Me class
Walk through lesson plans, see props and finished projects, gain valuable teaching experience. NOTE: Participants should purchase a copy of the Student, Counselor and Mentor books for this webinar. Books can be purchased from PRAY's online store atwww.praypub.org/store.
PDF of PowerPoint to follow along
God and Me Lesson Plan Grid
Note: There is a long pause at the beginning of the recording, please wait until the presenter begins.
Teaching God and Family
For people who want to teach a God and Family class
REC's Leading Religious Emblems/Duty to God Courses for
University of Scouting / Pow Wow, etc.
The How To's and Content for Leading An Hour-Long Session at Council and District Training Events
PDF of PowerPoint for making notes
Planning and Conducting a Duty to God Encampment
(Including On-line Planning Tools and Templates)
John Lyon, Council Religious Emblems Coordinator and Experienced Leader of Atlanta Area Council Duty to God Encampment
PDF of PowerPoint to follow along, Full PowerPoint Presentation (Right-Click, Save As)
Other resources for webinar (Zipped File, Right-Click, Save As)
For members of GSUSA, clergy, church lay leaders
This "mini session" is a teaser for the F.B.I. workshops offered by P.R.A.Y. in May and November 2013.
Unit Membership and Marketing Chairperson
Patriots' Path Council's vice president for membership and relations, Dr. Geoffrey Zoeller, points out that with "over a hundred years of documented proof that Scouting has a beneficial and ongoing positive impact on the lives of our youth, you would think that this would be reason enough for parents to want to have their children become Scouts." Unfortunately, without a strong and visible unit-level marketing strategy, hundred of boys in every community miss out on the opportunity to join this most-successful of youth movements. This is where the Unit Membership and Marketing Chairperson comes into play.
Area 2 of the Northeast Region has been instrumental in developing a comprehensive overview of this critical Pack Committee position. The chairperson is charged with the responsibility of educating available youth and parents in a given community and making sure that they are aware of the benefits of the Scouting program.
To view the complete Position Overview as developed by NER Area 2, please go to: http://ner-area2.org/wp-content/uploads/2013/03/Unit-MM-Chair-Overview-3-20-13.pdf
Ultimately, every leadership position in a unit needs to be concerned with membership. The new Unit M&M Chairperson, is a great way to keep everyone focused on the "Main Thing".
Major Changes Coming for Venturing
Patriots' Path Council's vice president for membership and relations, Dr. Geoffrey Zoeller, discusses the major changes that are coming out of the national BSA for Venturing. "These proposed changes should really help with Venturing's poor retention rate (53%) and the lack of sustained advancement of youth members (less than 1%!!!) ."
For some time, there has been a national committee looking to restructure the whole program. Their recommendations include a new national committee structure for venturing, a JTE approach to how units function and measure success, a completely new advancement system, as well as changes in reporting, training, uniforms, and more.
A very solid overview can be found in the Scouting magazine blog. Go to the following link to find out more about the exciting changes coming for the Venturing program: http://blog.scoutingmagazine.org/2013/05/28/venturing/
A beginning set of resources can also be found on the national BSA site by going to: http://www.scouting.org/scoutsource/Venturing/NewProgAnnouncement.aspx
Commissioner Role in Membership Support
Today, Dr. Geoffrey Zoeller, Patriots' Path Council vice president for membership and relations, shares some of the ideas regarding Commissioner Role in Membership Support found on the Commissioners of the Boy Scouts of America Facebook page, which can be found at: https://www.facebook.com/CommissionersOfTheBoyScoutsOfAmerica?hc_location=stream
We are all in the membership business, but let's talk about how you can use the Commissioner Corps to support your membership effort. The role of the Commissioner in membership will be an evolving one. The Commissioners own unit retention, and the Commissioners will continue to deliver unit service while we all work together to recruit and empower volunteers to lead the membership drive.
The Commissioners serve as the key link between the many resources of a functional district committee and the unique needs of each unit. In the absence of a strong district committee, the unit commissioner often is asked to fill many unit needs that should, in the best of all worlds, be provided by subject matter experts. Where those subject matter experts do not yet exist, our commissioners should support the efforts of the District Key 3 to recruit and deploy the right resource specialists. We have built this concept of linkage in our "simple and unified" mission statement and job description, outlined below. The concept of linkage is critical to our execution mission.
The Retention Mission of the Commissioner Corps is best achieved by providing an adequate number of trained unit commissioners who provide a link to District Committee resources in support of a quality unit program.
We are asking our unit commissioners to focus on the following four primary areas:
Supporting unit growth in the Journey to Excellence criteria: JTE measures performance characteristics that unlock the door to a successful unit. Unit Commissioners should analyze the unit's program and identify JTE areas where help is needed to move the unit to a higher level of JTE success.
Linking district committee resources to the unit: Unit Commissioners should support the district committee's delivery of a "catalog of services" to support the specific JTE elements needed for a particular unit's health and success.
Visiting units and logging the visits into the Unit Visit Tracking System (UVTS): Our core task remains visiting the unit. UVTS input feeds critical information to the district committee to help link resources to the unit.
Supporting on-time charter renewal: The commissioner's focus is the retention of the unit, though we should be especially mindful of supporting new youth membership efforts as we move more toward a volunteer-led, professionally guided approach to increasing membership.
Finally, the Commissioner Corps is fully supportive of the concept of the New Unit Commissioner, as outlined in the Unit Performance Guide http://www.scouting.org/filestore/membership/pdf/522-025_WB.pdf . The concept of assigning a New Unit Commissioner on a one-on-one basis to each new unit, through the first two recharter cycles, is a proven and approved best method that results in dramatically increased new unit retention.
Together, we can bridge the gap to increased membership in the BSA.
Resources for Commissioner Service
• Administration of Commissioner Service Manual, No. 34501 http://www.scouting.org/scoutsource/Commissioners/Manuals.aspx
• National BSA Commissioner Service website: www.scouting.org/commissioners
• Unit Performance Guide- http://www.scouting.org/filestore/membership/pdf/522-025_WB.pdf
• Unit Visit Tracking System (UVTS) help support. The Unit Visit Tracking System online help provides personal assistance from a BSA Help Desk operator. The phone number for commissioners only is 972-672-¬4389. The operator is on call to answer questions from 7 a.m. to 7 p.m. Monday through Saturday. (Leave a message and they will call you back.)
• UVTS Training Support includes a PowerPoint presentation, a FAQ, instructions, and a video:
—For professionals, go to MyBSA, click on the Knowledge Base tab, and then go to MyTraining and Unit Visit Tracking System.
—For volunteers, go to http://www.scouting.org/scoutsource/Commissioners.aspx , and then click on Unit Visit Tracking System 2.0—NEW. http://www.scouting.org/scoutsource/Commissioners/UnitVisitTrackingSystem.aspx
• MyScouting Support. Volunteers who have questions about MyScouting access or UVTS
access should send an e-mail to MyScouting@scouting.org with their user name, e-mail address, contact information, council name and headquarters city, and question.
Effective Cub Scout Program Planning
Patriots' Path Council's vice president for membership and relations, Dr. Geoffrey Zoeller, discusses the newly updated national resources for Cub Scout Pack Planning . "Research conducted by Eli Lilly of Indianapolis, Indiana, illustrated that strong packs all have in common good annual pack programs that are developed a year in advance. These plans are then placed in a calendar and shared with all families right in the beginning of the year. This shows good organization, avoids conflicts later, attracts more families, and you'll retain Cub Scouts for longer. Why wouldn't we all want to conduct strong and thoughtful program planning??"
Simple things a Council or Unit Commissioner can do to Increase Cub Scout Retention
Dr. Geoffrey Zoeller, Patriots' Path Council vice president for membership and relations, notes that "commissioners can make a huge difference in supporting and improving upon Cub Scouting programs. This in turn can lead to dramatic increases in Cub Scout retention within individual units and council wide."
Some of these ideas are shared in the Commissioners of the Boy Scouts of America Facebook page, which can be found at: https://www.facebook.com/CommissionersOfTheBoyScoutsOfAmerica?hc_location=stream
So, what are some simple things a Council or Unit Commissioner can do to increase Cub Scout retention?
1. Every new Cub has a handbook. Promote to all parents purchasing and using the appropriate Cub Scout handbook—Tiger Cub, Wolf, Bear, Webelos Scout. It is the manual to glean the most out of Cub Scouting for their child.
2. Studies show a trained Cub Scout leader retains Cubs longer. If den leader-specific tr...aining is not available before the first den meeting, have the new leader take Fast Start training. It is available on MyScouting.org
3. All new Cubs earn the Bobcat Rank. Promote earning the Bobcat Rank within 60 days of Joining, and presenting the badge at the First Pack Meeting
4. Promote Boy's Life Magazine. Subscribing and reading Boy's Life helps the boys experience Scouting even when he is not in a meeting
5. First Den Meeting within 7 Days of joining. Quality den meetings are a critical tool for retention. Make sure every Cub and parent knows the date, time and place of the first den meeting before they leave the joining meeting.
6. Communications. Promote frequent Communications between the Pack and parents on a regular basis: monthly newsletter, weekly contact using Facebook, Twitter, etc.
7. Help all Packs find and have a den chief for each den. A good Den Chief serves as a role model for Cub Scouts, they look up to him and follow his lead on how they should act and behave. He serves as the "Big Brother" to the Den
8. A well-planned program. Program planning is one of the most effective tools to retention. The pack must do their program planning in the spring of the year for 12 to 18 months, and distribute the plan to parents. Parents want to know what their Cubs will be doing
9. Summertime Pack program. Keeping Cubs active and participating in Scouting over the summer months is an important retention tools. Cubs and families that enjoy a Scouting summer are more likely to stay involved in the Fall
10. Orientation of all new Cubs and Parents. Successful parent orientation builds loyalty, lessens parent confusion, and reduces the potential of poor participation, which leads to retention through education. All families need to get the right information when they join.
11. Leader Succession Planning. Help the Pack Committee understand the importance of planning in advance who will replace outgoing Den leaders and other pack leadership
Retention is all about Fun. Retention begins and ends at the unit level. The den and pack is where the most program happens and where the Cubs are most affected. So all meetings have to be FUN!!!!!!
Be Highly Visible - Part I: Make it easy for fami...
Potential Cub Scouts come right to your door on Ha...
The Religious Emblems Program Can Create Stonger U...
Simple things a Council or Unit Commissioner can d...


'Fersen' has already seen fame in its appearance in the 2006 film 'Marie Antoinette' starring Kirsten Dunst and based on the Queens life leading up to the revolution. The film won an academy award for Milena Canonero's costume design, adding to her many great works in films such as Wes Anderson's 'The Life Aquatic' and 'The Grand Budapest Hotel'.
Edmond Petit's lampas fabric 'Fersen', with the stripes, flowers, ribbons and bows is in the style of Louis XVI (Queen Marie Antoinette's husband).
The detail in the pattern is quite remarkable and has me wondering how it can be produced with such delicate intricacy. As I gently pull away several weft threads, creating a little fringe at the end of the fabric, I am completely struck by how many thousands of fine strands lay within it.
The glory days of French Lampas go back to the time of Louis father, King Louis XV (1710- 1774). It is woven today in much the same way as it was with the invention of the jacquard loom back in 1801, only changed by some slight modernisation and electrification. It is created by adding extra 'weft' threads (the yarn that weaves through the fabrics from left to right, opposed to the vertical 'warp' threads) and is essentially one base fabric woven, then the extra weft threads picked up along the loom to come to the front and skip across the base fabric for as much as is needed to create the design. They are then, when not in use for the pattern, taken to the back, a stitch of the warp thread holding it before it loosely floats along the length of the back of the base fabric until it is once again needed for its contribution to the design. Each time that particular colour thread is needed, it is brought to the front.
I wish I could more adequately describe this process, to allow everyone reading to visualise how amazing the weaving of it is and how creating each part of this delicate design of tiny buds, roses, a little flower shaped like a violet, fine stems and leaves all woven in the most perfect shades of pink, gold, green and dark chocolate make it a work of art. Further to all this detail, trailing stems with the tiniest buds run up each side of the stripes.
It is only fit, for what is surely one of the prettiest fabrics, to be photographed in one of London's most charming afternoon tea establishments!


text:
\section{Introduction}

The main idea of this paper is to employ the classical
lemma of Abel and  Zeilberger's algorithm for hypergeometric
sums to verify and to discover various identities
on summations that are not hypergeometric.
Abel's lemma \cite{Abel-1826} on summation
by parts is stated as follows.

\begin{lem}
 For two arbitrary sequences
$\{a_k\}$ and $\{b_k\}$, we have
\[
\sum_{k=m}^{n-1}(a_{k+1}-a_k)b_k=\sum_{k=m}^{n-1}a_{k+1}(b_k-b_{k+1})+a_{n}b_{n}-a_{m}b_{m}.
\]
\end{lem}
For a sequence $\{\tau_k\}$, define the forward
difference operator $\Delta$ by
\[
\Delta\tau_k=\tau_{k+1}-\tau_k.
\]
Then Abel's lemma may be written as
\begin{equation}\label{Abel'lemma}
\sum_{k=m}^{n-1} b_k \Delta a_k =- \sum_{k=m}^{n-1} a_{k+1} \Delta b_k +
a_{n} b_{n} - a_{m}b_m.
\end{equation}

Graham, Knuth and Patashnik \cite{Knuth94} reformulated Abel's lemma in terms of finite calculus to evaluate indefinite sums.
Recently,
Chu \cite{Chu-w-ch2007-1} utilized Able's lemma to prove basic hypergeometric identities
including Bailey's very well poised $_6\psi_6$-series identity by finding pairs $(a_k,b_k)$.  Applying Abel's lemma to the
 pairs $(a_k,b_k)$, one obtains
 contiguous relations for the basic hypergeometric sums. Chen, Chen and Gu
\cite{Chen-2008} presented a systematic approach to finding  pairs $(a_k,b_k)$ by using the $q$-Gosper algorithm.

In this paper, we combine Abel's lemma and Zeilberger's algorithm  to verify and discover identities. Moreover, we use an extended version of Abel's lemma to deal with sums involving holonomic sequences. Let us illustrate our approach by considering identities involving harmonic numbers. The $k$-th harmonic number $H_k$ is given by
\begin{equation}\label{def-H}
H_k = \sum_{j=1}^k \frac{1}{j}.
\end{equation}
Note that by definition, $H_k=0$ whenever $k \le 0$.
Let $f_k$ be a hypergeometric term, i.e., $f_{k+1}/f_k$ is a rational function of $k$. We focus on the summation
\begin{equation}\label{eq-fH}
\sum_{k=m}^{n-1} f_k H_k .
\end{equation}
We can use Gosper's algorithm \cite{Gosper78} to determine whether there exists a hypergeometric term $a_k$ such that $\Delta a_k = f_k$. If such $a_k$ exists,   by Abel's lemma we get
\begin{equation}\label{H-trans}
\sum_{k=m}^{n-1} f_k H_k = \sum_{k=m}^{n-1} H_k \Delta a_k = -\sum_{k=m}^{n-1} \frac{a_{k+1}}{k+1}
+ a_n H_n - a_m H_m.
\end{equation}
Hence we can transform a summation involving harmonic numbers into a hypergeometric summation.
We call such an approach {\it the Abel-Gosper method.}

The same idea applies to the definite summation
\[
S(n) = \sum_{k=0}^n F(n,k) H_k,
\]
where $F(n,k)$ is a proper hypergeometric term such that $F(n,k)=0$ for $k>n$. In this case, we can apply
Zeilberger's algorithm to find a hypergeometric term $G(n,k)$
and polynomials $p_0(n), \dots, p_d(n)$ such that
\[
\sum_{j=0}^d p_j(n)F(n+j,k) =
G(n,k+1)-G(n,k).
\]
Taking
\[
f_k = \sum_{j=0}^d p_j(n)F(n+j,k) \quad \mbox{and} \quad a_k=G(n,k)
\]
in \eqref{H-trans} and summing over $k$ from $0$ to $n+d$, we deduce that
\begin{equation}\label{Abel-Z-H0}
\sum_{j=0}^d p_j(n) S(n+j) = - \sum_{k=0}^{n+d} \frac{G(n,k+1)}{k+1} + G(n,n+d+1) H_{n+d+1}.
\end{equation}
Moreover, we see that the summation on the right hand side of \eqref{Abel-Z-H0} is  again a hypergeometric sum. This approach is called {\it the Abel-Zeilberger method.}

In order to apply the above approach to general holonomic sequences, we extend Abel's lemma by
replacing the operator $\Delta$ with a linear difference operator $L$ of the form
\[
L a_k = r_0(k) a_k + r_1(k) a_{k+1} + \cdots + r_d(k) a_{k+d},
\]
where each $r_i(k)$ is a rational function of $k$.
By combining a variation of Zeilberger's algorithm and the extended Abel's lemma,
we can find a recurrence relation for a sum of the form
\begin{equation}\label{s-ab}
S(n) = \sum_k f(n,k) g(n,k),
\end{equation}
where $f(n,k)$ is hypergeometric and $g(n,k)$ satisfies two recurrence relations
\[
g(n,k+d) = r_1(n,k)g(n,k)+\cdots +r_d(n,k) g(n,k+d-1) + u(n,k),
\]
and
\[
g(n+1,k) = s(n,k) g(n,k) + v(n,k),
\]
where all the coefficients $r_i(n,k)$ and $s(n,k)$ are rational functions and
$u(n,k),v(n,k)$ are hypergeometric terms. The algorithm for finding recurrence relations for the sum \eqref{s-ab} is called the \emph{Abel-Zeilberger algorithm}.

The paper is organized as follows. In Section 2, we give examples to demonstrate that many indefinite sums involving harmonic numbers  can be
reduced to hypergeometric sums by the Abel-Gosper method. Section 3 shows how to
apply the Abel-Zeilberger method to find
recurrence relations of definite sums involving harmonic numbers.  For example, the Paule-Schneider identities fall into this
framework. In Section 4,
 by extending Abel's lemma and using a variation of Zeilberger's algorithm, we present
the Abel-Zeilberger algorithm. The last section provides several examples of the Abel-Zeilberger algorithm including  identities involving Fibonacci numbers and derangement numbers, as well some identities on multiple sums.

\section{The Abel-Gosper method}

In this section, we give several examples to illustrate how to combine Abel's lemma with Gosper's algorithm  to evaluate indefinite summations. We shall focus  on summations involving harmonic numbers.

We begin with a simple example. Consider the sum
\[
S(n) = \sum_{k=1}^{n} H_k.
\]
Expressing $1$ as $\Delta k$, we obtain
\begin{equation} \label{Harmonic-1}
S(n) = \sum_{k=1}^n H_k \Delta k = - \sum_{k=1}^n (k+1) \Delta H_k + (n+1)H_{n+1} - H_1 = (n+1) H_n - n.
\end{equation}

The idea of the above example can be generalized to  indefinite sums of products of polynomials and harmonic numbers. The following result is due to Spie{\ss} \cite{Spieb}. Here we give a derivation based on Abel's lemma.

\begin{theo}\label{Harmonic-3}
Let $H_k$ be the $k$-th harmonic number and $u(n)$ be a polynomial of degree $m$ in $n$. Then
\begin{equation}\label{eq-uH}
\sum_{k=0}^{n} u(k) H_k = p(n)H_n - q(n), \quad n=0,1,2,\ldots,
\end{equation}
where $p(n)$ and $q(n)$ are
both polynomials of degree $m+1$ in $n$. Moreover, $p(n)$ is divisible by $n+1$.
\end{theo}
\pf It is well-known that there exists a polynomial $f(k)$ of degree $m+1$ in $k$
such that
$\Delta f(k) = u(k)$ and the constant term of $f(k)$ is zero. Therefore, we may write $f(k)=k g(k)$, where   $g(k)$ is
 a polynomial of degree $m$. From \eqref{H-trans} it follows that
\[
\sum_{k=0}^{n} u(k) H_k = \sum_{k=0}^{n} \Delta (k g(k)) H_k = -\sum_{k=0}^{n}g(k+1) + (n+1)g(n+1)H_{n+1}.
\]
Since the sum $\sum_{k=0}^n g(k+1)$ is a polynomial of degree $m+1$ in $n$ and
\[
(n+1)g(n+1) H_{n+1} =  (n+1)g(n+1)H_n + g(n+1),
\]
we arrive at \eqref{eq-uH}. \qed

Setting $u(n)=1$ in \eqref{eq-uH}, we obtain \eqref{Harmonic-1}. When $u(n)=n^m$ for $m=1,2,3$, we have
\begin{align}
& \sum_{k=1}^{n}kH_k = \frac{n(n+1)}{2}H_n-\frac{(n-1)n}{4}, \label{Harmonic-p1}\\[6pt]
& \sum_{k=1}^{n}k^2H_k = \frac{n(n+1)(2n+1)}{6}H_n-\frac{(n-1)n(4n+1)}{36},\label{Harmonic-p2}\\[6pt]
& \sum_{k=1}^{n}k^3H_k = \frac{n^2(n+1)^2}{4}H_n-\frac{(n-1)n(n+1)(3n-2)}{48}.\label{Harmonic-p3}
\end{align}

The same idea also applies to the bonus problem 69 proposed by Graham, Knuth and Patashnik \cite[Chapter 6]{Knuth94}.

\begin{exam}
Find a closed form for
\[
\sum_{k=1}^{n} k^2 H_{n+k}.
\]
\end{exam}

The above sum can be rewritten as
\begin{equation}\label{BP69s}
\sum_{k=n+1}^{2n} (k-n)^2 H_k = \sum_{k=1}^{2n} (k-n)^2 H_k - \sum_{k=1}^{n} (k-n)^2 H_k.
\end{equation}
Expanding the summands and applying   formulas \eqref{Harmonic-1}, \eqref{Harmonic-p1}, \eqref{Harmonic-p2}, we obtain that
\begin{equation}\label{BP69a}
\sum_{k=1}^n k^2 H_{n+k} = \frac{n(n+1)(2n+1)}{6}(2H_{2n}-H_n)-\frac{n(n+1)(10n-1)}{36}.
\end{equation}

We may also apply Gosper's algorithm directly to $(k-n)^2$ and get its indefinite sum
\[
\frac{1}{3} k^3 - \frac{2n+1}{2} k^2 + \frac{6n^2+6n+1}{6} k + C.
\]
Setting $C=0$ and using \eqref{H-trans}, we also
obtain \eqref{BP69a}.

We remark that Chyzak \cite{Chyzak2000} and Schneider \cite{Schneider2000, Sch07} have proved \eqref{BP69a} by an extension of Zeilberger's algorithm and Karr's algorithm, respectively.

\begin{exam}
Evaluate the sum
\[
\sum_{k=0}^{n-1} \frac{1}{4^k} {2k \choose k} H_k.
\]
\end{exam}

By Gosper's algorithm, we find that
\[
\Delta \frac{2k}{4^k} {2k \choose k} = \frac{1}{4^k} {2k \choose k}.
\]
Therefore,
\begin{eqnarray*}
\sum_{k=0}^{n-1} \frac{1}{4^k} {2k \choose k} H_k & = &
- 2 \sum_{k=0}^{n-1} \frac{1}{4^{k+1}} {2k+2 \choose k+1} + \frac{2n}{4^n} {2n \choose n} H_n \\[6pt]
&=& 2 - \frac{n+1}{4^n} {2n+2 \choose n+1} + \frac{2n}{4^n} {2n \choose n} H_n.
\end{eqnarray*}

\begin{exam}
We have
\begin{equation}\label{Harmonic-6}
\sum_{k=0}^{n}H_k^2=(n+1)H_n^2-(2n+1)H_n+2n.
\end{equation}
\end{exam}
\pf
Setting $a_k=k$ and $b_k=H_k^2$ in  \eqref{Abel'lemma}, we deduce that
\begin{align*}
\sum_{k=0}^{n} H_k^2 & = -\sum_{k=0}^{n} (k+1) \Delta H_k^2 + (n+1) H_{n+1}^2 \\[6pt]
                     & = -2\sum_{k=0}^{n} H_k - \sum_{k=0}^{n} \frac{1}{k+1} + (n+1)H_{n+1}^2 \\[6pt]
                     & = -2(n+1)H_n+2n -H_{n+1}+(n+1)H_{n+1}^2 \tag*{\mbox{(By \eqref{Harmonic-1})}}\\[6pt]
                     & = (n+1)H_n^2-(2n+1)H_n+2n. \tag*{\qed}
\end{align*}

Similarly, by setting $b_k = H_k^3$, we deduce the following identities,
see \cite{Spieb, Chyzak2000},
\begin{align}
\sum_{k=0}^{n}H_k^3&=(n+1)H_n^3-\frac{3}{2}(2n+1)H_n^2+3(2n+1)H_n+\frac{1}{2}H_n^{(2)}-6n,\\
\sum_{k=0}^{n}(2k+1)H_k^3&=(n+1)^2H_n^3-\frac{3}{2}n(n+1)H_n^2+\frac{3n^2+3n+1}{2}H_n-\frac{3}{4}n(n+1).
\end{align}

\section{The Abel-Zeilberger method}

In this section, we use Abel's lemma and Zeilberger's algorithm to find recurrence relations of the definite summation of the form
\[
S(n) = \sum_{k=0}^n F(n,k) H_k,
\]
where $F(n,k)$ is a hypergeometric term in two variables $n$ and $k$. We shall give two examples
to explain the method.
First, we consider an identity due to Chu and De Donno \cite{Chu-w.ch2005}

\begin{exam}
 For $n\geq 0$, we have
\begin{equation}\label{Harmonic-2}
\sum_{k=0}^{n}{n\choose k}^2 H_k=(2H_n-H_{2n}){2n\choose n}.
\end{equation}
\end{exam}

\pf
By applying Zeilberger's algorithm to ${n \choose k}^2$, we obtain a skew
recurrence relation
\[
(n+1) {n+1 \choose k}^2 - 2 (2n+1) {n \choose k}^2 = G(n,k+1) - G(n,k),
\]
where
\[
G(n,k)=(-3-3n+2k){n\choose k-1}^2.
\]
Let
\[
S(n) = \sum_{k=0}^{n}{n\choose k}^2 H_k.
\]
Substituting $F(n,k)={n \choose k}^2$ and $G(n,k)=(-3-3n+2k){n\choose k-1}^2$ into \eqref{Abel-Z-H0}, we find that
\begin{equation}\label{rec-S}
(n+1)S(n+1)-2(2n+1)S(n)= \sum_{k=0}^{n+1} \frac{3n-2k+3}{k+1} {n \choose k}^2 = \frac{4n+1}{n+1}{2n\choose n},
\end{equation}
where the second equality can be justified  by applying Zeilberger's algorithm.

It is easy to verify that \[ R(n)= (2H_n-H_{2n}){2n\choose n}\] satisfies
the same recurrence relation \eqref{rec-S}. Since $S(0)=R(0)=0$, we  get \eqref{Harmonic-2}. This completes the proof. \qed

In \cite{Paule-Schneider2003}, Paule and Schneider considered the following summations:
\begin{equation}\label{eq-Tn}
T_n^{(\alpha)} = \sum_{k=0}^{n}(1+ \alpha (n-2k)H_k){n \choose k}^\alpha, \quad \alpha=1,2,\ldots.
\end{equation}
They found closed forms of $T_n^{(\alpha)}$ for $1 \le \alpha \le 4$ and derived recurrence relations of $T_n^{(\alpha)}$ for $5 \le \alpha \le 9$.
As will be seen, we can combine Abel's lemma and Zeilberger's algorithm to deal with the summations $T_{n}^{(\alpha)}$. As an example, let us consider the case $\alpha=3$.

\begin{exam} For $n\geq 0$, we have
\[
T_n^{(3)} = (-1)^n.
\]
\end{exam}
\pf
Let
\[
F(n,k) = (n-2k){n \choose k}^3.
\]
By Zeilberger's algorithm, we find that
\[
F(n,k)+F(n+1,k) = G(n,k+1)-G(n,k),
\]
where
\[
G(n,k) = (2n-k+2){n \choose k-1}^3.
\]
Let
\[
S(n) = T_n^{(3)} = \sum_{k=0}^n (1+3(n-2k)H_k){n \choose k}^3.
\]
By \eqref{Abel-Z-H0}, we deduce that
\[
S(n) + S(n+1) = - 3 \sum_{k=0}^{n+1} \frac{2n-k+1}{k+1} {n \choose k}^3 + \sum_{k=0}^n {n \choose k}^3 + \sum_{k=0}^{n+1} {n+1 \choose k}^3.
\]
By Zeilberger's algorithm, we find that the right hand side, denoted by $R(n)$, satisfies
\[
(n+1)R(n)+(n+2)R(n+1) = 0.
\]
Since $R(0)=0$, we have $R(n)=0$ for $n=0,1,\ldots$.  It is clear that  $S(0)=1$.
 So we get $S(n)=(-1)^n$. This completes the proof.  \qed

Moreover, as a direct consequence of \eqref{Abel-Z-H0}, we have the following property.

\begin{theo}
Let
\[
U_n^{(\alpha)}=\sum_{k=0}^{n}(n-2k){n\choose k}^{\alpha}.
\]
Assume that the minimal recurrence relation for $U_n^{(\alpha)}$ computed by Zeilberger's algorithm is
\[
\sum_{i=0}^{d}p_i(n)U_{n+i}^{(\alpha)}=0.
\]
Then the summation
\begin{equation}\label{SumT}
\sum_{i=0}^{d}p_i(n)T_{n+i}^{(\alpha)}
\end{equation}
is a hypergeometric summation.
\end{theo}

We find that the sum \eqref{SumT} equals zero for $\alpha=1,2,\ldots,9$ and
conjecture that it holds for any nonnegative integer $\alpha$. We note that this conjecture implies the conjecture of  Schneider and Paule  \cite{Paule-Schneider2003}, which says that
 $T_{n+i}^{(\alpha)}$ satisfies the minimal recurrence relation for $U_n^{(\alpha)}$ computed by Zeilberger's algorithm.

\section{The Abel-Zeilberger algorithm}

In this section, we give a description of the Abel-Zeilberger algorithm.
Notice that in the applications of Abel's lemma given in previous sections, the main idea lies in the fact that $\Delta H_k$ is a hypergeometric term. In fact, there are other  sequences satisfying similar properties that lead us to
consider an extension of Abel's lemma.

Let $\{a_k\}$ be an arbitrary sequence. We consider a linear operator $L$ of the form
\[
L a_k = r_0(k) a_k + r_1(k) a_{k+1} + \cdots + r_d(k) a_{k+d},
\]
where each $r_j(k)$ is a rational function of $k$. We associate the operator $L$ with a dual operator $L^*$ defined by
\[
L^* a_k  = r_0(k)a_k + r_1(k-1)a_{k-1} + \cdots + r_d(k-d) a_{k-d}.
\]
 In the above notation, Abel's lemma can be extended to the following form.

\begin{lem} For two arbitrary sequences
$\{a_k\}$ and $\{b_k\}$, we have
\begin{equation}\label{Generalized Abel's lemma}
\sum_{k=m}^{n-1} L^* a_k \cdot b_k = \sum_{k=m}^{n-1} a_k \cdot Lb_k
- T(n) + T(m),
\end{equation}
where
\begin{equation}\label{def-T}
T(k) = \sum_{i=1}^d \sum_{j=1}^i r_i(k-j)a_{k-j}b_{k+i-j}.
\end{equation}
\end{lem}

\pf It is easy to verify that
\begin{align*}
\sum_{k=m}^{n-1} L^* a_k \cdot b_k &=
\sum_{k=m}^{n-1} \sum_{i=0}^{d} r_i(k-i)a_{k-i} b_{k}\\[6pt]
&= \sum_{i=0}^{d}\sum_{k=m-i}^{n-1-i}r_i(k) a_k b_{k+i}\\[6pt]
&= \sum_{k=m}^{n-1}r_0(k)a_{k}b_{k}+\sum_{i=1}^{d}\left[
\sum_{k=m}^{n-1}+\sum_{k=m-i}^{m-1}-\sum_{k=n-i}^{n-1}\right]r_{i}(k)a_k b_{k+i}\\[6pt]
&= \sum_{k=m}^{n-1}a_{k}\cdot
L b_{k}+\sum_{i=1}^{d}\sum_{k=m-i}^{m-1}r_{i}(k) a_k b_{k+i} - \sum_{i=1}^{d} \sum_{k=n-i}^{n-1} r_{i}(k) a_k b_{k+i}.  \tag*{\qed}
\end{align*}

Let $f(n,k)$ be a bivariate hypergeometric term and $g(n,k)$ be a bivariate function. We aim to find a linear recurrence relation for the definite sum
\[
S(n) = \sum_{k=m}^{\ell}  f(n,k) g(n,k).
\]
Suppose that there exist rational functions $r_j(n,k)$ such that
\[
L g(n,k) = \sum_{j=0}^d r_j(n,k) g(n,k+j)
\]
is a bivariate hypergeometric term. We shall try to find polynomials $p_i(n)$, which are independent of $k$ and not all zero, together with hypergeometric terms $a(n,k)$ and $w(n,k)$ such that
\begin{equation}\label{paw}
\sum_{i=0}^I p_i(n) f(n+i, k) g(n+i, k) = g(n,k) L^* a(n,k) + w(n,k).
\end{equation}
Summing \eqref{paw} over $k$ and applying the extended Abel's lemma, we deduce that
\begin{align}
\sum_{i=0}^I p_i(n) S(n+i) & =  \sum_{i=0}^I \sum_{k=m}^{\ell}  p_i(n) f(n+i, k) g(n+i, k) \nonumber \\
&= \sum_{k=m}^{\ell} ( g(n,k) L^* a(n,k) + w(n,k) ) \nonumber \\
&= \sum_{k=m}^{\ell} a(n,k) L g(n,k) + \sum_{k=m}^\ell w(n,k) - T(\ell+1) + T(m), \label{nonhomrec}
\end{align}
where $T(k)$ is given by \eqref{def-T}. Notice that in the last expression, the two summands are hypergeometric. By Zeilberger's algorithm, the two sums satisfy linear recurrence relations, which lead to a non-homogenous linear recurrence relation for $S(n)$.

To solve equation \eqref{paw},
we impose the condition  that $g(n,k)$ satisfies
the relation
\[
g(n+1,k) = s(n,k)g(n,k) + v(n,k),
\]
where $s(n,k)$ is a rational function and $v(n,k)$ is a hypergeometric term. By induction, it is
easy to show that
there exist rational functions $s_i(n,k)$ and hypergeometric terms $v_i(n,k)$ such that
\begin{equation}\label{sivi}
g(n+i,k) = s_i(n,k) g(n,k) + v_i(n,k).
\end{equation}

Now we can solve the following equation for $p_i(n)$ and $a(n,k)$ by a variation of Zeilberger's algorithm
\begin{equation}\label{pa}
\sum_{i=0}^I p_i(n) f(n+i, k) s_i(n,k) = L^* a(n,k).
\end{equation}
It can be seen that $a(n,k)$ is similar to $f(n,k)$, that is,
\[
R(n,k)= \frac{a(n,k)}{f(n,k)}
\]
is a rational function of $n$ and $k$. Hence \eqref{pa} is equivalent to
\begin{equation} \label{eq-R}
\sum_{i=0}^I p_i(n) \frac{f(n+i, k)}{f(n,k)} s_i(n,k) = \sum_{j=0}^d \frac{f(n,k-j)}{f(n,k)} r_j(n,k-j) R(n,k-j).
\end{equation}
Since $f(n,k)$ is hypergeometric, both $f(n+i,k)/f(n,k)$ and $f(n,k-j)/f(n,k)$ are rational functions. Therefore, \eqref{eq-R} is a non-homogenous linear recurrence equation on $R(n,k)$ with parameters $p_i(n)$, which can be solved by Abramov's algorithm \cite{Abramov}.

Once we find a solution $(p_0(n), \ldots, p_I(n), a(n,k))$ to \eqref{pa}, it is easy to check that
\[
(p_0(n), \ldots, p_I(n), a(n,k), w(n,k))
\]
is a solution to \eqref{paw}, where
\begin{equation}\label{eq-w}
w(n,k) = \sum_{i=0}^I p_i(n) f(n+i, k) v_i(n,k).
\end{equation}

In summary,  the Abel-Zeilberger algorithm can be described as follows.

\begin{description}
\item
{\it Input:} a hypergeometric term $f(n,k)$ and a term $g(n,k)$ satisfying two recurrence relations
\begin{equation}\label{rec-1}
g(n,k+d) = r_1(n,k)g(n,k)+\cdots +r_d(n,k) g(n,k+d-1) + u(n,k),
\end{equation}
and
\begin{equation}\label{rec-2}
g(n+1,k) = s(n,k) g(n,k) + v(n,k),
\end{equation}
where $r_i(n,k)$ and $s(n,k)$ are rational functions, and
$u(n,k)$ and $v(n,k)$ are hypergeometric terms.

\item
{\it Output:} polynomials $p_i(n)$ that are independent of $k$, two hypergeometric terms $t_1(n,k)$, $t_2(n,k)$ and a term $T(k)$ satisfying
\[
\sum_{i=0}^I p_i(n) S(n+i) = \sum_{k=m}^{\ell} t_1(n,k) + \sum_{k=m}^{\ell} t_2(n,k) - T(\ell+1) + T(m),
\]
where
\[
S(n) = \sum_{k=m}^\ell f(n,k)g(n,k).
\]
\end{description}

The algorithm consists of the following steps.

\noindent Initially, we set $I=0$.

\noindent {Step 1.}
For $0 \le i \le I$, compute the rational functions $s_i(n,k)$ and the hypergeometric terms $v_i(n,k)$ defined by \eqref{sivi} by using the recurrence relations
\begin{align*}
& s_{i+1}(n,k)=s(n+i,k) s_i(n,k), \\[6pt] & v_{i+1}(n,k) = s(n+i,k)v_i(n,k) + v(n+i,k),
\end{align*}
with the initial values $s_0(n,k)=1$ and $v_0(n,k)=0$.

\noindent {Step 2.}
Let
\begin{equation}\label{def-L}
L g(n,k) = -r_1(n,k) g(n,k) - \cdots -r_d(n,k) g(n,k+d-1) + g(n,k+d).
\end{equation}
According to \eqref{eq-R}, construct an equation on $p_i(n)$ and $R(n,k)=a(n,k)/f(n,k)$. That is, compute polynomials
\[
P_j(n,k), \, 0 \le j \le d \quad \mbox{and} \quad Q_i(n,k), \, 0 \le i \le I
\]
such that
\begin{equation} \label{PRQ}
\sum_{j=0}^d P_j(n,k) R(n,k-j) = \sum_{i=0}^I p_i(n) Q_i(n,k).
\end{equation}

\noindent {Step 3.}
Solve equation \eqref{PRQ} for $R(n,k)$ and $p_i(n), 0 \le i \le I$ by using Abramov's algorithm.
If all the polynomials $p_i(n)$ are zeros, then we increase $I$ by one and repeat steps 1--3.

\noindent {Step 4.}
Compute $w(n,k)$ based on \eqref{eq-w}. For $L$ given by \eqref{def-L}, compute $T(k)$ according to \eqref{def-T}.
Finally, set
\[
t_1(n,k) = a(n,k) u(n,k) \quad \mbox{and} \quad  t_2(n,k) = w(n,k).
\]
Then $(p_i(n), t_1(n,k), t_2(n,k), T(k))$ is the desired output.

\section{Examples}
In this section, we provide several examples to
compute summations by using the Abel-Zeilberger algorithm.

We first consider the case when $g(n,k)$ is independent of $n$ in  the Abel-Zeilberger algorithm as described in the previous section, i.e., $s(n,k)=1$ and $v(n,k)=0$ in \eqref{rec-2}. In this case, we have $s_i(n,k)=1$ and $v_i(n,k)=0$ so that $t_2(n,k)=0$.

Let $F_n$ be the $n$-th \emph{Fibonacci number} which is defined by the recurrence relation
\[
F_{n+2}=F_{n+1}+F_n,
\]
 for $ n \ge 0$, with initial values $F_0=0, F_1 = 1$. Employing the Abel-Zeilberger algorithm, we can prove the following identities on the product of binomial coefficients and Fibonacci numbers, see \cite{Benjamin, Vajda1989}.
\begin{exam}\label{Fibonacci-1} We have
\begin{align}
& \sum_{k=0}^n {n\choose k}F_k = F_{2n}, \label{F1}\\[6pt]
& \sum_{k=0}^n (-1)^k {n\choose k} F_k = -F_{n}, \\[6pt]
& \sum_{k=0}^n {n\choose k}F_{3k}=2^nF_{2n}, \\
& \sum_{k=0}^n {n\choose k}F_{4k}=3^nF_{2n}.
\end{align}
\end{exam}

\pf For equation \eqref{F1}, taking $f(n,k)={n \choose k}$ and
$g(n,k)=F_k$ as the input of the Abel-Zeilberger algorithm, we
obtain
\[
p_0(n)=1,\ p_1(n)=-3,\ p_2(n)=1, \quad t_1(n,k)=t_2(n,k)=0,
\]
and
\[
T(k) = \big( (2k-n-3)F_k + (n+2-k)F_{k+1} \big) \frac{n!}{(n+2-k)!(k-1)!}.
\]
We see that $T(n+3)=T(0)=0$. Therefore, the summation
\[
S(n) = \sum_{k=0}^n {n \choose k} F_k
\]
satisfies the recurrence relation
\[
S(n)-3S(n+1)+S(n+2)=0.
\]
Since $F_{2n}$ satisfies the same recurrence relation with initial values $F_0 = 0$ and $F_2 = 1$,
we deduce that $S(n)=F_{2n}$.

The other three identities can be proved in the same fashion.
 The detailed arguments are omitted. This completes the proof.
\qed

It is not difficult to see that the Abel-Zeilberger algorithm could be used to verify identities involving general $C$-finite sequences. For instance, suppose that $\{G_k\}_{k \ge 0}$ satisfies a recurrence relation
\[
G_{k+2}=b G_{k+1} + c G_k, \quad k\geq 0,
\]
where $b$ and $c$ are constants. Then the Abel-Zeilberger algorithm generates the recurrence relation
\[
(b+1-c)S(n)-(b+2)S(n+1)+S(n+2)=0
\]
for the summation
\[
S(n)=\sum_{k=0}^{n} {n\choose k} G_k.
\]

The next example involves the $n$-th \emph{derangement number}
$D_n$ as given by
\[
D_n = n! \sum_{k=0}^{n}\frac{(-1)^k}{k!}.
\]
Using the method of MacMahon's partition analysis,
 Andrews and Paule \cite{G.E.Andrews1999}
 the following identity
 on $D_n$.
We shall  give a derivation
 by applying the Abel-Zeilberger
algorithm.

\begin{exam}

\begin{equation}\label{G.E.Andrews-1999}
\sum_{j\geq0} {k \choose j} \frac{(k+n-j)!}{(k+N-j)!}
D_{k+N-j} = (-1)^n \sum_{j\geq0} (-1)^j {n\choose
j} (j+k)!,
\end{equation}
for  $n\geq N\geq n-k$.
\end{exam}

\pf
Substituting $k+N-j$ for $j$,  the left hand side of \eqref{G.E.Andrews-1999} can be rewritten as
\[
S(N,n) = \sum_{j=N}^{k+N} {k\choose j-N}\frac{(n+j-N)!}{j!} D_j.
\]
Because of the recurrence relation
\[
D_n = n D_{n-1} + (-1)^n
\]
for $D_n$, we may put
\[
f(N,j)={k\choose j-N}\frac{(n+j-N)!}{j!}
 \quad \mbox{and} \quad g(N,j)=D_j
\]
as the input of the Abel-Zeilberger algorithm.
Then we obtain
\[
S(N,n)-S(N+1,n) = \sum_{j\geq N}(-1)^j{k\choose
j-N}\frac{(n+j-N)!}{(j+1)!} .
\]
Denote the right hand side by $G(k)$.
 By Zeilberger's algorithm,
we find that for $ k \ge 0$,
\[
(1+N-n+k)G(k)-(2+k+N)G(k+1)=0,
\]
which implies that
$G(k) = 0$ for  $k \ge n-N \ge 0$. Thus we get
$S(N,n)=S(N+1,n)$. In particular,
\[
S(N,n)=S(n,n)=\sum_{j\geq n}{k\choose j-n} D_j.
\]
Applying the Abel-Zeilberger algorithm to
\[
f(n,j) = {k \choose j-n} \quad \mbox{and} \quad g(n,j)=D_j,
\]
we find
\[
(n+1)S(n,n)+(n+k+1)S(n+1,n+1)-S(n+2,n+2) = \sum_{j\geq n}(-1)^j {k+1\choose
j-n} = 0.
\]
By Zeilberger's algorithm, we see that
 the right hand side of \eqref{G.E.Andrews-1999}
 satisfies the same recurrence relation.
 Finally, from the identity
\[
\sum_{j=0}^k {k \choose j} D_j = k!,
\]
we deduce that $S(0,0)=k!$ and $S(1,1)=(k+1)!-k!$,
which coincides with the initial values of
 the right hand side of \eqref{G.E.Andrews-1999}.
  Thus \eqref{G.E.Andrews-1999}
   holds for $n \ge N \ge n-k$. \qed

The following identity was found by Schmidt and has been proved in several
ways, see \cite{Strehl-94, Wegschaider1997}.

\begin{exam}
[The Ap\'{e}ry-Schmidt-Strehl Identity]
 For $n\geq 0$, we have
\begin{equation}\label{Wei-P71-Identity}
\sum_{k}\sum_{j}{n\choose k}{n+k\choose k}{k\choose
j}^3=\sum_{k}{n\choose k}^2{n+k\choose k}^2.
\end{equation}
\end{exam}

\pf
Let
\[
f(n,k) = {n\choose k} {n+k\choose k},  \quad
g(n,k) =\sum_{j=0}^k {k\choose j}^3,
\]
and
\[
S(n) = \sum_{k=0}^n f(n,k)g(n,k).
\]
By Zeilberger's algorithm, $g(n,k)$ is annihilated by the operator
\[
-8(k+1)^2-(7k^2+21k+16)K+(k+2)^2K^2.
\]
where $K$ denotes the shift operator on $k$. Now applying the Abel-Zeilberger algorithm to
$f(n,k)$ and $g(n,k)$, we obtain
\[
S(n)-\frac{(3+2n)(39+51n+17n^2)}{(n+1)^3}S(n+1)+\frac{(n+2)^3}{(n+1)^3}S(n+2)=0,
\]
Meanwhile, using Zeilberger's algorithm, we find that the Ap\'{e}ry
numbers \[ A(n) = \sum_{k=0}^n {n\choose k}^2{n+k\choose k}^2\]  satisfy
the same recurrence relation.
Finally, by comparing the initial values, we arrive at \eqref{Wei-P71-Identity}. \qed

To conclude this paper, we consider the following summations,
\[
S_n^{(\alpha)}=\sum_{k=0}^{n}\left(\sum_{j=0}^{k}{n\choose j}\right)^{\alpha}.
\]
For $\alpha=1,2$ and $3$, closed forms for $S_n^{(\alpha)}$ have been derived by Andrews and Paule \cite{G.E.Andrews1999} by using the
method of MacMahon's partition analysis.
It is easy to see that these formulas can be
derived by using the Abel-Zeilberger algorithm.

\begin{exam}\label{ex-Calkin}
We have
\begin{equation}\label{Calkin's identity-1}
S_n^{(1)} = \sum_{k=0}^{n}\sum_{j=0}^{k}{n\choose j} = n2^{n-1}+2^n,
\end{equation}
and
\begin{equation}\label{Calkin's identity-2}
S_n^{(2)} = \sum_{k=0}^{n}\left(\sum_{j=0}^{k}{n\choose
j}\right)^2=\left(\frac{n}{2}+1\right)2^{2n}-\frac{n}{2}{2n\choose n}.
\end{equation}
\end{exam}
\pf Let
\[
f(n,k)=1, \quad  g(n,k)=\sum_{j=0}^{k}{n\choose j}.
\]
It is clear that
\begin{equation}\label{fg-rec}
g(n,k+1)=g(n,k)+{n\choose k+1} \quad \mbox{and} \quad
g(n+1,k)=2g(n,k)-{n\choose k}.
\end{equation}
Applying the Abel-Zeilberger algorithm to $f(n,k)$ and $g(n,k)$, we find that
\[
S_n^{(1)} = \sum_{k=0}^n (C-k){n \choose k+1} - T(n+1) + T(0),
\]
where $T(k) = (C+1-k) g(n,k)$ and $C$ is a constant. Setting $C=-1$, we get $T(0)=0$. Thus
\begin{align*}
S_n^{(1)}&=-\sum_{k=0}^{n}(k+1){n\choose k+1}+(n+1)g(n,n+1)\\[6pt]
         &=-n\sum_{k=0}^{n}{n-1\choose k}+(n+1)\sum_{j=0}^{n+1}{n\choose j}\\[6pt]
         &=-n2^{n-1}+(n+1)2^n\\[6pt]
         &=n2^{n-1}+2^n.
\end{align*}

Now we consider the evaluation of $S_n^{(2)}$. Let $f(n,k), g(n,k)$ be given as above, and let $h(n,k)=g(n,k)^2$. From \eqref{fg-rec}, we see that
\begin{align*}
& h(n,k+1)=h(n,k)+u(n,k), \\[6pt]
& h(n+1,k) = 4h(n,k) + v(n,k),
\end{align*}
where
\begin{align*}
& u(n,k) = 2{n\choose k+1}g(n,k)+{n\choose k+1}^2,
\\[6pt]
& v(n,k) = -4{n\choose k}g(n,k)+{n\choose k}^2.
\end{align*}
It should be mentioned that there is actually
no need to impose the condition for $u(n,k)$ and $v(n,k)$ to be hypergeometric in the Abel-Zeilberger algorithm. Therefore, we can still apply the Abel-Zeilberger algorithm to $f(n,k)$ and $h(n,k)$ to deduce that
\[
S_n^{(2)} =\sum_{k=0}^{n}(C-k) \left( 2 {n\choose
k+1}g(n,k)+{n\choose
k+1}^2 \right) + (n-C) h(n,n+1)+(C+1)h(n,0),
\]
where $C$ is a  constant. Setting $C=n/2-1$ and applying the Abel-Zeilberger algorithm, we find that
\[
\sum_{k=0}^{n}(n/2-1-k) {n\choose
k+1}g(n,k) = \sum_{k=0}^{n} \frac{-n+k+1}{2} {n\choose k+1}^2 - \frac{n}{2} g(n,0).
\]
Hence
\begin{align*}
S_n^{(2)} &=\sum_{k=0}^{n}(-n+k+1){n\choose k+1}^2-n +
       \sum_{k=0}^{n} \left( \frac{n}{2}-1-k \right) {n\choose k+1}^2 +
       \left( \frac{n}{2}+1 \right) 2^{2n} + \frac{n}{2}\\[6pt]
    &=-\frac{n}{2}\sum_{k=0}^{n}{n\choose k+1}^2 + \left( \frac{n}{2}+1 \right) 2^{2n}-\frac{n}{2}\\[6pt]
    &=-\frac{n}{2}\left({2n\choose n}-1\right)+ \left( \frac{n}{2}+1 \right) 2^{2n}-\frac{n}{2}\\[6pt]
    &=\left( \frac{n}{2}+1 \right)2^{2n} - \frac{n}{2}{2n\choose n}. \tag*{\qed}
\end{align*}

Using the same argument, we can deduce Calkin's identity \cite{Calkin}
\begin{equation}
S_n^{(3)} = n2^{3n-1}+2^{3n}-3n2^{n-2}{2n\choose n}.\label{Calkin's
identity}
\end{equation}

\noindent\textbf{Acknowledgments.}  We wish to thank Professor Lu Yang for valuable comments.
 This work was supported by the
973 Project, the PCSIRT Project of the Ministry of Education, and the National Science
Foundation of China.

redpajama_set_name: RedPajamaArXiv


Forty-five 6th-8th grade Future Cavemen participated in fall skills camp on October 7th.
HECK'S PICNIC Saturday, October 14th 2-4pm @ Twin Branch Park ALL PAST, PRESENT, FUTURE CAVEMEN, AND THEIR FAMILIES ARE INVITED! FREE FOOD WILL BE PROVIDED BY THE COACHING STAFF AND MAROON SWISH CLUB! MEET THE COACHING STAFF AND PLAYERS, GET A PREVIEW OF OUR SEASON, AND LEARN ABOUT OUR BOOSTER CLUB!


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text:
Parents charged with starving their 15-year-old to death, severely injuring another child
topical top story
REEDSBURG | TEENAGER STARVED
Shelley K. Mesch | Wisconsin State Journal , Rob Schultz | Wisconsin State Journal
The right-side unit of this duplex on Alexander Avenue in Reedsburg is where the Omosebi family lived.
ROB SCHULTZ, STATE JOURNAL
BARABOO — The parents of a 15-year-old boy who allegedly died of starvation last week in Reedsburg were charged Tuesday in Sauk County with causing the child's death through neglect.
Kehinde Omosebi, 49, and Titilayo Omosebi, 47, were also charged with child neglect causing great bodily harm after another son, 11, was found alive but extremely emaciated following what the boys' father, Kehinde Omosebi, called a religious fast that he told police began on July 17, according to a criminal complaint.
Reedsburg Police Chief Timothy Becker said police had to break through padlocked doors to reach the boys at a residence on Alexander Avenue on the west side of the city after Kehinde Omosebi alerted them Sunday that his son had died there. Once inside, the police immediately encountered the distinct smell of a corpse, he said.
"It wasn't a fast, it was neglect because the statutes make it clear that you have to provide necessary food (to children)," Becker said. "When you lock your kids in the house and the father is the only one who can leave, it stops being a fast and starts being starvation and neglect."
Officers found the 15-year-old boy's body in a chair in a bedroom, according to Becker. The criminal complaint said the boy was wearing a gray hooded sweatshirt with the hood pulled over his head. The boy's backbones and ribs were visible under his skin, it said.
"It wasn't like the kid was healthy on Thursday and then died on Friday," Becker said. "His death was a long process and his parents did nothing to stop it. That's the most concerning thing about this."
The Omosebis' 11-year-old son was so weak when police arrived that he couldn't walk out of the residence on his own and had a hard time talking, according to the criminal complaint. The boy was holding onto a Bible and an envelope that police said contained pamphlets about death, which were handed out to officers when they arrived at the residence, the complaint said.
The boy also signed his name to a handwritten letter that police found at the residence, the complaint said. It was addressed to "Lawyers of Sauk County Circuit Court" and included statements such as "The hunger is too much. Please help me now so I may eat," "I can't continue in such a life with no food" and "If I don't get food now I'll probably die of hunger."
Becker said the boy was initially taken to Reedsburg Area Medical Center but was later moved to the American Family Children's Hospital in Madison, where he has been placed in protective custody.
"My heart goes out to the kids because the people they trusted the most betrayed them," he said.
The entire residence was barren of furniture and other belongings with the exception of three other chairs in the bedroom, Becker said.
"The residence had no phone, no power, no food," said Becker, who has been in law enforcement for 30 years and Reedsburg's police chief for 13 years. "I've never seen anything like it before. It's tragic."
Kehinde Omosebi, who is from Nigeria, made up a story about being the minister of a church called the "Cornerstone Reformation Ministries," according to Becker.
"He is not affiliated with any church that has any public records proving their existence," Becker said.
Sauk County Circuit Judge Wendy J.N. Klicko set bail at $5,000 for Kehinde Omosebi, who made his initial court appearance via closed circuit television from the Sauk County Jail in Baraboo. As a condition of bail, Klicko ordered him to have no contact with his surviving son. An initial court hearing for Titilayo Omosebi was delayed.
Both parents are eating again, according to Becker.
Assistant District Attorney Linda A. Hoffman asked for the $5,000 bail because she said the Omosebis are a flight risk.
Hoffman said Kehinde Omosebi told police he believes he has done nothing wrong and that his family was planning to move to Atlanta.
He also sold all of the family's belongings and hadn't paid rent in several months, Hoffman told the court.
Omosebi's attorney, Leonie Dolch, told the court that Omosebi hasn't worked since February. He had been employed at Grede Foundries in Reedsburg.
Dolch said Omosebi has lived in the United States for 12 years. Becker said the family moved to Reedsburg about a year ago from Madison. Hoffman said they previously lived in Missouri and Iowa.
Becker said Kehinde Omosebi walked a few miles from the Alexander Avenue residence to the Reedsburg police station Sunday and became very animated when he told police that his oldest son had died Friday and that his family had just finished two days of prayer for him in accordance with their religious beliefs.
The residence is about 100 yards from a community service agency that specializes in programs for children. The Reedsburg campus for Madison College and an elementary school are also on the same block.
"We never saw hide nor hair of any children playing there," said Sheila Pickel, the early childhood supervisor for the Head Start facility in the community service agency. "There are a lot of kids that come down here to play, too, but they never were part of the group."
Vicki Sherd, a bus driver for Head Start, said that it seems like nobody knows anybody who lives in their neighborhood anymore. "We've all lost the sense of neighborhood," she said. "I just wish we could see through some of these houses so we could see the troubles going on inside them and save the children who are being hurt."
According to the criminal complaint:
When police arrived at the residence they found every door that leads outside or to the garage padlocked in a way to keep anyone inside from getting out. They immediately noticed there were no pictures on the walls, the kitchen cabinets and refrigerator were empty, and no eating utensils, plates or any pots or pans were visible. The bathrooms had no personal hygiene items and no toilet paper, shower curtains or towels.
A push mower was found in the garage and another one was found in a hallway closet on the first floor.
Kehinde Omosebi told police that his family had fasted previously but this one was the longest. He said they were waiting for the "blessing from God" and that the fast was supposed to end on Friday, which was the day his son died.
He also told police that they were moving to Atlanta because God told him to move there. Titilayo Omosebi told police that they needed God's blessing before they could move. She said the fasting began July 19 while the boy said it began on July 20.
The boy told police that he was allowed only to drink water during the fast.
Kehinde Omosebi
REEDSBURG POLICE DEPARTMENT
Titilayo Omosebi
Shelley K. Mesch | Wisconsin State Journal
Shelley K. Mesch is a business and technology reporter for the Wisconsin State Journal. She can be contacted at (608)252-6143 or smesch@madison.com
Follow Shelley K. Mesch | Wisconsin State Journal
Rob Schultz | Wisconsin State Journal
Rob Schultz has won multiple writing awards at the state and national levels and covers an array of topics for the Wisconsin State Journal in south-central and southwestern Wisconsin.
Follow Rob Schultz | Wisconsin State Journal
Reedsburg parents arrested after 15-year-old dies during 40-day fast, police say
An 11-year-old child was also found extremely emaciated but alive when police arrested the mother and father on Sunday, the Reedsburg Police Department said.
Suspect arrested in overnight stabbing on State Street, Madison police say
Ephrom Walker, 58, stabbed the 62-year-old victim when their argument on the 200 block of State Street turned violent at about 2:40 a.m.
Man taken to hospital after being shot by Madison police
A Madison police officer shot a 35-year-old man Saturday afternoon on the East Side after the man threatened the officer with a knife, police said.
Man arrested for 4th OWI and firearm charge after wading into Yahara River, Sheriff's Office says
Cody J. Goen, 38, of Waunakee, had left his vehicle in a parking lot at 5360 Westport Road shortly before 7 p.m. Saturday
Nigerian mother reaches deal with Sauk County prosecutors in fasting death case
A Nigerian native charged with starving her two sons — one of them to death — during a religious fast has reached an agreement with prosecutors and is scheduled for a plea hearing Dec. 27.
Nigerian mother pleads guilty in son's fasting death in Reedsburg
As part of a deal with prosecutors, Titilayo A. Omosebi pleaded guilty to one felony count of child neglect resulting in death. She will be sentenced April 1, and faces up to 25 years of prison.
'Tragic mistake': Nigerian mother gets 3 years in prison in Reedsburg fasting death case
"I was not careful enough to see these consequences," 48-year-old Titilayo A. Omosebi said softly as she read from a prepared statement just before learning her sentence. "This was a very tragic mistake on my part and I am very sorry."
Father accused of starving son to death ruled not competent to stand trial
Sauk County Circuit Court Judge Wendy Klicko found Kehinde Omosebi mentally unfit to stand trial Sept. 5 after a second psychiatric exam.


Herbrand gave early signs of his mathematical gifts, entering the École Normale Supérieure at the exceptional age of seventeen and ranking first in the entering class. He completed his doctoral dissertation in April 1929. That October he began a year of service in the French army. He then went to Germany on a Rockefeller fellowship, studying in Berlin (until May 1931) with John von Neumann, then in Hamburg (May-June) with Emil Artin, and in Göttingen (June-July) with Emmy Noether. He left Göttingen for a vacation in the Alps and a few days later was killed in a fall at the age of twenty-three.
The Herbrand theorem establishes an unexpected bridge between quantification theory and sentential logic. Testing a formula for sentential validity is a purely mechanical operation. Given a formula F of quantification theory, one tests the kth Herbrand disjunction of F successively for k = 1, k = 2, and so on; if F is provable, one eventually reaches a number k for which the kth Herbrand disjunction is valid. If F is not provable, there is, of course, no such k; and one never learns that there is no such k (in accordance with the fact that there is no decision procedure for quantification theory). Besides yielding a very convenient proof procedure, the Herbrand theorem has many applications (a field explored by Herbrand himself) to decision and reduction problems and to proofs of consistency. Almost all the methods for proving theorems by machine rest upon the Herbrand theorem.
In modern algebra Herbrand's contributions are in class-field theory, the object of which is to gain knowledge about Abelian extensions of a given algebraic number field from properties of the field. Initiated by Leopold Kronecker and developed by Heinrich Weber, David Hilbert, Teiji Takagi, and Emil Artin, the theory received essential contributions from Herbrand in 1930–1931. He wrote ten papers in this field, simplifying previous proofs, generalizing theorems, and discovering important new results.
Herbrand's logical writings have been reprinted in his Écrits logiques, Jean van Heijenoort, ed. (Paris, 1968); the pref. includes reference to the paper by B. Dreben, P. Andrews, and S. Aanderaa, "False Lemmas in Herbrand," in Bulletin of the American Mathematical Society, 69 (1963), 699–706, as well as further information about the gap in Herbrand's demonstration of his theorem. Also see The Logical Writings of Jacques Herbrand, Warren D. Goldfarb, ed. (Reidel, 1971).
The list of Herbrand's papers on class-field theory can be found in Ernest Vessiot's intro. to Helmut Hasse, Über gewisse Ideale in einer einfachen Algebra (Paris, 1934). Herbrand's Le développement moderne de la théorie des corps algébriques was published posthumously and edited by his friend Claude Chevalley (Paris, 1936).
"Herbrand, Jacques." Complete Dictionary of Scientific Biography. . Encyclopedia.com. 26 Apr. 2019 <https://www.encyclopedia.com>.