Truth matters, especially when it comes to worship. That ought to be obvious; you can't properly praise the Lord if you don't know who He is. Christ Himself was unequivocal on that point—He said true worshippers "must worship in spirit and truth" (John 4:24, emphasis added).
However, much of modern worship music seems to aim at taming the one true God. Some popular "worship songs" are nothing more than artificial praise offered to a different god altogether. In his book Worship, John MacArthur describes the fallout of the biblical illiteracy that permeates the church today.
"Worship" aims to be as casual and as relaxed as possible, reflecting an easy familiarity with God unbefitting His transcendent majesty. This type of "worship" seems to aim chiefly at making sinners comfortable with the idea of God—purging from our thoughts anything like fear, trembling, reverence, or profound biblical truth. . . .
Most diners and restaurants across the country can serve you fried chicken, but only KFC has the Colonel's original recipe of eleven herbs and spices. The same goes for other brand names you recognize—Coca Cola and Krispy Kreme stand out from their competitors because of the uniqueness of their products.
In each case, the secret recipe is the key to their success. Through trial and error, each of them has developed a specific formula for its product that appeals to the widest-possible audience.
But what if you're not selling chicken, soda, or doughnuts? What if, as prosperity preacher T.D. Jakes once said, "Jesus is the product"? Tragically, too many in the church today have adopted that mindset—they've developed their own formulas to make Christ and the gospel more appealing to the world.
Finney was a pragmatist driven solely by results. He held no strong allegiance to any theological framework. That's why his preaching was such a mixed bag—he was only interested in refining his sales pitch. In his day, that meant crusades of fire and brimstone preaching, as he worked to scare sinners into the arms of the Savior.
Finney's methods live on in the hellfire and damnation preachers we see on busy street corners today. Their promise of fire insurance against God's impending wrath echoes the tone and topic of much of Finney's teaching.
But Finney's legacy extends beyond modern prophets of doom. The seeker-sensitive movement—while seemingly antithetical to fire and brimstone preaching—owes just as much to Finney's influence, with its emphasis on emotion, pragmatism, and developing widespread appeal.
Just like Finney, seeker-sensitive gurus are devoted to developing the latest and greatest formula for selling the gospel. Every aspect of the church experience, from the style of music and teaching to design aesthetics—even the kind of clothes the pastor wears—are carefully chosen to make the message as user-friendly and enticing as possible.
The truth is that the gospel doesn't need to be cleverly packaged—it simply needs to be preached.
The pure gospel message has never been a seductive sales pitch; on the contrary, it is foolishness to the unbeliever.
There's no clever gimmick that makes Christ's sacrificial death more believable. The truth of the gospel is inherently offensive to fleshly ears. Making it attractive to the world would require altering the message itself; and that's what often happens in seeker-sensitive churches. Unappealing fundamental truths about sin, hell, repentance, faith, submission, and holiness are buried under layers of worldly entertainment, pop culture trends, and man-centered self-improvement.
The bottom line is simple—either God's Word is sufficient to bring people to repentance and faith or it isn't. And while many seeker-sensitive gurus wouldn't openly question the sufficiency of Scripture, their dependence on extrabiblical methodologies and manipulation unmistakably indicate where their confidence truly lies.
In the end, the fatal flaw of seeker-sensitivity—and every other movement that carries on Finney's legacy—is that they usurp a role that does not belong to them. Salvation is not up to us. No one—no matter how clever or cool—can coerce a sinner into God's kingdom. God alone is responsible for rescuing sinners from hell, as He intervenes into lives and illuminates hearts to the truth of His Word.
As Jonah succinctly declared, "Salvation belongs to the Lord" (Jonah 2:9 ESV). Christ Himself confirmed that very point in John 6:44, explaining that "No one can come to Me unless the Father who sent Me draws him." Only God can resurrect the dead and grant them new life in Him (Ephesians 2:4-5).
In God's gracious plan, we do have a part to play in bringing the gospel to a lost and dying world (Matthew 28:19-20; Romans 10:14). But we need to keep our role in proper perspective: We're called to be messengers, not manipulators. With that in mind, we need to measure the success of our gospel preaching not by the number of professed converts, but by our faithfulness to the truth of Scripture.
Standing together for the gospel compels us to stand together against any movement that overestimates the importance of the messenger and his methods. It means we need to boldly oppose anyone who seeks to usurp God's role in the work of salvation, and that we faithfully proclaim the sufficiency of God's Word.
This article"Together against Extrabiblical Methods"originally appeared at Grace to You. Copyright 2016, Grace to You. All rights reserved. Used by permission.
Imagine you see a drowning man, and the life preserver in your hand is all that stands between him and certain death. You wouldn't disguise the flotation device to resemble the water in which he was drowning. You wouldn't openly question the efficacy of the life-preserver, or debate the relative merits of other floating objects. And you certainly wouldn't try to convince him that he could manipulate the power of the life-preserver if he simply had enough faith.
Sadly, too many in the church today treat the life preserver of the gospel in similar, wrong-headed ways.
Lost in a sea of self-deception, self-righteousness, rebellion, lust, and all sorts of depravity, the world's only hope is the transforming work of the Lord through His Spirit and Word. The church exists as God's witness on earth, testifying to His truth in word and deed, and doing what it can to rescue lost sinners before it is too late.
However, one of the greatest hindrances to the work of the church is often the church itself.
The reasons vary. Sometimes the church excuses and tolerates sin to the point it becomes a hypocritical laughingstock, rendering its testimony meaningless and its ministry impossible. It's hard to overstate how dangerous the church's dalliances with unchecked sin can be. We've already discussed depravity in detail, and if the church can't or won't set a moral and spiritual example, it effectively has nothing to offer the world.
But it's not just sin that impedes the reach and the usefulness of the church. Today, the church's effectiveness is often cut short by wrong priorities. By placing a high priority on things like numerical growth, worldly credibility, or extra-biblical authority, the church diminishes the value of biblical fidelity and cripples its ability to be salt and light in the world (Matthew 5:13-16). Let's consider some prominent examples of that tragic pattern.
Seeker-sensitivity is not a new phenomenon, but its chameleonic qualities have kept it a driving force in church culture for several decades. And while the trends it imitates may change, the underlying priority—to make the church more attractive, entertaining, and agreeable to unsaved audiences—is still misleading congregations across the country and around the world.
If anything, the market-driven philosophy of ministry has become a much more pervasive problem in recent years. Whereas seeker-sensitivity once had a fairly consistent look and feel—primarily appealing to upper-middle class suburban sensibilities—today there is no end of sub-cultures and interest groups that churches cater to.
But all those variations come down to the same basic compromise, and, as John MacArthur explains in Ashamed of the Gospel, they all share the same fatal flaw.
If the church is the only hope there is for the world, why would it spend so much time and effort trying to imitate the world? And if a sinner was convinced of his need for the Savior, why would he turn to a congregation that so closely resembled the world he needed to be saved out of?
If the church wants to confuse, weaken, or shroud the world's only hope, seeker-sensitive methodology is the way to go.
Vying for worldly acceptance and credibility isn't solely the territory of seeker-sensitive churches. The postmodern church is a kind of seeker-sensitive model tuned specifically to cynics, skeptics, and anti-authority types.
The postmodern church embodies the world's approach to truth—that it is relative and subjective. The result is that biblical doctrine—including fundamental and historical doctrine—is open for debate and deconstruction. They hold even their own convictions with an open hand, and rather than establishing and fighting for the truth, they reflect the attitude of the Athenian philosophers Paul encountered, always looking for something new and novel (Acts 17:21).
All of that is undergirded by a feigned humility—as though conviction could only be the product of close-minded arrogance. In that regard, the postmodern church reduces faith and Bible study to academic pursuits, and kicks open the door to any and all spurious doctrines. But that kind of doctrinal pluralism isn't producing more refined theology or more confidence in the Word of God. Instead it's propagating heresy, and, as John MacArthur explains in The Truth War, encouraging unbelief.
The idea that the Christian message should be kept pliable and ambiguous seems especially attractive to young people who are in tune with the culture and in love with the spirit of the age and can't stand to have authoritative biblical truth applied with precision as a corrective to worldly lifestyles, unholy minds, and ungodly behavior. And the poison of this perspective is being increasingly injected into the evangelical church body.
There is no hope to offer the world if the church refuses to be clear about what it believes and steadfast in its commitment to God's Word. God's truth faces enough attacks from the outside world—we need not tear it down from the inside, too.
Another way the church has fostered its own misdirection is exemplified in the charismatic church. By overemphasizing the apostolic gifts and taking an unbiblical view on the work of the Holy Spirit—not to mention the believer's supposed ability to wield and direct the Spirit—the charismatic movement has largely derailed from biblical orthodoxy and plunged millions worldwide into blind idolatry.
Perhaps the most spiritually deadly aspect of the charismatic church is how it insulates its followers from biblical discernment. With miracles and messages from the Lord supposedly abounding, there is no consistency or continuity to the manifestations of the Spirit, and no fixed standard by which all miraculous claims are measured. In spite of the evidence of hoaxes and charlatans, the movement continues to suspend disbelief.
Moreover, any attempt to police the movement is met with accusations of trying to quench the Spirit (1 Thessalonians 5:19-20) or charges of committing the unpardonable sin (Matthew 12:31-32).
Impervious to critique, charismatic leaders can do and say whatever they like, and their capacity for inventing new heresies knows no bounds. As John MacArthur describes in Strange Fire, the charismatic movement has funneled false teaching into the rest of the church.
Unsurprisingly, the world can see through the lies and the corruption of the charismatic church. So its continued dominance isn't just a threat to the people caught under its influence—it also tarnishes the reputation of the rest of the church, as it turns the Lord, His miraculous power, and faith in Him into a pathetic joke for the watching world. And until the charismatic movement begins to police itself and ceases to be a breeding ground for heretics, charlatans, and con artists, it will continue to be a significant hindrance to the reach of the true gospel.
God's people need to get serious about His truth—its sufficiency, its inerrancy, and its authority. We need to be clear about what it says, and what it means by what it says. And we need to be unhindered in our ability to bring that truth to a corrupt and dying world.


text:
\section{INTRODUCTION}

Half-metallic ferromagnetism, which involves metallic conductivity only in
one spin channel, renowned to be a key phenomenon in spintronics.
Its first-principles introduction by de Groot \textit{et al}. \cite{deGroot1983}
raised enthusiasm among the community to explore this property within materials.
For then, several reports have been published regarding the prediction and/or observation
of a half-metallic electronic structure in different materials including metallic oxides
\cite{Jemeda2001,Lewis1997}, full and half Heusler alloys
\cite{Borca2000,Ambrose2000,Galanakis2002,Fujii1990},
and diluted magnetic semiconductors \cite{Akai1998,Ogawa1999,Yao2005}.
In spite of several half-metallic compound appeared in the literature,
the search for novel ferromagnetic half-metals is an active research area.
In this context, a significant attention is attracted to the transition metal
pnictides and chalcogenides with zincblende (ZB) structure
\cite{Sanyal2003,Yao2005-2,Galanakis2003,Xie2003,Xu2002}.
Although, many of transition metal compounds exhibit half-metallic property in the ZB structure,
these materials crystallize in a different crystal structure including hexagonal NiAs or
orthorhombic structures; hence realization of their half-metallic behavior
requires special requirement for synthesis of metastable ZB structure.
It is claimed that the ZB structure MnAs \cite{Kim2006}, CrAs \cite{Mizuguchi2002},
and CrSb \cite{Zhao2001} have been successfully synthesized in the form of thin films.

In 2004 Kusakabe \textit{et al.} \cite{Kusakabe2004} showed that calcium pnictides
in the ZB structure, in absence of any transition metal element,
exhibit half-metal ferromagnetism.
These compounds expose an unconventional type of ferromagnetism
where the spin polarization is in the anion $p$ electrons without any direct
involvement of $d$ orbital, while in the conventional transition metals magnets,
$d$ electrons are responsible for magnetic ordering.
This $p$ orbital magnetization is intrinsic and is not triggered by
crystal defects of any type. The spirit of the exchange is neither
double exchange nor $p-d$ exchange.
It is explained that these compounds belong to classes of $d^0$ ferromagnets
which have been recently observed experimentally and/or theoretically\cite{Kusakabe2004}.
Ferromagnetism in the compounds with no valance $d$ or $f$ electron
has challenged our understanding of magnetism.

Later on, Sieberer \textit{et al.} \cite{Sieberer2006} and
Volnianska \textit{et al.} \cite{Volnianska2007} studied ferromagnetism in
the binary compounds of I$^A$/II$^A$-V elements in tetrahedrally and
octahedrally coordinated structures.
They argued that half-metallic ferromagnetism originates from the flat band
magnetism of holes and may occurs in more structures including RS and NiAs.
It was explained that large cell volume, high ionicity and a slight
hybridization of anion $p$ orbitals and induced $d$ states around
the Fermi level effectively enhance formation of ferromagnetic (FM) state in the system.

Our specific aims in this work is to apply accurate first-principles electronic calculations
to investigate magnetic properties  and structural stability
of two members of binary p magnetic compounds (CaN and CaAs)
in six different structures.
In section III we will discuss structural and magnetic properties of CaN and CaAs. Electronic structures
of these two compounds are described in section IV.

\section{COMPUTATIONAL METHOD}

We performed first-principles calculations based on Density functional theory (DFT) using
pseudopotential technique and spin-dependent generalized gradient approximation (GGA)
in the scheme of Perdew, Burke, and Ernzerhof \cite{PhysRevLett.77.3865}
by means of {\sc Quantum ESPRESSO} package \cite{QE-2009}.
The relativistic effects are taken into account in the scalar limit,
neglecting the spin-orbit coupling which is expected to be small in light atoms like Ca, N, and As.
Using conventional GGA calculations of electronic and magnetic properties
of materials which do not consist of atoms with inner $d$ or $f$
electrons are thought to be accurate and give reliable results.
We used ultrasoft pseudopotentials\cite{PhysRevB.59.1758}, a kinetic energy cut off of 35 Ryd
for the plane wave expansion of Kohn-Sham orbitals, and a kinetic energy cut off of
400 Ry for the Fourier expansion of electron density.
The Brillouin-zone integrations were performed using a Monkhorst-Pack mesh of
$12\times12\times12$ k points for cubic structures and equivalent meshes for hexagonal structures.
Atomic relaxation of the structures with internal parameters was accurately
performed to achieve residual forces less than 0.01 mRy/Bohr and energy
accuracy of better than 0.001 mRy/$fu$ ($formula~unit$).

\begin{center}
\begin{table}[t]
\newcommand{\multicolumn}{\multicolumn}
\caption{\label{tab:parameters}
 Computed spin polarization energies $\Delta E^{FM-NM}$ (meV/$fu$),
 lattice parameters (\AA), and bulk moduli (GPa) for CaN and CaAs in six studied structures.
 For hexagonal structures, the lattice parameter $c/a$ is given below the first lattice
 parameter.
}
\begin{ruledtabular}
\begin{tabular}{lcccccc}
          &  RS  &  ZB  & NiAs &  WZ  & AsNi & NaO  \\
\hline                                              \\
          &   \multicolumn{6}{c}{$\Delta E^{FM-NM}$}        \\
\\%
     CaN  & -102 & -214 & -177 & -124 &  0   &   0  \\
     CaAs & 0    & -110 &  0   &   0  &  0   &   0  \\
\\
          &\multicolumn{6}{c}{Equilibrium lattice parameters} \\
\\%
     CaN  & 5.00 & 5.45 & 3.39 & 4.19 & 3.73 & 6.39 \\
          &      &      & 1.88 & 1.18 & 1.34 & 0.77 \\
     CaAs & 6.00 & 6.73 & 3.88 & 4.11 & 3.82 & 7.90 \\
          &      &      & 2.10 & 2.22 & 2.16 & 0.75 \\
\\
          &    \multicolumn{6}{c}{Bulk modulus}             \\
\\%
     CaN  & 78.4 & 55.4 & 79.8 & 62.3 & 86.9 & 92.1 \\
     CaAs & 40.9 & 24.9 & 39.3 & 30.5 & 41.4 & 49.4 \\
\end{tabular}
\end{ruledtabular}
\end{table}
\end{center}

\section{STRUCTURAL AND MAGNETIC PROPERTIES}

In this work, we take into account six different structures for binary
CaN and CaAs compounds; ZB, wurtzite (WZ), rock-salt (RS), hexagonal NiAs,
hexagonal anti NiAs (AsNi), and NaO structures.
The tetracoordinated cubic ZB structure is important because several binary compound
exhibit half-metallic ferromagnetism in this structure.
The fact that many binary semiconductors crystallize in the ZB structure
increases the importance of study of this structure for binary half-metals,
because these half-metals are expected to be ideal ferromagnetic
sources for spin injection into semiconductors.
Also the tetracoordinated WZ structure is the hexagonal analogous of the cubic ZB structure
and hence is included in our study.
The significant electronegativity differences in Ca-N (2.0 Pauling) and Ca-As (1.0 Pauling) pairs
\cite{periodic-table} shows the significant contribution of ionic bonding in CaN and CaAs and
hence these compounds may favor a higher coordination atomic arrangement
to reduce their total energy.
Therefore, the hexacoordinated cubic RS structure is also considered in this work.
Recently Liu \textit{et al.} have claimed that they have grown a layer
of RS-CaN on Cu(0 0 1) by means of a new unclear self-assembly mechanism \cite{Liu2008}.
Since some of magnetic transition metal pnictides (II$^{TM}$-V compounds)
crystallize in the hexagonal NiAs structure, both NiAs and AsNi (anti-NiAs)
structures were considered for CaN and CaAs.
The natural crystal structure of CaAs seems to be a NaO type lattice \cite{Iandelli1973},
which is derived from the AsNi structure, by a $30^\circ$ rotation clockwise around the z axis,
followed by a shift of $[0,0,0.25]$ in the z direction, and finally an increase of the $a$ and $b$
cell parameters by a factor of $\sqrt{3}$.

The difference between the minimized total energies of the FM
and non-magnetic (NM) states, $\Delta E^{FM-NM}$, of CaN and CaAs in all
considered structures are presented in table \ref{tab:parameters}.
To find the minimized total energies, we have optimized lattice constants
of all structures and atomic positions of the low symmetry NaO structure.
The negative values indicate more stability of the FM phase.
It is seen that CaN has a FM ground state and non-vanishing magnetic moment
in the RS, ZB, NiAs, and WZ structures while the FM behavior of CaAs
is limited to the only ZB structure. Both compounds are NM in the NaO phase.
Regardless of the crystal structure, in all magnetic ground states,
the total magnetic moment per chemical formula unit is 1 $\mu_{B}$.
In contrast to the transition metal pnictides, the small cation spin moment in
the magnetic structures of CaN and CaAs is parallel to anion moment
which may be explained by the model of covalent polarization \cite{Lalanne1996}.
Furthermore, table \ref{tab:parameters} lists the equilibrium lattice parameter and
bulk modulus for both compounds.
It is generally seen that magnetic structures, compared with the nonmagnetic systems,
have higher volume and lower bulk modulus.
As it was mentioned in the introduction, a large cell volume enhances spin polarization
of binary $p$ magnetic materials.

\begin{figure}[!t]
\includegraphics*[scale=0.7]{ev}
\caption{\label{fig:E-V}
 Calculated total energy-volume curves of CaN (top) and CaAs (bottom)
 in different crystal structures.
}
\end{figure}

The calculated total energies in different volumes for CaN and CaAs are plotted
in figure \ref{fig:E-V}. According to the figure, the structures with hexacoordinated
anions are energetically more favorable.
These structures have smaller cell volumes and higher bulk moduli, indicating that their bonds
are stronger due to the smaller interaction distances.
The ZB structure has the highest total energy and the
experimentally observed structure for CaAs, NaO, with a difference of
about 0.1 Ry/$fu$ is the most favorable one for both compounds.
The calculated equilibrium lattice constants of NaO structure for CaAs are 14.93 and
11.15 Bohr for a and c, respectively, which is in good agreement with
the experimentally found values of 14.85 and 11.19 Bohr.
The comparison of the ZB and WZ structures shows that the hexagonal arrangement gives rise
to lower energy for both compounds.

\begin{figure}
\includegraphics*[scale=0.8]{dos-1}
\caption{\label{fig:DOS}
  Calculated total density of states (DOS) of CaN and CaAs in six structures.
  The shaded area shows the contribution of the Ca $d$ orbital an
  the remained white area belongs to the anion $p$ states.
  Dashed lines show the Fermi energy. Positive and negative values hold for spin up
  and spin down states, respectively.}
\end{figure}

\begin{figure}
\includegraphics*[scale=.9]{band}
\caption{\label{fig:band}
  Obtained band structures of CaN and CaAs in the ZB and NaO structures.
  The horizontal dashed lines show the Fermi energy.
  For the ZB structure spin up and spin down band structures are shown
  on the left and right hand side, respectively.}
\end{figure}

\section{ELECTRONIC STRUCTURE}

In this section, the detailed investigation of the electronic structure
of all considered crystal structures is presented, and the essence of ferromagnetism in these
materials is discussed by analyzing their electronic structure.
According to a previous study \cite{Volnianska2007},
in agreement with ours, for Ca pnictides (CaN, CaP, CaAs, CaSb) in the ZB phase,
the energy of spin polarization is the largest for CaN and decreases
with the increase in atomic number of the anion, and almost vanishes for CaSb.
Therefore, anions should have an important contribution in governing
magnetization in these compounds.
For better understanding, we calculated free atom energies in the spin unpolarized and
polarized states and found that the spin polarization energy of a free N atom (-3.32 eV)
is significantly higher than a free As atom (-1.58 eV).
This observation may explain stability of FM state in more structures for CaN,
compared with CaAs.

In Fig \ref{fig:DOS}, the spin resolved total density of states (DOS)
of the systems are presented.
It is seen that the crystal field splitting between bonding and antibonding states
is more pronounced in the tetra coordinated ZB and WZ structures, compared with other systems.
The magnetic structures including NiAs, WZ, ZB, and RS structure of CaN and
ZB structure of CaAs obviously exhibit half-metallic property,
as in their majority spin channel there is an energy gap around
the Fermi level while their minority spin bands cut the Fermi level.
This is different from the half-metallic transition metal pnictides and chalcogenides,
where their majority spin channel is metallic and the half-metallic gap
appears in their minority spin channel.
The spin flip gap, the distance between the majority valence band maximum
and the Fermi level, measures the minimum required energy for creating
a majority spin carrier at the top of the majority valence band.
Hence, systems with larger spin flip gap are expected to be
better source of fully spin polarized current.
Among the studied systems, the largest spin flip gap (0.8 eV) is observed in the ZB-CaN.

It has been argued \cite{Geshi2004} that the half-metallicity in the ZB structure
is mainly caused by the local bonding environment.
These alkaline pnictides involve seven valence electrons per formula unit.
Since the anion $s$ states are the lowest energy valence states,
they must be occupied by two electrons,
and the rest five electron occupy the anion $p$ states which can accommodate six electrons.
Because of spin splitting, the majority $p$ states lie lower in energy than
the minority $p$ states, so the majority spin $p$ bands are fulfilled
by three electrons and the minority spin $p$ bands are partially filled.
As a result of the hole appears in the minority spin valence bands, a total spin moment
of 1 $\mu_B$ is left on the anion atom.
But in the case of transition metal pnictides, there are enough valence electrons
to make the anion $p$ states fulfilled, and the antibonding $t_2-p$
bands and the $e$ bands for majority spin channel are also pulled down
to accommodate the additional valence electrons,
the Fermi level cut the majority spin bands and a half-metallic gap remains in
the minority spin channel.
The magnetic moments in these systems are mainly from the transition metals $d$ electrons.

The loss of the half-metallicity can be understood by checking
the change of the Fermi level position and $p-d$ hybridization with the lattice constants. While the
lattice constants decrease, the Fermi level will be push up to higher energy and the spin flip gap
increases, but the $p-d$ hybridization will be strengthened and the valence bands characterized by $p$
states are expanded which would reduce the spin flip gap. When the lattice constants are compressed to
smaller than the critical value, the expanded majority $p$-character bands would cut the Fermi level and
the half-metallicity is destroyed.

The inter-metallic CaN and CaAs compounds are made of a strong cationic metal and weak anionic
elements, which are known as the Zintle phase. The Zintle phases stand somewhere in-between metallic and
ionic compounds. In Zintle phases anions do not reach the stable octet state as isolated species,
and hence they need additional bonds to become more stable.
Therefore, in these compounds the anion atoms usually connect covalently together
and form dimers.
Formation of anion dimers in the Zintle phases, electronically balances the system
and leads to closed-shell compounds, i.e. the number of electrons provided by the
constituting elements equals the number of electrons needed for covalent bonding in the structure.

The spin-dependent band structures along high symmetry lines in
the Brillouin zone for the best half-metallic structure (ZB)
and the most stable structure (NaO) are given in Fig \ref{fig:band}.
We know that the anion $p$ states are localized around the nuclei
and hence the valance bands are flat and low dispersed.
the anion atoms and according to DOS profiles the hybridization between
the anion $p$ states and the Ca $d$ states is weak.
reflecting the weak polarized-covalent bonds of the anions
$p$ orbitals and Ca $d$ states.

\section{CONCLUSION}

Conventional magnetism occurs in systems which contain transition metals or rare earth ions with
partially filled $d$ or $f$ shells. Recent experimental and/or theoretical observations of magnetism in
systems with no valance $d$ or $f$ electron challenge our classical understanding of magnetism. It is
theoretically predicted that compounds of groups IA and IIA with IV and V, in some structural
phases are ferromagnetic half-metals which made them new candidates for spintronics applications.
Employing density functional theory (DFT) we investigate magnetism in binary compounds CaN
and CaAs. Regarding the structure of analogous magnetic materials and experimental results of
CaAs synthesis, we have considered two cubic structures: rocksalt (RS) and zinc-blende (ZB), and
four hexagonal structures: NiAs, wurtzite (WZ), anti-NiAs, and NaO.
The calculated results show that CaN in cubic, NiAs, and wurtzite structures, and CaAs only in
zincblende phase have ferromagnetic ground states with a magnetic moment of 1 $\mu_B$ . Electronic structure
analysis of these materials indicates that magnetism originates from anion $p$ states. Existence
of flat $p$ bands and consequently high density of states at the Fermi level of magnetic structures
gives rise to Stoner spin splitting and spontaneous ferromagnetism. Larger exchange interaction in
$2p$ orbitals of nitrogen respect to $4p$ orbitals of arsenic causes that CaN has more structures with
ferromagnetic ground state than CaAs. Slight hybridization with calcium $d$ states increase density
of $p$ states at the Fermi level and enhance ferromagnetism in the system.

redpajama_set_name: RedPajamaArXiv


text:
Where does Lindy Hop come from?
Lindy Hop is nearly a century old! It comes from the 1920s and 30s, and – like the jazz music that inspired it – Lindy Hop has deep roots as a uniquely Black American art form.
It's also important to know that Lindy Hop specifically, and in general Jazz music and dance, was borne of a culture of oppression and racism. Nowadays when we talk about the 1920s and 30s we might think of movies like The Great Gatsby, or of the Great Depression, but jazz culture is hugely informed by the racism and segregation that Black Americans faced at that time – swing music, and Lindy Hop, were a defiant reaction to the oppression and hardships that people were facing, a joyous resistance and expression of freedom.
Check out this video to learn more about how the Lindy Hop was created, including some of the ragtime and jazz-era dances that came before it:
For more of these history videos please click here
Hellzapoppin' (1941)
The dancing: This might be the most famous lindy hop clip of all time, and deservedly so. The dancers, the top performance troupe Whitey's Lindy Hoppers, choreographed by Frankie Manning, are explosive and energetic and just all-around amazing. In this version, the clip has been colorized, and our friend Andy Lewis has resynchronized the sound.
Historical note: Aside from the impressive and inspiring dancing, this clip is a sobering reminder of racism. You'll notice that the dancers are dressed as servants instead of fancy performance clothes; this is in order to make it palatable for white people of the time to watch Black people on film. Also, this film, as many back then, was created so that the entire scene with Black people could be edited out for showing in the southern U.S.
Al Minns and Leon James dancing Charleston (1961)
Al and Leon were two of the most prominent Lindy Hoppers of the original era. They were, like most dancers of their day, equally accomplished as solo and as partnered dancers. In this clip, they are mostly dancing the Charleston solo; toward the end they also demonstrate some partnered Charleston and a few comic elements they seemed to love doing.
This video footage is from well after the swing era was over, but it gives us a great chance to see Al and Leon strut their stuff. Al and Leon didn't stop Lindy Hopping – they continued dancing, and spreading the dance, even touring and appearing on 1960s talk shows such as in this clip.
Dean and Jewel in Buck Privates (1941)
In the 1930s and into the 40s, swing music, and Lindy Hop, became more and more popular, spreading from Black communities, to be adopted by Whites and other communities across the country, and eventually the globe. It was a huge craze, and became a feature in many Hollywood movies of the time. Here are two of the most famous Hollywood dancers: Dean Collins (who had learned to Lindy Hop in Harlem before moving to Los Angeles) and his partner Jewel McGowan. Dean and Jewel, as White dancers, would have had many more show business opportunities than Black dancers such as the Whitey's Lindy Hoppers in the clip above.
White dancers' prevalence on film can sometimes make it seem as though the dance was more popular with White people, but actually this is simply a result of opportunities given to them and denied to Black dancers whose access was often actively suppressed by Hollywood and other industries.
What Happened to Lindy Hop after the Swing Era?
In the late 1940s the big band era died out and popular music moved on. Swing dancing was no longer a huge craze across the globe, but it didn't die out. Social dancing evolved along with popular music to create new dance styles, and in the meanwhile dancers like Mama Lou Parks were important culture bearers, keeping swing alive. There is a popular myth that swing dancing "died" in this period, and that our modern Lindy Hop scene is a "revival". Check out Katie's myth-busting video!
(Extra reading: here's the Swungover blog post that Katie refers to in her video.)

redpajama_set_name: RedPajamaCommonCrawl


Do you want to share your exciting travel adventures with your friends and family? Click here to learn how to craft the perfect postcard messages.
Do you want to continue a thoughtful tradition that will always hold value? Get in the habit of writing postcards. Even in the modern internet age, postcards are still an amazing way to reach out to those you love.
This communication method is important if you constantly travel. You can find postcards that represent your adventures and send the postcards to loved ones.
But why send a blank postcard to a family member? Writing a small memento will mean so much to those who love you.
If you don't think you're the best writer, there are ways to write great postcard messages. Here are 8 tips to get you started.
All good writers start with one basic rule: formatting. Before you actually write your postcard messages, know where to write your message.
Ideally, a postcard has two different sides. One is for the receiver's and sender's addresses, and you place a stamp at the top corner. The area to the left is for your message.
But if you're delivering your postcard in-person or plan to send postcards online, you have more wiggle room.
Be creative when formatting. Maybe write long blocks of texts that take up both sides. Or write a simple message on one side with your signature on the other side.
What's the best way to get you in the writing mood? Recounting your travels, of course.
Traveling is exciting; no matter where you visit, traveling opens you up to new experiences and opportunities. The thrill of traveling also boosts creativity and keeps your pen moving.
When writing postcard messages, share your traveling stories with your loved ones. List everything you saw, the people you met, describe the food and even share some off-the-beaten-path adventures you experienced.
When picking out a postcard, find an image that best represents your traveling experiences.
How many people will receive a postcard? Even if you're sending mass postcards to multiple people, you likely won't send the same postcard messages to all recipients.
When writing a postcard, be personable. Refer to the recipient by name and discuss topics only you two would know about.
For example, you can share special experiences you know the recipient will love.
Maybe you can discuss a museum or other interesting place you visit. Or maybe the recipient has visited the place you're in, and you can compare your experiences.
Postcards aren't meant to be long and descriptive. If you want to write a memoir for your recipient, you can type it up and print it out. But postcards are meant to be short and sweet.
Before the internet, postcards were used as a simple greeting or a "saying hi from while traveling" message. To keep traditions stagnant, use the postcards to write a simple message.
This also helps improve your writing; lots of writers get descriptive and wordy. A postcard challenges you to write an effective message while keeping your message short.
This is a fact all writers learn: write for your audience.
The message you would write for your spouse is different than the one you would write to your grandparents.
Identify the person you're writing to and use effective language. For a spouse, you can share romantic things to do when you two visit the destination together.
For your parents or grandparents, give endearing messages and recount on all of the exciting adventures you experienced.
No vacation is complete without gifts! Souvenirs are some of the most exciting aspects of traveling. You can buy items in other places you can't buy back home. If you bought gifts for your loved ones, hint at their gifts in the message.
But don't give the gifts away. Use a message such as, "I spent all day shopping in Paris. Don't worry, you'll see all of the amazing items I bought." This way, the recipient knows you went shopping but doesn't know you bought them anything.
Are you a traveler who plans your vacation?
Share your plans with your recipients. When you arrive at your destination, find a postcard and immediately send it. Share your schedule and all of the exciting things you have planned.
Are you a traveler who goes with the flow? You can still use this writing trick.
Instead of sending a schedule, mention you've been getting ideas from locals or found some great advice online. You can even describe the hotel and everything you've seen so far.
Would you rather wait to write the postcard? This is a great way to recount your adventure and let the good memories live on. Saving your postcard for last also helps kill time on the flight back home.
If you're unsure what exactly to write, discuss your favorite day or your favorite adventure.
Recount everything — from the moment you woke up to the moment when you fell asleep. Share everything you did and all of your adventures.
Don't have a favorite day? Share your favorite activity or the best sight you saw.
Maybe you loved scuba diving the best or maybe you loved checking out another country's local music scene.
Even describing the nature and weather will transport your reader to the destination.
Traveling is exciting and you want to share your experiences with those you love.
Even in the modern internet age, writing postcards is an amazing way to reach out to those you love.
But not everyone is a wordsmith. You can simply describe your trip, or you can write a more personalized and endearing message.
Even if you're not a writer, you can still write an incredible postcard message.
Take this writing advice and find the most amazing postcard message.
Are you looking for future travel ideas? Take a look at our resources.


Spirulina is one of Earth's oldest life forms, an algae. It provides a huge range of nutrition and has been enjoyed for centuries by several Asian cultures. Modern science has found that it is of wide nutritional value, and the Japanese government classifies it as a functional food in its own right. Rainforest Foods Spirulina is popular among customers combining improvements in their dietary regime with exercise programmes.
These tablets contain Chlorella and Spirulina - two different species of algae and two of the most complete foods in existence.
Spirulina is an algae rich in various vitamins and minerals. The protein content makes it a popular fitness dietary supplement.
With all the benefits of Spirulina Powder, our Spirulina Tablets give you a healthy dose of nutrition at home or on the go.


Happy Wednesday! I hope your week has been going well and you have had a chance to get creative. Ericka here, and I have three examples of ways you can use cut files on your cards. Today I'm using the beautiful April Missy Whidden Color Kit and the butterfly cut file included in it.
For this glittery butterfly card, I placed double stick adhesive on one side of the the paper and then proceed to use the cut file in my Silhouette machine, sizing it to fit on an A2 card. Once it was finished cutting, I removed the adhesive on the butterfly portion and used glitter to cover the sticky adhesive.
I glued the cut file onto the base and then added this sentiment from Reverse Confetti.
I used Shimmerz Bumble Bee and 1 Hot Mama Creameez for the background on watercolor paper. It's difficult to get a good picture of the texture that the vellum cut file gives to the card but here is a closer look.
This was a quick fun card that I added a Reverse Confetti Butterfly to with some blue sequins. I love the shimmer the Jeni B Bleu adds to any project.


text:
Тарас Поликарпович Харитонов (15 марта 1899 — 16 июня 1970) — сапёр 51-го отдельного сапёрного батальона (13-й гвардейский стрелковый корпус, 43-й армия, 1-й Прибалтийский фронт) красноармеец - на момент представления к награждению орденом Славы 1-й степени., участник Великой Отечественной войны, кавалер ордена Славы трёх степеней.

Биография
Родился 15 марта 1899 года в деревне Лужки, ныне Дмитриевского района Курской области, в семье крестьянина. Русский. Образование начальное. В 1920-1923 годах походил службу в Красной армии.

Работал в колхозе. Жил в Волновахском районе Сталинской (Донецкой) области. В первые годы войны оставался на оккупированной территории.

В сентябре 1943 года, после освобождения района частями Красной армии, был вновь призван в армию Волновахским райвоенкоматом. С октября 1943 года участвовал в боях с захватчиками. Воевал на 4-м Украинском и 1-м Прибалтийском фронтах. Сражался сапёром 51-го отдельного сапёрного батальона 13-го гвардейского стрелкового корпуса. Красноармеец Харитонов особо отличился в боях за освобождение Прибалтики и в Восточной Пруссии.

Летом 1944 года 2-я гвардейская армия, в состав которой входил 13-й гвардейский стрелковый корпус на 1-ом Прибалтийском фронте. Здесь сапёр Харитонов принимал участие в Вильнюсской и Шяуляйской наступательных операциях, затем отражала контрудары противника западнее и северо-западнее Шяуляя. В октябре участвовал в Мемельской наступательной операции.

7 октября 1944 года у населённого пункта Геткойце (2 км южнее города Пагегяй, Литва) красноармеец Харитонов в составе группы сопровождения танков, при преследовании противника на лесной дроге, разминировал и растащил два завала, освободив дорогу для танков.

21 октября в боях за станцию Пагегяй (Таурагский уезд, Литва) под огнём противника проделал проход в проволочном заграждении. Вместе с пехотой ворвавшись в опорный пункт, участвовал в уничтожении оборонявших его гитлеровцев.

Приказом по войскам 13-го гвардейского стрелкового корпуса № 89/н от 14 декабря 1944 года красноармеец Харитонов Тарас Поликарпович награждён орденом Славы 3-й степени.

К началу 1945 года 13-й гвардейский стрелковый корпус входил в состав 43-й армии. 13 января началась Восточно-Прусская операция. 43-я армия помогала наступлению 3-го Белорусского фронта ударом на Тильзит.

20-21 февраля 1945 года у населённого пункта Куменен (Восточная Пруссия, ныне посёлок Кумачёво Зеленоградский район Калининградской области) под огнём противника установил 24 мины, закрыв проход в минном поле на направлении вражеской контратаки. В последующих боях по закреплению рубежей неоднократно проявил образцы отваги. В бою за важную высоту ринулся в атаку, не отставая от молодых бойцов. В рукопашной схватке уничтожил гитлеровца. Был представлен к награждению медалью «За отвагу».

Приказом по войскам 43-й армии № 75 от 21 марта 1945 года красноармеец Харитонов Тарас Поликарпович награждён орденом Славы 2-й степени.

5-7 апреля 1945 года при штурме города Кёнигсберг (Восточная Пруссия, ныне город Калининград) красноармеец Харитонов неоднократно участвовал в отражении контратак противника, минировании, разрушении оборонительных сооружений и подавлении огневых точек. 7 апреля в составе штурмовой группы блокировал опорный пункт противника - укреплённый дом с огневой точкой. Был ранен, но продолжил установку мин. Совместно с тремя сапёрами взорвал дом, в котором находилась огневая точка немцев. Только после второго ранения был эвакуирован в тыл.

Указом Президиума Верховного Совета СССР  от 19 апреля 1945 года красноармеец Харитонов Тарас Поликарпович награждён орденом Славы 1-й степени. Стал полным кавалером ордена Славы.

День Победы встретил в госпитале. В августе 1945 года старшина Харитонов был демобилизован.

Жил на хуторе Шевченко Волновахского района Донецкой области Украины. Трудился в колхозе. Скончался 16 июня 1970 года. Похоронен .

Награды

орден Славы I степени(19.04.1945)
орден Славы II степени(21.03.1945)
орден Славы III степени (14.12.1944)
 медали, в том числе:
 «За победу над Германией в Великой Отечественной войне 1941—1945 гг.» (1945)
 «За взятие Кёнигсберг» (09.06.1945)

Примечания

Литература
 .

 Лобода В. Ф. Солдатская слава. М., 1967. Кн. 2. С. 181-182;
 Костенков А. Г. Славен их путь боевой. Калининград, 1987. С. 80;

Ссылки

Полные кавалеры ордена Славы
Сапёры Великой Отечественной войны

redpajama_set_name: RedPajamaWikipedia


Gotta drop 'em all?! Seriously, phones are slippery and iPhones tend to hit the ground and shatter at a moment's notice. That means you need a case that can protect your phone. The best part is with this Pokémon themed Phone Case your Phone doesn't have to look bad while wearing its armor!
Coming in two Pokémon themed styles, these cases conjure up smiles wherever you are and look fantastic when you are playing Pokémon Go! Choose between the traditional Pokeball color pattern, or our personal favorite, a Pokedex themed pattern! This case is designed for the 4.7 inch iPhone 7 and made to protect it through some pretty severe drops. At under $10, it's an incredible value as well. The exterior is made from hardened Polycarbonate to give it a strong, sturdy feel, while also being non-slip. We all know holding iPhones can be like holding onto a greased pig. As a premium quality case, it has a thin lip to secure the phone while also managing to have excellent design that allows all buttons to function comfortably. If you have butterfingers, or a naked phone. Invest in this case, it is FAR cheaper than repairing a broken screen, and looks sweet too!


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Singer – songwriter and multi – instrumentalist, æmma, is set to play a special headline show at Bush Hall in Shepherd's Bush on Wednesday the 24th October 2018.
The 'Hunters Moon' night will see her accompanied on grand piano by a string quartet, driving beats and synths together with a female choir. The evening will also feature a live performance from Nick Francis & The Quill; an eighteen piece folk orchestra.
æmma recently released a debut single entitled 'House Of Hym' on the fledgling LGM Records. Making it's premiere on The Line Of Best Fit, the track is beautifully crafted dark pop that's both unsettling and magical and is the first indication of what to expect from the forthcoming debut album.
Thou now London based, æmma was born and raised in the remote and bleak yet inspirational wilderness of Dartmoor. The multi – talented classically trained composer is a bold, songwriting visionary that evokes the sound of Fiona Apple, PJ Harvey's 'White Chalk' album and early Goldfrapp meets LA native Julia Holter. A sense of magic, romance and wonder is created through complex haunting melodies and arrangements, brought together by powerful cerebral lyrics that encourage sensitivity, romance and vulnerability in a society that often celebrates the opposite.
London based imprint LGM Records was officially launched at a sell-out event in Soho on the 27th of March 2017. Now home to a roster of talent such as 2018's breakthrough act Amaroun, indie legends Goldheart Assembly and Steve Lamaq favourites Mono Club (feat. Lyla Foy), they are set to announce further new signings and live nights in the very near future.


I will prepare a translation of the entire section, as time permits, and post it here on the NSDF as a separate thread. I hope, soon, to be able to post the Dec 1 1812 list of officers by location as mentioned above. It should be helpful as a partial answer to the OP.


text:
Q: Como fazer no HQL uma consulta por ID usando o operador LIKE? Como faço essa consulta em HQL?
select * from entidadeQualquer where  id::text like '%12';

Já tentei o código abaixo mas não funcionou:
select c from entidadeQualquerc  where  c.id like '%12';

A: Sem o código da entidade não dá para ter certeza, mas você pode tentar algo assim:
String hql = "select c from entidadeQualquer c where CAST(id as text) like :id";

E para definir o parâmetro:
Query query = em.createQuery(hql);
query.setParameter("id", "%12");
List<EntidadeQualquer> lista = query.getResultList()

A: Pra mim, funcionou assim:
String sql = "select t from Tabela t where str(t.id) like :id";

parametro:
query.setParameter("id", "%" + parteDoIdPassadoComoParametro + "%");

redpajama_set_name: RedPajamaStackExchange


text:
Paopao, French Polynesia weather in September
Historical temperature average in September
In Paopao, French Polynesia
F/inches/miles
C/inches/miles
C/millimeters/kilometers
Planning a trip to Paopao?
General weather summary
Pleasantly warm with a gentle breeze
Chance of rain
12.0 hrs
Daylight length
Planning a trip to Paopao in September?
Build, organize, and map your itinerary to Paopao with our free trip planner.
Start planning a trip
Scan the QR code or enter your mobile number.
We'll text you a link to download the app.
What is the average temperature
How much does it rain
How cloudy is it
When is sunrise and sunset
How humid is it
How windy is it
What is the average temperature in September
The average temperature in Paopao in September for a typical day ranges from a high of 80°F (26°C) to a low of 77°F (25°C). Some would describe it as pleasantly warm with a gentle breeze.
For comparison, the hottest month in Paopao, April, has days with highs of 83°F (28°C) and lows of 81°F (27°C). The coldest month, August has days with highs of 79°F (26°C) and lows of 77°F (25°C). This graph shows how an average day looks like in Paopao in September based on historical data.
Highs and lows in Paopao in Sep
How much does it rain in September
In Paopao in September, there's a 18% chance of rain on an average day. And on the average day it rains or snows, we get 0.62 in (15.8 mm) of precipitation. In more common terms of how much that is, some would describe it as moderate rain.
The wettest month in Paopao is January where a typical day has a 54% chance of precipitation and gets 0.10 inches (2.5 mm) of precipitation, while the dryest month in Paopao is July where a typical day has a 14% chance of precipitation and gets 0.13 inches (3.3 mm) of precipitation. These graphs show the probability of it raining/snowing in September and the amount of rainfall.
Chance of rain or snow on an average Paopao day by month
Average precipitation by hour of day
How cloudy is Paopao in September
The average amount of time that the sky is clear or sunny (partly cloudy or less) in Paopao during September is 14.8 hours (62% of the day). For comparison, the month with the most clear, sunny days in Paopao is August with an average of 15.0 hours per day while January has the shortest amount of cloud-free hours with only 6.0 hours per day. This graph shows the average amount of sunshine in Paopao in September based on historical data.
Monthly clear skies percent of time in Paopao
When is sunrise and sunset in September
The average day in Paopao during September has 12.0 hours of daylight, with sunrise at 5:54 AM and sunset at 5:53 PM.
The day with the longest amount of daylight in Paopao is December 15th with 13.2 hours while June 24th has the shortest amount of daylight with only 11.1 hours.
This graph shows the average amount of daylight in Paopao in September based on historical data.
Average hours of daylight in Paopao by month
How humid is it in September
In September, Paopao is moderately humid with an average amount of 73% (relative humidity), which could be described as humid but cool. February has the highest relative humidity at 77% and is the least humid in August at 73%. This graph shows the average amount of humidity in Paopao throughout September based on historical data.
Average relative humidity in Paopao by month
How windy is it in September
Historically, the wind in Paopao during September blows at an average speed of 14.8 mph (23.8 kph). The windiest month is July with an average wind speed of 15.2 mph (24.4 kph), while the calmest month is March with an average wind speed of 11.5 mph (18.5 kph). This graph shows the average wind speed in Paopao in September based on historical data.
Average monthly wind speed in Paopao
What to wear in September
With an average high of 80°F (26°C) and a low of 77°F (25°C), it generally feels pleasantly warm with a gentle breeze. With that in mind, most people would dress in something very light like a sleeveless shirt.
What's the weather like in Paopao the rest of the year
We've collected the weather data for Paopao during all other months of the year too:
Weather in Paopao in January
Weather in Paopao in February
Weather in Paopao in March
Weather in Paopao in April
Weather in Paopao in May
Weather in Paopao in June
Weather in Paopao in July
Weather in Paopao in August
Weather in Paopao in October
Weather in Paopao in November
Weather in Paopao in December
Where does this data come from
Weather data for Paopao was collected from the MERRA-2 project from NASA, which used a climate model combined with historical data from weather stations around the world to estimate what the conditions were like for every point on the Earth.
For all data based on historical data, we've averaged the data from the past 11 years (2010-2020). For example, for the hourly temperature at 10am, we've looked at the temperature at 10am on every day in September (e.g., September 1, September 2, etc. in 2010, 2011, etc.) and took the arithmetic mean. We did not smooth the data, so for example, our daily temperature line will have some randomness due to the fact that weather is random in the first place.
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text:
Q: how can i make insert into table and overwrite if is not empty? I would like to know how can i make this code: if is not empty insert into table, else overwrite the rows:
$affectedRows = $pdo->exec(
    "insert into table_1 (name, money, month, year)
    select name, count(*), MONTH(STR_TO_DATE(data, '%d/%m/%Y')),
    YEAR(STR_TO_DATE(data, '%d/%m/%Y'))
    from table_2
    group by name");

Thanks

A: Use the following query:
INSERT INTO table_1 (name, money, month, year)
SELECT t.name, t.money, t.month, t.year
FROM
(
    SELECT name,
           COUNT(*) AS money
           MAX(MONTH(STR_TO_DATE(data, '%d/%m/%Y'))) AS month,
           MAX(YEAR(STR_TO_DATE(data, '%d/%m/%Y')))  AS year
    FROM table_2
    GROUP BY name
) t
ON DUPLICATE KEY UPDATE name=t.name, money=t.money, month=t.month, year=t.year

Note that I wrapped your calls to MONTH() and YEAR() with MAX(), an aggregate function, because it is incorrect to select a non aggregate column when using GROUP BY.  If you show us actual data for table_2, then perhaps this could be improved upon.
Update:
Note that the subquery whose result set is used for insertion above can never generate duplicate records, because there can only ever be a single record per group.  So to test this answer, you would already have to have some data in table_1 which would be duplicated by the subquery.


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text:
Welsh Water warns of the dangers of unauthorised swimming in reservoirs on the fifteenth anniversary of teen's tragic death
ON the fifteenth anniversary of the devastating death of teenager Reuben Morgan, his mother Maxine has joined Welsh Water to remind the public of the potentially fatal dangers of unauthorised swimming in reservoirs this summer.
The warning comes after the tragic news of another fatality which saw the body of a 27-year-old man recovered from Ponden reservoir in Bradford yesterday.
Each year, large numbers of people, predominantly young men, enter the water in reservoirs across Wales in unauthorised conditions, putting their lives and the lives of others at risk. This is despite the fact that statistics from Water Safety Wales estimate that 45 people in Wales lose their lives to drowning each year in coastal and inland waters.
Welsh Water has heightened concerns this year with more members of the public flocking to beauty spots to enjoy the warmer weather and new freedoms as lockdown restrictions ease and more people stay local this year.
Since May, rangers patrolling Welsh Water reservoirs have had to approach 37 different people who had attempted to enter the water to swim or use personal canoes or paddleboards on the water. Despite the dangers, these numbers are expected to rise throughout the summer.
The dangers of swimming in reservoirs include:
Hidden structures made of concrete or metal beneath the surface of the water which can often operate without warning.
Freezing currents that can cause even strong swimmers to find themselves in cold water shock and unable to swim.
Reduced chance of rescue due to the remote location of many reservoirs, with little or no mobile reception.
Welsh Water's annual reservoir safety campaign which also starts today raises awareness of these dangers and serves to highlight the enormous impact unauthorised swimming could have on friends, family, communities, and emergency services.
Reuben Morgan was just fifteen years old when he visited Pontsticill reservoir with friends. Five of the group decided to swim across the reservoir to cool down. While Reuben's friends made it across, after disappearing under the water three times, Reuben did not resurface.
Reuben's mother, Maxine said: "Reuben has been gone now for as long as he was alive. At the time, I never thought Reuben would enter the water. He died because of cold water shock from the freezing temperatures and if he'd have known more about the dangers his death could have been prevented. Losing Reuben has had a massive impact on so many of us; his friends, family, and teachers. We as a family are heartbroken and we continue to miss him every day.
"Educating younger people and spreading the word about the very real dangers of unauthorised swimming in reservoirs is absolutely essential."
Head of Visitor Attractions at Welsh Water, Mark Davies said: "Our reservoirs are absolutely beautiful places to relax and unwind. From the surface they look calm and peaceful, and people are tempted to swim, paddle, or even jump into the water. But safety is paramount; there are many hidden dangers underneath the surface of the water and unauthorised swimming can be fatal. One life lost is one too many.
"We know that enjoying the water safely can provide immense benefits for mental and physical wellbeing which is why we have been working with Swim Wales to introduce safe, supervised, and bookable sessions at a small number of our Visitor Attractions. An incredible amount of work has gone into ensuring these sessions are supervised and safe and we've positioned clear signage at our sites to remind visitors that unauthorised swimming is strictly prohibited. It is essential people understand that unauthorised access to the water is highly dangerous and can be fatal."
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text:
\section{Introduction}

As is well known, the Chebyshev polynomials of the first kind, $T_{n}\left(x\right)$,
$\left(n\ge0\right)$, are defined by the generating function
\begin{equation}
\frac{1-t^{2}}{1-2xt+t^{2}}=\sum_{n=0}^{\infty}T_{n}\left(x\right)\frac{t^{n}}{n!},\quad\left(\text{see \cite{key-1,key-3,key-5,key-7,key-17,key-21}}\right).\label{eq:1}
\end{equation}

The higher-order Chebyshev polynomials are given by the generating
function
\begin{equation}
\left(\frac{1-t^{2}}{1-2xt+t^{2}}\right)^{\alpha}=\sum_{n=0}^{\infty}T_{n}^{\left(\alpha\right)}\left(x\right)t^{n},\label{eq:2}
\end{equation}
and Chebyshev polynomials of the second kind are denoted by $U_{n}$
and given by generating function
\begin{equation}
\frac{1}{1-2xt+t^{2}}=\sum_{n=0}^{\infty}U_{n}\left(x\right)t^{n},\quad\left(\text{see \cite{key-1,key-8,key-12,key-17}}\right).\label{eq:3}
\end{equation}

The higher-order Chebyshev polynomials of the second kind are also
defined by
\begin{equation}
\left(\frac{1}{1-2xt+t^{2}}\right)^{\alpha}=\sum_{n=0}^{\infty}U_{n}^{\left(\alpha\right)}\left(x\right)t^{n}.\label{eq:4}
\end{equation}

The Chebyshev polynomials of the third kind are defined by the generating
function
\begin{equation}
\frac{1-t}{1-2xt+t^{2}}=\sum_{n=0}^{\infty}V_{n}\left(x\right)t^{n},\quad\left(\text{see \cite{key-1,key-7,key-8,key-17}}\right).\label{eq:5}
\end{equation}
and the higher-order Chebyshev polynomials of the third kind are also
given by the generating function
\begin{equation}
\left(\frac{1-t}{1-2xt+t^{2}}\right)^{\alpha}=\sum_{n=0}^{\infty}V_{n}^{\left(\alpha\right)}\left(x\right)t^{n}.\label{eq:6}
\end{equation}

Finally, we introduce the Chebyshev polynomials of the fourth kind
defined by the generating function
\begin{equation}
\frac{1+t}{1-2xt+t^{2}}=\sum_{n=0}^{\infty}W_{n}\left(x\right)t^{n}.\label{eq:7}
\end{equation}

The higher-order Chebyshev polynomials of the fourth kind are defined
by
\begin{equation}
\left(\frac{1+t}{1-2xt+t^{2}}\right)^{\alpha}=\sum_{n=0}^{\infty}W_{n}^{\left(\alpha\right)}\left(x\right)t^{n}.\label{eq:8}
\end{equation}

It is well known that the Legendre polynomials are defined by the
generating function
\begin{equation}
\frac{1}{\sqrt{1-2xt+t^{2}}}=\sum_{n=0}^{\infty}p_{n}\left(x\right)t^{n},\quad\left(\text{see \cite{key-2,key-19}}\right).\label{eq:9}
\end{equation}

Chebyshev polynomials are important in approximation theory because
the roots of the Chebyshev polynomials of the first kind, which are
also called Chebyshev nodes, are used as nodes in polynomial nodes
(see \cite{key-18}).

The Chebyshev polynomials of the first kind and of the second kind are
solutions of the following Chebyshev differential equations
\begin{equation}
\left(1-x^{2}\right)y^{\prime\prime}-xy^{\prime}+n^{2}y=0,\label{eq:10}
\end{equation}
and
\begin{equation}
\left(1-x^{2}\right)y^{\prime\prime}-3xy^{\prime}+n\left(n+2\right)y=0.\label{eq:11}
\end{equation}

These equations are special cases of the Strum-Liouville differential
equation (see \cite{key-1,key-2,key-3}).

The Chebyshev polynomials of the first kind can be defined by the
contour integral
\begin{equation}
T_{n}\left(z\right)=\frac{1}{4\pi i}\oint\frac{\left(1-t^{2}\right)}{1-2tz+t^{2}}t^{-n-1}dt,\label{eq:12}
\end{equation}
where the contour encloses the origin and is traversed in a counterclockwise
direction (see \cite{key-1,key-18,key-21}). The formula for $T_{n}\left(x\right)$
is given by
\begin{equation}
T_{n}\left(x\right)=\sum_{m=0}^{\left[\frac{n}{2}\right]}\binom{n}{2m}x^{n-2m}\left(x^{2}-1\right)^{m}.\label{eq:13}
\end{equation}

From (\ref{eq:3}), we note that
\begin{equation}
2\left(x-t\right)\left(1-2xt+t^{2}\right)^{-2}=\sum_{n=0}^{\infty}nU_{n}\left(x\right)t^{n-1}.\label{eq:14}
\end{equation}

Thus, by (\ref{eq:14}), we get
\begin{equation}
\left(2xt-2t^{2}\right)\left(1-2xt+t^{2}\right)^{-2}=\sum_{n=0}^{\infty}nU_{n}\left(x\right)t^{n}.\label{eq:15}
\end{equation}

From (\ref{eq:3}) and (\ref{eq:15}), we can derive the following
equation:
\begin{align}
\frac{\left(2xt-2t^{2}\right)+\left(1-2xt+t^{2}\right)}{\left(1-2xt+t^{2}\right)^{2}} & =\frac{1-t^{2}}{\left(1-2xt+t^{2}\right)^{2}}\label{eq:16}\\
 & =\sum_{n=0}^{\infty}\left(n+1\right)U_{n}\left(x\right)t^{n}.\nonumber
\end{align}

Note that
\begin{align}
 & \frac{1-t^{2}}{\left(1-2xt+t^{2}\right)^{2}}\label{eq:17}\\
 & =\left(\frac{1-t^{2}}{1-2xt+t^{2}}\right)\left(\frac{1}{1-2xt+t^{2}}\right)\nonumber \\
 & =\left(\sum_{l=0}^{\infty}T_{l}\left(x\right)t^{l}\right)\left(\sum_{m=0}^{\infty}U_{m}\left(x\right)t^{m}\right)\nonumber \\
 & =\sum_{n=0}^{\infty}\left(\sum_{l=0}^{n}T_{l}\left(x\right)U_{n-l}\left(x\right)\right)t^{n}.\nonumber
\end{align}

From (\ref{eq:16}) and (\ref{eq:17}), we have
\[
U_{n}\left(x\right)=\frac{1}{n+1}\sum_{l=0}^{n}T_{l}\left(x\right)U_{n-l}\left(x\right).
\]

The Chebyshev polynomials have been studied by many authors in the several
areas (see \cite{key-1,key-2,key-3,key-4,key-5,key-6,key-7,key-8,key-9,key-10,key-11,key-12,key-13,key-14,key-15,key-16,key-17,key-18,key-19,key-20,key-21}).

In \cite{key-10}, Kim-Kim studied non-linear differential equations
arising from Changhee polynomials and numbers related to Chebyshev poynomials.

In this paper, we study non-linear differential equations arising
from Chebyshev polynomials and give some new and explicit formulas
for those polynomials.

\section{Differential equations arising from Chebyshev polynomials and their
applications}

Let
\begin{equation}
F=F\left(t,x\right)=\frac{1}{1-2tx+t^{2}}.\label{eq:18}
\end{equation}

Then, by (\ref{eq:1}), we get
\begin{equation}
F^{\left(1\right)}=\frac{d}{dt}F\left(t,x\right)=2\left(x-t\right)F^{2}.\label{eq:19}
\end{equation}

From (\ref{eq:19}), we note that
\begin{equation}
2F^{2}=\left(x-t\right)^{-1}F^{\left(1\right)}.\label{eq:20}
\end{equation}

By using (\ref{eq:20}) and (\ref{eq:19}), we obtain the following
equations:
\begin{align}
2^{2}\cdot2F^{3} & =\left(x-t\right)^{-3}F^{\left(1\right)}+\left(x-t\right)^{-2}F^{\left(2\right)},\label{eq:20-1}\\
2^{3}\cdot2\cdot3F^{4} & =3\left(x-t\right)^{-5}F^{\left(1\right)}+3\left(x-t\right)^{-4}F^{\left(2\right)}+\left(x-t\right)^{-3}F^{\left(3\right)}\label{eq:21}
\end{align}
and
\begin{align}
2^{4}\cdot2\cdot3\cdot4F^{5} & =3\cdot5\left(x-t\right)^{-6}F^{\left(1\right)}+3\cdot5\left(x-t\right)^{-6}F^{\left(2\right)}\label{eq:22}\\
 & \relphantom =+\left(3\cdot2\right)\left(x-t\right)^{-5}F^{\left(3\right)}+\left(x-t\right)^{-4}F^{\left(4\right)},\nonumber
\end{align}
where
\[
F^{N}=\underset{N-\text{times}}{\underbrace{F\times\cdots\times F}}\quad\text{and}\quad F^{\left(N\right)}=\left(\frac{d}{dt}\right)^{N}F\left(t,x\right).
\]

Continuing this process, we set
\begin{equation}
2^{N}N!F^{N+1}=\sum_{i=1}^{N}a_{i}\left(N\right)\left(x-t\right)^{i-2N}F^{\left(i\right)},\label{eq:23}
\end{equation}
where $N\in\mathbb{N}$.

From (\ref{eq:23}), we note that
\begin{align}
 & \relphantom =2^{N}N!F^{N}\left(N+1\right)F^{\left(1\right)}\label{eq:24}\\
 & =\sum_{i=1}^{N}a_{i}\left(N\right)\left(2N-i\right)\left(x-t\right)^{i-2N-1}F^{\left(i\right)}+\sum_{i=1}^{N}a_{i}\left(N\right)\left(x-t\right)^{i-2N}F^{\left(i+1\right)}.\nonumber
\end{align}

By (\ref{eq:19}) and (\ref{eq:24}), we get
\begin{align}
 & \relphantom =2^{N}N!\left(N+1\right)F^{N}\left(2\left(x-t\right)F^{2}\right)\label{eq:25}\\
 & =\sum_{i=1}^{N}a_{i}\left(N\right)\left(2N-i\right)\left(x-t\right)^{i-2N-1}F^{\left(i\right)}\nonumber \\
 & \relphantom =+\sum_{i=1}^{N}a_{i}\left(N\right)\left(x-t\right)^{i-2N}F^{\left(i+1\right)}.\nonumber
\end{align}

Thus, from (\ref{eq:25}), we have
\begin{align}
 & \relphantom =2^{N+1}\left(N+1\right)!F^{N+2}\label{eq:26}\\
 & =\sum_{i=1}^{N}a_{i}\left(N\right)\left(2N-i\right)\left(x-t\right)^{i-2\left(N+1\right)}F^{\left(i\right)}\nonumber \\
 & \relphantom =+\sum_{i=2}^{N+1}a_{i-1}\left(N\right)\left(x-t\right)^{i-2\left(N+1\right)}F^{\left(i\right)}.\nonumber
\end{align}

On the other hand, by replacing $N$ by $N+1$, in (\ref{eq:23}),
we get
\begin{equation}
2^{N+1}\left(N+1\right)!F^{N+2}=\sum_{i=1}^{N+1}a_{i}\left(N+1\right)\left(x-t\right)^{i-2\left(N+1\right)}F^{\left(i\right)}.\label{eq:27}
\end{equation}

Comparing the coefficients on both sides of (\ref{eq:26}) and
(\ref{eq:27}), we have
\begin{align}
a_{1}\left(N+1\right) & =\left(2N-1\right)a_{1}\left(N\right),\label{eq:28}\\
a_{N+1}\left(N+1\right) & =a_{N}\left(N\right),\label{eq:29}
\end{align}
and
\begin{equation}
a_{i}\left(N+1\right)=a_{i-1}\left(N\right)+\left(2N-i\right)a_{i}\left(N\right),\quad\left(2\le i\le N\right).\label{eq:30}
\end{equation}

Moreover, by (\ref{eq:20-1}) and (\ref{eq:23}), we get
\begin{equation}
2F^{2}=\left(x-t\right)^{-1}F^{\left(1\right)}=a_{1}\left(1\right)\left(x-t\right)^{-1}F^{\left(1\right)}.\label{eq:31}
\end{equation}

By comparing the coefficients on both sides of (\ref{eq:31}),
we get
\begin{equation}
a_{1}\left(1\right)=1.\label{eq:32}
\end{equation}

Now, by (\ref{eq:28}) and (\ref{eq:32}), we have
\begin{align}
a_{1}\left(N+1\right) & =\left(2N-1\right)a_{1}\left(N\right)\label{eq:33}\\
 & =\left(2N-1\right)\left(2N-3\right)a_{1}\left(N-1\right)\nonumber \\
 & =\left(2N-1\right)\left(2N-3\right)\left(2N-5\right)a_{1}\left(N-2\right)\nonumber \\
 & \vdots\nonumber \\
 & =\left(2N-1\right)\left(2N-3\right)\left(2N-5\right)\cdots1\cdot a_{1}\left(1\right)\nonumber \\
 & =\left(2N-1\right)!!,\nonumber
\end{align}
where $\left(2N-1\right)!!$ is Arfken's double factorial.

From (\ref{eq:29}), we easily note that
\begin{equation}
a_{N+1}\left(N+1\right)=a_{N}\left(N\right)=\cdots=a_{1}\left(1\right)=1.\label{eq:34}
\end{equation}

For $2\le i\le N$, from (\ref{eq:30}), we can derive the following
equation:
\begin{align}
a_{i}\left(N+1\right) & =a_{i-1}\left(N\right)+\left(2N-i\right)a_{i}\left(N\right)\label{eq:35}\\
 & =a_{i-1}\left(N\right)+\left(2N-i\right)a_{i-1}\left(N-1\right)+\left(2N-i\right)\left(2N-2-i\right)a_{i}\left(N-1\right)\nonumber \\
 & \vdots\nonumber \\
 & =\sum_{k=0}^{N-i}\left(\prod_{l=0}^{k-1}\left(2\left(N-l\right)-i\right)\right)a_{i-1}\left(N-k\right)+\prod_{l=0}^{N-i}\left(2\left(N-l\right)-i\right)a_{i}\left(i\right)\nonumber \\
 & =\sum_{k=0}^{N-i}2^{k}\left(N-\frac{i}{2}\right)_{k}a_{i-1}\left(N-k\right)+2^{N-i+1}\left(N-\frac{i}{2}\right)_{N-i+1}\nonumber \\
 & =\sum_{k=0}^{N-i+1}2^{k}\left(N-\frac{i}{2}\right)_{k}a_{i-1}\left(N-k\right),\nonumber
\end{align}
where $\left(x\right)_{n}=x\left(x-1\right)\cdots\left(x-n+1\right)$,
$\left(n\ge1\right)$ and $\left(x\right)_{0}=1$.

As the above is also valid for $i=N+1$, by (\ref{eq:35}), we get
\begin{equation}
a_{i}\left(N+1\right)=\sum_{k=0}^{N+1-i}2^{k}\left(N-\frac{i}{2}\right)_{k}a_{i-1}\left(N-k\right),\label{eq:36}
\end{equation}
where $2\le i\le N+1$.

Now, we give an explicit expression for $a_{i}\left(N+1\right)$.

From (\ref{eq:33}) and (\ref{eq:36}), we can derive the following
equations:
\begin{align}
a_{2}\left(N+1\right) & =\sum_{k_{1}=0}^{N-1}2^{k_{1}}\left(N-\frac{2}{2}\right)_{k_{1}}a_{1}\left(N-k_{1}\right)\label{eq:37}\\
 & =\sum_{k_{1}=0}^{N-1}2^{k_{1}}\left(N-\frac{2}{2}\right)_{k_{1}}\left(2\left(N-k_{1}-1\right)-1\right)!!,\nonumber
\end{align}
\begin{align}
a_{3}\left(N+1\right) & =\sum_{k_{2}=0}^{N-2}2^{k_{2}}\left(N-\frac{3}{2}\right)_{k_{2}}a_{2}\left(N-k_{2}\right)\label{eq:38}\\
 & =\sum_{k_{2}=0}^{N-2}\sum_{k_{1}=0}^{N-2-k_{2}}2^{k_{1}+k_{2}}\left(N-\frac{3}{2}\right)_{k_{2}}\left(N-k_{2}-\frac{4}{2}\right)_{k_{1}}\left(2\left(N-2-k_{1}-k_{2}\right)-1\right)!!,\nonumber
\end{align}
and
\begin{align}
a_{4}\left(N+1\right) & =\sum_{k_{3}=0}^{N-3}2^{k_{3}}\left(N-\frac{4}{2}\right)_{k_{3}}a_{3}\left(N-k_{3}\right)\label{eq:39}\\
 & =\sum_{k_{3}=0}^{N-3}\sum_{k_{2}=0}^{N-3-k_{3}}\sum_{k_{1}=0}^{N-3-k_{3}-k_{2}}2^{k_{1}+k_{2}+k_{3}}\left(N-\frac{4}{2}\right)_{k_{3}}\left(N-k_{3}-\frac{5}{2}\right)_{k_{2}}\left(N-k_{3}-k_{2}-\frac{6}{2}\right)_{k_{1}}\nonumber \\
 & \relphantom =\times\left(2\left(N-3-k_{1}-k_{2}-k_{3}\right)-1\right)!!.\nonumber
\end{align}

Thus, we see that, for $2\le i\le N+1$,
\begin{align}
a_{i}\left(N+1\right) & =\sum_{k_{i-1}=0}^{N-i+1}\sum_{k_{i-2}=0}^{N-i+1-k_{i-1}}\cdots\sum_{k_{1}=0}^{N-i+1-k_{i-1}-\cdots-k_{2}}2^{\sum_{j=1}^{i-1}k_{j}}\label{eq:40}\\
 & \relphantom =\times\prod_{j=2}^{i}\left(N-\sum_{l=j}^{i-1}k_{l}-\frac{2i-j}{2}\right)_{k_{j-1}}\left(2\left(N-i+1-\sum_{j=1}^{i-1}k_{j}\right)-1\right)!!.\nonumber
\end{align}

Therefore, we obtain the following theorem.
\begin{thm}
\label{thm:1} The nonlinear differential equations
\[
2^{N}N!F^{N+1}=\sum_{i=1}^{N}a_{i}\left(N\right)\left(x-t\right)^{i-2N}F^{\left(i\right)},\quad\left(N\in\mathbb{N}\right)
\]
has a solution $F=F\left(t,x\right)=\frac{1}{1-2tx+t^{2}}$, where
\begin{align*}
a_{1}\left(N\right) & =\left(2N-3\right)!!,\\
a_{i}\left(N\right) & =\sum_{k_{i-1}=0}^{N-i}\sum_{k_{i-2}=0}^{N-i-k_{i-1}}\cdots\sum_{k_{1}=0}^{N-i-k_{i-1}-\cdots-k_{2}}2^{\sum_{j=1}^{i-1}k_{j}}\\
 & \relphantom =\times\prod_{j=2}^{i}\left(N-\sum_{l=j}^{i-1}k_{l}-\frac{2i+2-j}{2}_{k_{j-1}}\right)\left(2\left(N-i-\sum_{j=1}^{i-1}k_{j}\right)-1\right)!!
\end{align*}
$\left(2\le i\le N\right)$.
\end{thm}
From (\ref{eq:3}) and (\ref{eq:9}), we note that
\begin{align}
 & \sum_{n=0}^{\infty}U_{n}\left(x\right)t^{n}\label{eq:41}\\
 & =\frac{1}{1-2xt+t^{2}}\nonumber \\
 & =\left(\frac{1}{\sqrt{1-2xt+t^{2}}}\right)^{2}\nonumber \\
 & =\left(\sum_{l=0}^{\infty}p_{l}\left(x\right)t^{l}\right)\left(\sum_{m=0}^{\infty}p_{m}\left(x\right)t^{m}\right)\nonumber \\
 & =\sum_{n=0}^{\infty}\left(\sum_{l=0}^{n}p_{l}\left(x\right)p_{n-l}\left(x\right)\right)t^{n}.\nonumber
\end{align}

Thus, from (\ref{eq:41}), we have
\begin{equation}
U_{n}\left(x\right)=\sum_{l=0}^{n}p_{l}\left(x\right)p_{n-l}\left(x\right).\nonumber
\end{equation}

From (\ref{eq:4}), we obtain
\begin{equation}
2^{N}N!F^{N+1}=2^{N}N!\sum_{n=0}^{\infty}U_{n}^{\left(N+1\right)}\left(x\right)t^{n}.\label{eq:42}
\end{equation}

On the other hand, by Theorem \ref{thm:1}, we get
\begin{align}
2^{N}N!F^{N+1} & =\sum_{i=1}^{N}a_{i}\left(N\right)\left(x-t\right)^{i-2N}F^{\left(i\right)}\label{eq:43}\\
 & =\sum_{i=1}^{N}a_{i}\left(N\right)\left(\sum_{m=0}^{\infty}\binom{2N+m-i-1}{m}x^{i-2N-m}t^{m}\right)\left(\sum_{l=0}^{\infty}U_{i+l}\left(x\right)\left(l+i\right)_{i}t^{l}\right)\nonumber \\
 & =\sum_{i=1}^{N}a_{i}\left(N\right)\sum_{n=0}^{\infty}\left\{ \sum_{l=0}^{n}\binom{2N+n-l-i-1}{n-l}x^{i-2N-n+l}U_{l+i}\left(x\right)\left(l+i\right)_{i}\right\} t^{n}\nonumber \\
 & =\sum_{n=0}^{\infty}\left\{ \sum_{i=1}^{N}a_{i}\left(N\right)\sum_{l=0}^{n}\binom{2N+n-l-i-1}{n-l}x^{i+l-2N-n}U_{i+l}\left(x\right)\left(l+i\right)_{i}\right\} t^{n}.\nonumber
\end{align}

Comparing the coefficients on the both sides of (\ref{eq:42}) and (\ref{eq:43}),
we obtain the following theorem.
\begin{thm}
\label{thm:2} For $N\in\mathbb{N}$, and $n\in\mathbb{N}\cup\left\{ 0\right\} $,
the following identity holds.

\[
U_{n}^{\left(N+1\right)}\left(x\right)=\frac{1}{2^{N}N!}\sum_{i=1}^{N}a_{i}\left(N\right)\sum_{l=0}^{n}\binom{2N+n-l-i-1}{n-l}U_{l+i}\left(x\right)x^{i+l-2N-n}\left(l+i\right)_{i}.
\]

\end{thm}
The higher-order Legendre polynomials are given by the generating
function
\begin{equation}
\left(\frac{1}{\sqrt{1-2xt+t^{2}}}\right)^{\alpha}=\sum_{n=0}^{\infty}p_{n}^{\left(\alpha\right)}\left(x\right)t^{n}.\label{eq:44}
\end{equation}

Thus, by \ref{eq:4} and (\ref{eq:43}), we get
\begin{align}
 & \sum_{n=0}^{\infty}U_{n}^{\left(\alpha\right)}\left(x\right)t^{n}\label{eq:45}\\
 & =\left(\frac{1}{1-2xt+t^{2}}\right)^{\alpha}\nonumber \\
 & =\left(\frac{1}{\sqrt{1-2xt+t^{2}}}\right)^{2\alpha}\nonumber \\
 & =\left(\sum_{l=0}^{\infty}p_{l}^{\left(\alpha\right)}\left(x\right)t^{l}\right)\left(\sum_{m=0}^{\infty}p_{m}^{\left(\alpha\right)}\left(x\right)t^{m}\right)\nonumber \\
 & =\sum_{n=0}^{\infty}\left(\sum_{l=0}^{n}p_{l}^{\left(\alpha\right)}\left(x\right)p_{n-l}^{\left(\alpha\right)}\left(x\right)\right)t^{n}.\nonumber
\end{align}

From (\ref{eq:45}), we note that
\begin{equation}
U_{n}^{\left(\alpha\right)}\left(x\right)=\sum_{l=0}^{n}p_{l}^{\left(\alpha\right)}\left(x\right)p_{n-l}^{\left(\alpha\right)}\left(x\right).\label{eq:46}
\end{equation}

Therefore, we obtian the following corollaries.
\begin{cor}
\label{cor:3} For $N\in\mathbb{N}$ and $n\in\mathbb{N}\cup\left\{ 0\right\} $,
we have
\begin{align*}
 & \sum_{l=0}^{n}p_{l}^{\left(N+1\right)}p_{n-l}^{\left(N+1\right)}\left(x\right)\\
 & =\frac{1}{2^{N}N!}\sum_{i=1}^{N}a_{i}\left(N\right)\sum_{l=0}^{n}\binom{2N+n-l-i-1}{n-l}U_{l+i}\left(x\right)\left(l+i\right)_{i}x^{i+l-2N-n}.
\end{align*}

\end{cor}

\begin{cor}
\label{cor:4} For $N\in\mathbb{N}$ and $n\in\mathbb{N}$, we have
\begin{align}
& U_{n}^{\left(N+1\right)}\left(x\right)\nonumber\\
& =\frac{1}{2^{N}N!}\sum_{i=1}^{N}a_{i}\left(N\right)\sum_{l=0}^{n}\sum_{j=0}^{l+i}\binom{2N+n-l-i-1}{n-l}x^{i+l-2N-n}\left(l+i\right)_{i}\left(x\right)p_{l+i-j}\left(x\right).\nonumber
\end{align}

\end{cor}
By (\ref{eq:6}), we get
\begin{align}
 & 2^{N}N!F^{N+1}\label{eq:47}\\
 & =2^{N}N!\left(1-t\right)^{-N-1}\left(\frac{1-t}{1-2xt+t^{2}}\right)^{N+1}\nonumber \\
 & =2^{N}N!\left(\sum_{m=0}^{\infty}\binom{N+m}{m}t^{m}\right)\left(\sum_{l=0}^{\infty}V_{l}^{\left(N+1\right)}\left(x\right)t^{l}\right)\nonumber \\
 & =2^{N}N!\sum_{n=0}^{\infty}\left(\sum_{l=0}^{n}\binom{N+n-l}{n-l}V_{l}^{\left(N+1\right)}\left(x\right)\right)t^{n}.\nonumber
\end{align}

On the other hand, by Theorem \ref{thm:1}, we have
\begin{align}
2^{N}N!F^{N+1} & =\sum_{i=1}^{N}a_{i}\left(N\right)\left(x-t\right)^{i-2N}F^{\left(i\right)}\label{eq:48}\\
 & =\sum_{i=1}^{N}a_{i}\left(N\right)\left(x-t\right)^{i-2N}\left(\frac{d}{dt}\right)^{i}\left(\frac{1}{1-t}\cdot\frac{1-t}{1-xt+t^{2}}\right).\nonumber
\end{align}

From Leibniz formula, we note that
\begin{align}
 & \left(\frac{d}{dt}\right)^{i}\left(\frac{1-t}{1-2xt+t^{2}}\cdot\frac{1}{1-t}\right)\label{eq:49}\\
 & =\sum_{l=0}^{i}\binom{i}{l}\left(\left(\frac{d}{dt}\right)^{i-l}\frac{1}{1-t}\right)\left(\left(\frac{d}{dt}\right)^{l}\frac{1-t}{1-2xt+t^{2}}\right)\nonumber \\
 & =\sum_{l=0}^{i}\binom{i}{l}\left(i-l\right)!\left(1-t\right)^{-i+l-1}\left(\frac{d}{dt}\right)^{l}\left(\frac{1-t}{1-2xt+t^{2}}\right)\nonumber \\
 & =\sum_{l=0}^{i}\binom{i}{l}\left(i-l\right)!\sum_{s=0}^{\infty}\binom{i-l+s}{s}t^{s}\sum_{p=0}^{\infty}V_{p+l}\left(x\right)\left(p+l\right)_{l}t^{p}\nonumber \\
 & =\sum_{l=0}^{i}\frac{i!}{l!}\sum_{s=0}^{\infty}\binom{i-l+s}{s}t^{s}\sum_{p=0}^{\infty}V_{p+l}\left(x\right)\left(p+l\right)_{l}t^{p}.\nonumber
\end{align}

By (\ref{eq:48}) and (\ref{eq:49}), we get
\begin{align}
 & 2^{N}N!F^{N+1}\label{eq:50}\\
 & =\sum_{n=0}^{\infty}\left\{ \sum_{i=1}^{N}\sum_{l=0}^{i}a_{i}\left(N\right)\frac{i!}{l!}\sum_{m+s+p=n}\binom{2N+m-i-1}{m}\binom{i-l+s}{s}\right.\nonumber \\
 & \relphantom =\left.\times\left(p+l\right)_{l}x^{i-2N-m}V_{p+l}\left(x\right)\right\} t^{n}.\nonumber
\end{align}

Therefore, by (\ref{eq:47}) and (\ref{eq:50}), we obtain the following
theorem.
\begin{thm}
\label{thm:5} For $N\in\mathbb{N}$ and $n\in\mathbb{N}\cup\left\{ 0\right\} $,
we have the following identity:
\begin{align*}
 & \sum_{l=0}^{n}\binom{N+n-l}{n-l}V_{l}^{\left(N+1\right)}\left(x\right)\\
 & =\frac{1}{2^{N}N!}\sum_{i=1}^{N}\sum_{l=0}^{i}a_{i}\left(N\right)\frac{i!}{l!}\sum_{m+s+p=n}\binom{2N+m-i-1}{m}\binom{i-l+s}{s}\left(p+l\right)_{l}\\
 & \relphantom =\times x^{i-2N-m}V_{p+l}\left(x\right).
\end{align*}

\end{thm}
From (\ref{eq:8}), we note that
\begin{align}
 & 2^{N}N!F^{N+1}\label{eq:51}\\
 & =2^{N}N!\left(1+t\right)^{-N-1}\left(\frac{1+t}{1-2xt+t^{2}}\right)^{N+1}\nonumber \\
 & =2^{N}N!\left(\sum_{m=0}^{\infty}\binom{N+m}{m}\left(-1\right)^{m}t^{m}\right)\left(\sum_{l=0}^{\infty}W_{l}^{\left(N+1\right)}\left(x\right)t^{l}\right)\nonumber \\
 & =2^{N}N!\sum_{n=0}^{\infty}\left(\sum_{l=0}^{n}\left(-1\right)^{n-l}\binom{N+n-l}{n-l}W_{l}^{\left(N+1\right)}\left(x\right)\right)t^{n}.\nonumber
\end{align}

On the other hand, by Theorem \ref{thm:1}, we get
\begin{equation}
2^{N}N!F^{N+1}=\sum_{i=1}^{N}a_{i}\left(N\right)\left(x-t\right)^{i-2N}\left(\frac{d}{dt}\right)^{i}\left\{ \frac{1}{1+t}\cdot\frac{1+t}{1-2xt+t^{2}}\right\} .\label{eq:52}
\end{equation}

Now, we observe that
\begin{align}
 & \left(\frac{d}{dt}\right)^{i}\left\{ \left(\frac{1}{1+t}\right)\left(\frac{1+t}{1-2xt+t^{2}}\right)\right\} \label{eq:53}\\
 & =\sum_{l=0}^{i}\binom{i}{l}\left(-1\right)^{i-l}\left(i-l\right)!\left(\frac{1}{1+t}\right)^{i-l+1}\left(\frac{d}{dt}\right)^{l}\left(\frac{1+t}{1-2xt+t^{2}}\right)\nonumber \\
 & =\sum_{l=0}^{i}\binom{i}{l}\left(-1\right)^{i-l}\left(i-l\right)!\sum_{s=0}^{\infty}\binom{i-l+s}{s}\left(-1\right)^{s}t^{s}\sum_{p=0}^{\infty}W_{p+l}\left(x\right)\left(p+l\right)_{l}t^{p}.\nonumber
\end{align}

From (\ref{eq:52}) and (\ref{eq:53}), we have
\begin{align}
 & 2^{N}N!F^{N+1}\label{eq:54}\\
 & =\sum_{n=0}^{\infty}\left\{ \sum_{i=1}^{N}a_{i}\left(N\right)\sum_{l=0}^{i}\left(-1\right)^{i-l}\frac{i!}{l!}\sum_{m+s+p=n}\left(-1\right)^{s}\binom{2N+m-i-1}{m}\right.\nonumber \\
 & \relphantom =\left.\times\binom{i-l+s}{s}\left(p+l\right)_{l}x^{i-2N-m}W_{p+l}\left(x\right)\right\} t^{n}.\nonumber
\end{align}

Therefore, by (\ref{eq:51}) and (\ref{eq:54}), we obtain the following
theorem.
\begin{thm}
\label{thm:6} For $N\in\mathbb{N}$ and $n\in\mathbb{N}\cup\left\{ 0\right\} $,
the following identity is valid:
\begin{align*}
 & \sum_{l=0}^{n}\left(-1\right)^{n-l}\binom{N+n-l}{n-l}W_{l}^{\left(N+1\right)}\left(x\right)\\
 & =\frac{1}{2^{N}N!}\sum_{i=1}^{N}\sum_{l=0}^{i}\left(-1\right)^{i-l}a_{i}\left(N\right)\frac{i!}{l!}\sum_{m+s+p=n}\left(-1\right)^{s}\binom{2N+m-i-1}{m}\\
 & \relphantom =\times\binom{i-l+s}{s}\left(p+l\right)_{l}x^{i-2N-m}W_{p+l}\left(x\right).
\end{align*}

\end{thm}
From (\ref{eq:1}), we have
\begin{align}
 & 2^{N}N!F^{N+1}\label{eq:55}\\
 & =2^{N}N!\left(\frac{1}{1-t^{2}}\cdot\frac{1-t^{2}}{1-2xt+t^{2}}\right)^{N+1}\nonumber \\
 & =2^{N}N!\left(\frac{1}{1-t}\right)^{N+1}\left(\frac{1}{1+t}\right)^{N+1}\left(\frac{1-t^{2}}{1-2xt+t^{2}}\right)^{N+1}\nonumber \\
 & =2^{N}N!\left(\sum_{l=0}^{\infty}\binom{N+l}{l}t^{l}\right)\left(\sum_{m=0}^{\infty}\binom{m+N}{m}\left(-1\right)^{m}t^{m}\right)\left(\sum_{p=0}^{\infty}T_{p}^{\left(N+1\right)}\left(x\right)t^{p}\right)\nonumber \\
 & =2^{N}N!\sum_{n=0}^{\infty}\left(\sum_{l+m+p=n}\binom{N+l}{l}\binom{m+N}{m}\left(-1\right)^{m}T_{p}^{\left(N+1\right)}\left(x\right)\right)t^{n}.\nonumber
\end{align}

On the other hand, by Theorem \ref{thm:1}, we get
\begin{align}
 & 2^{N}N!F^{N+1}\label{eq:56}\\
 & =\sum_{i=1}^{N}a_{i}\left(N\right)\left(x-t\right)^{i-2N}F^{\left(i\right)}\nonumber \\
 & =\frac{1}{2}\sum_{i=1}^{N}a_{i}\left(N\right)\left(x-t\right)^{i-2N}\left(\frac{d}{dt}\right)^{i}\left\{ \left(\frac{1}{1-t}+\frac{1}{1+t}\right)\frac{1-t^{2}}{1-2xt+t^{2}}\right\} .\nonumber
\end{align}

From Leibniz formula, we note that the following equations:
\begin{align}
 & \left(\frac{d}{dt}\right)^{i}\left\{ \left(\frac{1}{1-t}\right)\cdot\left(\frac{1-t^{2}}{1-2xt+t^{2}}\right)\right\} \label{eq:57}\\
 & =\sum_{l=0}^{i}\binom{i}{l}\left(i-l\right)!\sum_{s=0}^{\infty}\binom{i+s-l}{s}t^{s}\sum_{p=0}^{\infty}T_{p+l}\left(x\right)\left(p+l\right)_{l}t^{p},\nonumber
\end{align}
and
\begin{align}
 & \left(\frac{d}{dt}\right)^{i}\left\{ \left(\frac{1}{1+t}\right)\left(\frac{1-t^{2}}{1-2xt+t^{2}}\right)\right\} \label{eq:58}\\
 & =\sum_{l=0}^{i}\binom{i}{l}\left(i-l\right)!\left(-1\right)^{i-l}\sum_{s=0}^{\infty}\binom{i-l+s}{s}\left(-1\right)^{s}t^{s}\sum_{p=0}^{\infty}T_{p+l}\left(x\right)\left(p+l\right)_{l}t^{p}.\nonumber
\end{align}

By (\ref{eq:56}), (\ref{eq:57}), and (\ref{eq:58}), we obtain
\begin{align}
 & 2^{N}N!F^{N+1}\label{eq:59}\\
 & =\frac{1}{2}\sum_{i=1}^{N}a_{i}\left(N\right)\left(x-t\right)^{i-2N}\sum_{l=0}^{i}\binom{i}{l}\left(i-l\right)!\sum_{s=0}^{\infty}\binom{i+s-l}{s}t^{s}\sum_{k=0}^{\infty}T_{p+l}\left(x\right)\left(p+l\right)_{l}t^{p}\nonumber \\
 & \relphantom =+\frac{1}{2}\sum_{i=1}^{N}a_{i}\left(N\right)\left(x-t\right)^{i-2N}\sum_{l=0}^{i}\binom{i}{l}\left(i-l\right)!\left(-1\right)^{i-l}\sum_{s=0}^{\infty}\binom{i-l+s}{s}\left(-1\right)^{s}t^{s}\nonumber \\
 & \relphantom =\times\sum_{p=0}^{\infty}T_{p+l}\left(x\right)\left(p+l\right)_{l}t^{p}\nonumber \\
 & =\frac{1}{2}\sum_{n=0}^{\infty}\sum_{i=1}^{N}\sum_{l=0}^{i}a_{i}\left(N\right)\frac{i!}{l!}\sum_{m+s+p=n}\binom{2N+m-i-1}{m}\binom{i+s-l}{s}\left(p+l\right)_{l}\nonumber\\
 & \relphantom =\times x^{i-2N-m}T_{p+l}\left(x\right)t^{n}+\frac{1}{2}\sum_{n=0}^{\infty}\sum_{i=1}^{N}\sum_{l=0}^{i}a_{i}\left(N\right)\frac{i!}{l!}\left(-1\right)^{i-l}\nonumber\\
 & \relphantom =\times\sum_{m+s+p=n}\left(-1\right)^{s}\binom{2N+m-i-1}{m}\binom{i+s-l}{s}\left(p+l\right)_{l}x^{i-2N-m}T_{p+l}\left(x\right)t^{n}.\nonumber
\end{align}

Therefore, by (\ref{eq:55}) and (\ref{eq:59}), we obtain the following
theorem.
\begin{thm}
\label{thm:7} For $n\in\mathbb{N}\cup\left\{ 0\right\} $ and $N\in\mathbb{N}$,
we have the following identity
\begin{align*}
 & 2^{N+1}N!\sum_{s+m+p=n}\binom{N+s}{s}\binom{m+N}{m}\left(-1\right)^{m}T_{p}^{\left(N+1\right)}\left(x\right)\\
 & =\sum_{i=1}^{N}\sum_{l=0}^{i}a_{i}\left(N\right)\frac{i!}{l!}\sum_{m+s+p=n}\binom{2N+m-i-1}{m}\binom{i+s-l}{s}\left(p+l\right)_{l}\nonumber\\
 & \relphantom = \times x^{i-2N-m}T_{p+l}\left(x\right)+\sum_{i=1}^{N}\sum_{l=0}^{i}a_{i}\left(N\right)\frac{i!}{l!}\left(-1\right)^{i-l}\sum_{m+s+p=n}\left(-1\right)^{s}\binom{2N+m-i-1}{m}\nonumber\\
 & \relphantom =\times\binom{i+s-l}{s}\left(p+l\right)_{l}x^{i-2N-m}T_{p+l}\left(x\right).
\end{align*}
\end{thm}

Acknowledgements. This paper is supported by grant NO 14-11-00022 of Russian Scientific Fund.

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A 35-year-old Veblen man has been selected as interim executive director of the South Dakota Republican Party. Lucas Lentsch is a South Dakota State University graduate and a veteran who served in Iraq. He was a founding member and the first chairman of Glacial Lakes Area Development, which continues to serve Pierpont, Langford, Eden, Roslyn, Lake City, Veblen and Britton. Lentsch ran for the state Senate from District 1 last year, losing to Democrat Gary Hanson. Karl Adam, state Republican Party chairman, said Lentsch has energy, experience and passion.