Analysis of a second order thermodynamic model for the performance analysis of a Stirling engine prototype
Miguel Torres García, Elisa Carvajal Trujillo, José Antonio Vélez Godiño, David Sánchez Martínez
Subject: Engineering, Energy & Fuel Technology Keywords: Stirling engine, combustion, themodynamic model
The Stirling engine is a simple type of external-combustion engine with an external combustion engine based on cyclic compression and expansion of gas at different temperature levels. It has high efficiency; low vibration levels, simple structure and can run on any combustible fuels. It has been the object of numerous studies. This paper presents an analysis of a Stirling engine model GENOA 03 for electric power generation, of 3 KW of nominal power with pressurized air as working fluid, currently under development. To improve its performance and ensure a good operational reliability, it is necessary to carry out a modelling of the engine in all its operating range. This requires complex numerical models that simulate the behaviour of any element of the engine in a cycle. Two typologies of thermodynamic models are developed in this work: isothermal and adiabatic. The main benefits and shortcomings of each model are mentioned. The geometry and conditions of the engine have been adapted through the Matlab ® tool, in order to obtain the operative conditions of the cooler that you want to replace, as well as an approximation to the expected behaviour.
Preprint SHORT NOTE | doi:10.20944/preprints201703.0055.v1
A Closed Form for the Stirling Polynomials in Terms of the Stirling Numbers
Feng Qi, Bai-Ni Guo
Subject: Mathematics & Computer Science, Analysis Keywords: closed form; Stirling polynomial; Stirling number; Bernoulli number; Faá di Bruno's formula; Bell polynomial
In the paper, by virtue of the Faá di Bruno formula and two identities for the Bell polynomial of the second kind, the authors find a closed form for the Stirling polynomials in terms of the Stirling numbers of the first and second kinds.
Modified Degenerate Carlitz's q-Bernoulli Polynomials and Numbers with Weight (α, β)
Ugur Duran, Mehmet Acikgoz
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Carlitz's q-Bernoulli polynomials; Stirling numbers of the first kind; Stirling numbers of the second kind; p-adic q-integral
The main goal of the present paper is to construct some families of the Carlitz's q-Bernoulli polynomials and numbers. We firstly introduce the modified Carlitz's q-Bernoulli polynomials and numbers with weight (α, β) and investigate their some explicit properties and identities arising from the bosonic q-Volkenborn integral on ℤp. We then define the modified degenerate Carlitz's q-Bernoulli polynomials and numbers with weight (α, β) and obtain some recurrence relations and other identities. Moreover, we derive some correlations with the modified Carlitz's q-Bernoulli polynomials with weight (α, β), the modified degenerate Carlitz's q-Bernoulli polynomials with weight (α, β), the Stirling numbers of the first kind and second kind.
Relationships between Mahler Expansion and Higher Order q-Daehee polynomials
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: q-numbers, p-adic numbers, p-adic gamma function, Mahler expansion, Bernoulli polynomials, Daehee polynomials, Stirling numbers of the first kind, Stirling numbers of the second kind.
In this paper, we derive multifarious relationships among the two types of higher order q-Daehee polynomials and p-adic gamma function via Mahler theorem. Also, we compute some weighted p-adic q-integrals of the derivative of p-adic gamma function related to the Stirling numbers of the both kinds and the q-Bernoulli polynomials of order k.
Some Identities for a Sequence of Unnamed Polynomials Connected with the Bell Polynomials
Feng Qi, Da-Wei Niu, Bai-Ni Guo
Subject: Mathematics & Computer Science, Analysis Keywords: identity; Bell polynomial; unnamed polynomial; explicit formula; inversion theorem; Stirling number; binomial coefficient
In the paper, using two inversion theorems for the Stirling numbers and binomial coe cients, employing properties of the Bell polynomials of the second kind, and utilizing a higher order derivative formula for the ratio of two di erentiable functions, the authors present two explicit formulas, a determinantal expression, and a recursive relation for a sequence of unnamed polynomials, derive two identities connecting the sequence of unnamed polynomials with the Bell polynomials, and recover a known identity connecting the sequence of unnamed polynomials with the Bell polynomials.
Some Identities Relating to Eulerian Polynomials and Involving Stirling Numbers
Feng Qi, Dongkyu Lim, Bai-Ni Guo
Subject: Mathematics & Computer Science, Computational Mathematics Keywords: identity; generating function; Eulerian polynomial; higher order Eulerian polynomial; Stirling number; open problem
In the paper, the authors establish two identities, which can be regarded as nonlinear differential equations, for the generating function of Eulerian polynomials, find two identities for the Stirling numbers of the second kind, and present two identities for Eulerian polynomials and higher order Eulerian polynomials, pose two open problems about summability of two finite sums involving the Stirling numbers of the second kind. Some of these conclusions meaningfully and significantly simplify several known results.
Simplifying Two Families of Nonlinear Ordinary Differential Equations
Feng Qi, Jing-Lin Wang, Bai-Ni Guo
Subject: Mathematics & Computer Science, Analysis Keywords: nonlinear ordinary differential equation; Stirling number of the first kind; simplifying; technique; combinatorial analysis
In the paper, by virtue of techniques in combinatorial analysis, the authors simplify two families of nonlinear ordinary differential equations in terms of the Stirling numbers of the first kind.
Degenerate Poisson Distribution
Subject: Mathematics & Computer Science, Probability And Statistics Keywords: Poisson distribution; raw moments; Bell polynomials; degenerate exponential function; unsigned Stirling number of the first kind.
The main purpose of this paper is to introduce and investigate degenerate Poisson distrib- ution which is a new extension of the Poisson distribution including the degenerate expo- nential function. We then provide several properties of the degenerate Poisson distribution such as the first and the second raw moments and di¤erence operator property. Moreover, we acquired the skewness and the kurtosis for the degenerate Poisson distribution. We also derive its moment generating function by which we define the degenerate Bell polynomials and give a connection for these polynomials related to the unsigned Stirling numbers of the …rst kind.
On (q, r, w)-Stirling Numbers of the Second Kind
Ugur Duran, Mehmet Acikgoz, Serkan Araci
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: q-calculus; stirling numbers of the second kind; bernoulli polynomials and numbers; generating function; cauchy product
In this paper, we introduce a new generalization of the r-Stirling numbers of the second kind based on the q-numbers via an exponential generating function. We investigate their some properties and derive several relations among q-Bernoulli numbers and polynomials, and newly de…ned (q, r, w)-Stirling numbers of the second kind. We also obtain q-Bernstein polynomials as a linear combination of (q, r, w)-Stirling numbers of the second kind and q-Bernoulli polynomials in w.
Simplifying Coefficients in Differential Equations Associated with Higher Order Bernoulli Numbers of the Second Kind
Subject: Mathematics & Computer Science, Analysis Keywords: simplification; coefficient; ordinary differential equation; higher order Bernoulli number of the second kind; Stirling number of the first kind; Stirling number of the second kind; inversion formula; Bell polynomial of the second kind; Faà di Bruno formula.
In the paper, by virtue of the Faà di Bruno formula, some properties of the Bell polynomials of the second kind, and an inversion formula for the Stirling numbers of the first and second kinds, the authors establish meaningfully and significantly two identities which simplify coefficients in a family of ordinary differential equations associated with higher order Bernoulli numbers of the second kind.
Combinatorial Models of the Distribution of Prime Numbers
Vito Barbarani
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: set partitions; Stirling numbers of the second kind; harmonic numbers; prime number distribution; Riemann hypothesis; Gumbel distribution
This work is divided into two parts. In the first one the combinatorics of a new class of randomly generated objects, exhibiting the same properties as the distribution of prime numbers, is solved and the probability distribution of the combinatorial counterpart of the n-th prime number is derived, together with an estimate of the prime-counting function π(x). A proposition equivalent to the Prime Number Theorem (PNT) is proved to hold, while the equivalent of the Riemann Hypothesis (RH) is proved to be false with probability 1 (w.p. 1) for this model. Many identities involving Stirling numbers of the second kind and harmonic numbers are found, some of which appear to be new. The second part is dedicated to generalizing the model to investigate the conditions enabling both PNT and RH. A model representing a general class of random integer sequences is found, for which RH holds w.p. 1. The prediction of the number of consecutive prime pairs, as a function of the gap d, is derived from this class of models and the results are in agreement with empirical data for large gaps. A heuristic version of the model, directly related to the sequence of primes, is discussed and new integral lower and upper bounds of π(x) are found.
Construction of the Type 2 Degenerate Multi-Poly-Euler Polynomials and Numbers
Waseem Ahmad Khan, Mehmet Acikgoz, Ugur Duran
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Euler polynomials; Degenerate multi-polyexponential functions; Degenerate multi-poly-Euler polynomials; Degenerate Stirling numbers; Degenerate Whitney numbers
In this paper, we consider a new class of polynomials which is called the multi-poly-Euler polynomials. Then, we investigate their some properties and relations. We provide that the type 2 degenerate multi-poly-Euler polynomials equals a linear combination of the degenerate Euler polynomials of higher order and the degenerate Stirling numbers of the first kind. Moreover, we provide an addition formula and a derivative formula. Furthermore, in a special case, we acquire a correlation between the type 2 degenerate multi-poly-Euler polynomials and degenerate Whitney numbers.
A Note on Type 2 Degenerate Multi-Poly-Bernoulli Polynomials and Numbers
Waseem A Khan, Aysha Khan, Ugur Duran
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Bernoulli polynomials; Degenerate multi-polyexponential functions; Degenerate multi-poly-Bernoulli polynomials; Degenerate Stirling numbers; Degenerate Whitney numbers
Inspired by the definition of degenerate multi-poly-Genocchi polynomials given by using the degenerate multi-polyexponential functions. In this paper, we consider a class of new generating function for the degenerate multi-poly-Bernoulli polynomials, called the type 2 degenerate multi-poly-Bernoulli polynomials by means of the degenerate multiple polyexponential functions. Then, we investigate their some properties and relations. We show that the type 2 degenerate multi-poly-Bernoulli polynomials equals a linear combination of the weighted degenerate Bernoulli polynomials and Stirling numbers of the first kind. Moreover, we provide an addition formula and a derivative formula. Furthermore, in a special case, we acquire a correlation between the type 2 degenerate multi-poly-Bernoulli numbers and degenerate Whitney numbers.
Viewing Some Ordinary Differential Equations from the Angle of Derivative Polynomials
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: viewpoint; ordinary differential equation; solution; derivative polynomial; identity; Stirling numbers; Bernoulli number; Bernoulli polynomial; Frobenius-Euler polynomial
In the paper, the authors view some ordinary differential equations and their solutions from the angle of (the generalized) derivative polynomials and simplify some known identities for the Bernoulli numbers and polynomials, the Frobenius-Euler polynomials, the Euler numbers and polynomials, in terms of the Stirling numbers of the first and second kinds.
Hermite based Poly-Bernoulli Polynomials with a q-parameter
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Hermite-Kampé de Fériet polynomials; Hermite Bernoulli polynomials; poly Bernoulli polynomials; Stirling numbers of second kind; Polylogarithm functions.
We introduce the Hermite based poly-Bernoulli polynomials with a q parameter and give someof their basic properties including not only addition property, but also derivative properties and integralrepresentations. We also de.ne the Hermite based λ -Stirling polynomials of the second kind, and thenprovide some relations. Moreover, we derive several correlations and identities including the Hermite-Kampéde Fériet (or Gould-Hopper) family of polynomials, the Hermite based poly-Bernoulli polynomials with a qparameter and the Hermite based λ -Stirling polynomials of the second kind.
Unified Degenerate Central Bell Polynomials
Mehmet Acikgoz, Ugur Duran
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Central Bell polynomials; central factorial numbers; degenerate central Bell polynomials; Stirling numbers of the first kind; special numbers; special polynomials.
In this paper, we firstly consider extended degenerate central factorial numbers of the second kind and provide some properties of them. We then introduce unified degenerate central Bell polynomials and numbers and investigate many relations and formulas including summation formula, explicit formula and derivative property. Moreover, we derive several correlations for the fully degenerate central Bell polynomials associated with the degenerate Bernstein polynomials and the degenerate Bernoulli, Euler and Genocchi numbers.
Some Relationships Including p-Adic Gamma Function and q-Daehee Polynomials and Numbers
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: p-adic numbers; p-adic gamma function; p-adic Euler constant; q-Daehee polynomials; Stirling numbers of the first kind
In this paper, we investigate p-adic q-integral (q-Volkenborn integral) on Zp of p-adic gamma function via their Mahler expansions. We also derived two q-Volkenborn integrals of p-adic gamma function in terms of q-Daehee polynomials and numbers and q-Daehee polynomials and numbers of the second kind. Moreover, we discover q-Volkenborn integral of the derivative of p-adic gamma function. We acquire the relationship between the p-adic gamma function and Stirling numbers of the first kind. We finally develop a novel and interesting representation for the p-adic Euler constant by means of the q-Daehee polynomials and numbers.
The Boole Polynomials Associated with the p-adic Gamma Function
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: p-adic numbers, p-adic gamma function, p-adic Euler constant, Mahler expansion, Boole polynomials, Stirling numbers of the first kind.
The main aim of this paper is to set some correlations between Boole polynomials and p-adic gamma function in conjunction with p-adic Euler contant. We develop diverse formulas for p-adic gamma function by means of their Mahler expansion and fermionic p-adic integral on ℤ_{p}. Also, we acquire two fermionic p-adic integrals of p-adic gamma function in terms of Boole numbers and polynomials. We then provide fermionic p-adic integral of the derivative of p-adic gamma function and a representation for the p-adic Euler constant by means of the Boole polynomials. Furthermore, we investigate an explicit representation for the aforesaid constant covering Stirling numbers of the first kind.
Unified (p, q) -analog of Apostol type polynomials of order a
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: (p, q)-calculus; apostol-bernoulli polynomials; apostol-euler polynomials; apostol- genocchi polynomials; stirling numbers of second kind; generating function; cauchy product
Motivated by Kurt's work [Filomat 30 (4) 921-927, 2016], we first consider a class of a new generating function for (p, q)-analog of Apostol type polynomials of order a including Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials of order a. By making use of their generating function, we derive some useful identities. We also introduce (p, q)-analog of Stirling numbers of second kind of order v by which we construct a relation including aforementioned polynomials.
Some Inequalities of the Bell Polynomials
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Bell polynomial; Bell number; Bell polynomial of the second kind; higher order derivative; generating function; Faa di Bruno formula; inversion theorem; Stirling number of the first kind; Stirling number of the second kind; explicit formula; inversion formula; logarithmically absolute monotonicity; logarithmically complete monotonicity; determinantal inequality; product inequality
In the paper, the author (1) presents an explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials, with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds; (2) recovers an explicit formula and its inversion formula for the Bell polynomials in terms of the Stirling numbers of the first and second kinds, with the aid of the above explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials; (3) constructs some determinantal and product inequalities and deduces the logarithmic convexity of the Bell polynomials, with the assistance of the complete monotonicity of generating functions of the Bell polynomials. These inequalities are main results of the paper.
On Mahler Expansion of p-adic Gamma Function Affiliated with the q-Boole Polynomials
Subject: Keywords: p-adic numbers, p-adic gamma function, p-adic Euler constant, Mahler expansion, q-Boole polynomials, Stirling numbers of the first kind.
In this paper, we investigate several relations for p-adic gamma function by means of their Mahler expansion and fermionic p-adic q-integral on ℤ_{p}. We also derive two fermionic p-adic q-integrals of p-adic gamma function in terms of q-Boole polynomials and numbers. Moreover, we discover fermionic p-adic q-integral of the derivative of p-adic gamma function. We acquire a representation for the p-adic Euler constant by means of the q-Boole polynomials. We finally develop a novel, explicit and interesting representation for the p-adic Euler constant including Stirling numbers of the first kind.
On p-adic Gamma Function Related to q-Daehee Polynomials and Numbers
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: p-adic numbers, p-adic gamma function, p-adic Euler constant, Mahler expansion, q-Daehee polynomials, Stirling numbers of the first kind.
In this paper, we investigate p-adic q-integral (q-Volkenborn integral) on ℤ_{p} of p-adic gamma function via their Mahler expansions. We also derived two q-Volkenborn integrals of p-adic gamma function in terms of q-Daehee polynomials and numbers and q-Daehee polynomials and numbers of the second kind. Moreover, we discover q-Volkenborn integral of the derivative of p-adic gamma function. We acquire the relationship between the p-adic gamma function and Stirling numbers of the first kind. We finally develop a novel and interesting representation for the p-adic Euler constant by means of the q-Daehee polynomials and numbers.
Some Properties of $Q$-Hermite Fubini Numbers and Polynomials
Waseem A. Khan
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: $q$-Hermite polynomials; $q$-Hermite-Fubini polynomials; $q$-Bernoulli polynomials; $q$-Euler polynomials; $q$-Genocchi polynomials; Stirling numbers of the second kind
The main purpose of this paper is to introduce a new class of $q$-Hermite-Fubini numbers and polynomials by combining the $q$-Hermite polynomials and $q$-Fubini polynomials. By using generating functions for these numbers and polynomials, we derive some alternative summation formulas including powers of consecutive $q$-integers. Also, we establish some relationships for $q$-Hermite-Fubini polynomials associated with $q$-Bernoulli polynomials, $q$-Euler polynomials and $q$-Genocchi polynomials and $q$-Stirling numbers of the second kind.
Certain Results on (p,q)-Hermite Based Apostol Type Frobenius-Euler Polynomials
Waseem Khan, Idrees Ahmad Khan, Ugur Duran, Mehmet Acikgoz
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Hermite polynomials, Apostol type Frobenius-Euler polynomials, Hermite based Apostol type Frobenius Euler polynomials, Stirling numbers of the second kind, (p,q)-numbers.
In the present paper, the (p,q)-Hermite based Apostol type Frobenius-Euler polynomials and numbers are firstly considered and then diverse basic identities and properties for the mentioned polynomials and numbers, including addition theorems, difference equations, integral representations, derivative properties, recurrence relations. Moreover, we provide summation formulas and relations associated with the Stirling numbers of the second kind.
An Odd-Power Identity Involving Discrete Convolution
Kolosov Petro
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: power identities; polynomials; convolution; convolution power; integral transforms; power function; binomial theorem; multinomial theorem; Faulhaber's formula; Bernoulli numbers; Worpitzky identity; Stirling numbers; falling factorial
Let be a power function $f_{r,M}(s)$ defined for every $s$ within the finite set $M$ as follows $$f_{r,M}(s)= \begin{cases} s^r, \ &s\in M,\\ 0, \ &\mathrm{otherwise}. \end{cases} $$ Let a discrete convolution of $f_{r,M}(s)$ be denoted as follows $\mathrm{Conv}_{r,M}[n]=(f_{r,M}*f_{r,M})[n]$. Let a real coefficients $A_{m,j}$ be given by the following recurrence $$ A_{m,j} = \begin{cases} 0, & \mathrm{if } \ j<0 \ \mathrm{or } \ j>m, \\ (2j+1)\binom{2j}{j} \sum_{d=2j+1}^{m} A_{m,d} \binom{d}{2j+1} \frac{(-1)^{d-1}}{d-j} B_{2d-2j}, & \mathrm{if } \ 0 \leq j < m, \\ (2j+1)\binom{2j}{j}, & \mathrm{if } \ j=m. \end{cases} $$ In this paper we show that for every $n>0$ the following odd-power identities involving coefficients $A_{m,j}$ and convolution transform $\mathrm{Conv}_{r,M}[n]$ hold $$ \begin{split} n^{2m+1}+1&=\sum_{r=0}^{m}A_{m,r}\mathrm{Conv}_{r,\mathbb{N}}[n],\\ n^{2m+1}-1&=\sum_{r=0}^{m}A_{m,r}\mathrm{Conv}_{r,\mathbb{Z}_{>0}}[n],\\ n^{2m+1}&=\sum_{r=0}^{m}A_{m,r}\sum_{k=1}^{n} k^r(n-k)^r\\ &=\sum_{r=0}^{m}A_{m,r}\sum_{k=0}^{n-1} k^r(n-k)^r. \end{split} $$
Generalizations of the Bell Numbers and Polynomials and Their Properties
Feng Qi, Da-Wei Niu, Dongkyu Lim, Bai-Ni Guo
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: Bell number; Bell polynomial; generalization; explicit formula; inversion formula; inversion theorem; Stirling number; Bell polynomial of the second kind; determinantal inequality; product inequality; completely monotonic function; logarithmic convexity
In the paper, the authors present unified generalizations for the Bell numbers and polynomials, establish explicit formulas and inversion formulas for these generalizations in terms of the Stirling numbers of the first and second kinds with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem connected with the Stirling numbers of the first and second kinds, construct determinantal and product inequalities for these generalizations with aid of properties of the completely monotonic functions, and derive the logarithmic convexity for the sequence of these generalizations.
Some Properties of a Solution to a Family of Inhomogeneous Linear Ordinary Differential Equations
Subject: Mathematics & Computer Science, Analysis Keywords: explicit form; inhomogeneous linear ordinary differential equation; derivative; Lerch transcendent; absolute monotonicity; complete monotonicity; Bernstein function; inequality; diagonal recurrence relation; Stirling numbers of the first kind; logarithmic function
In the paper, the authors present an explicit form for a family of inhomogeneous linear ordinary differential equations, find a more significant expression for all derivatives of a function related to the solution to the family of inhomogeneous linear ordinary differential equations in terms of the Lerch transcendent, establish an explicit formula for computing all derivatives of the solution to the family of inhomogeneous linear ordinary differential equations, acquire the absolute monotonicity, complete monotonicity, the Bernstein function property of several functions related to the solution to the family of inhomogeneous linear ordinary differential equations, discover a diagonal recurrence relation of the Stirling numbers of the first kind, and derive an inequality for bounding the logarithmic function.
A Note on $(P,Q)$-Analogue Type of Fubini Numbers and Polynomials
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: $(p,q)$-calculus; $(p; q)$-Bernoulli polynomials; $(p; q)$-Euler polynomials; $(p; q)$-Genocchi polynomials; $(p; q)$-Fubini numbers and polynomials; $(p; q)$ Stirling numbers of the second kind
In this paper, we introduce a new class of $(p,q)$-analogue type of Fubini numbers and polynomials and investigate some properties of these polynomials. We establish summation formulas of these polynomials by summation techniques series. Furthermore, we consider some relationships for $(p,q)$-Fubini polynomials associated with $(p,q)$-Bernoulli polynomials, $(p,q)$-Euler polynomials and $(p,q)$-Genocchi polynomials and $(p,q)$-Stirling numbers of the second kind.
Series Representation of Power Function
Petro Kolosov
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: series representation; power function; monomial; binomial theorem; multinomial theorem; worpitzky identity; stirling numbers of second kind; faulhaber's sum; finite difference; faulhaber's formula; central factorial numbers; binomial coefficients; binomial distribution; binomial transform; bernoulli numbers; oeis; multinomial coefficients
In this paper we discuss a problem of generalization of binomial distributed triangle, that is sequence A287326 in OEIS. The main property of A287326 that it returns a perfect cube n as sum of n-th row terms over k; 0 ≤ k ≤ n−1 or 1 ≤ k ≤ n, by means of its symmetry. In this paper we have derived a similar triangles in order to receive powers m = 5; 7 as row items sum and generalized obtained results in order to receive every odd-powered monomial n2m+1; m ≥ 0 as sum of row terms of corresponding triangle. In other words, in this manuscript are found and discussed the polynomials Dm(n,k) and Um(n,k), such that, when being summed up over k in some range with respect to m and n returns the monomial n2m+1.
On Multi-Order Logarithmic Polynomials and their Explicit Formulas, Recurrence Relations, and Inequalities
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: multi-order logarithmic number; multi-order logarithmic polynomial; explicit formula; identity; recurrence relation; inversion theorem; Bell polynomial of the second kind; Stirling number; determinantal inequality; product inequality; completely monotonic function; logarithmic convexity; Faa di Bruno formula
In the paper, the author introduces the notions "multi-order logarithmic numbers" and "multi-order logarithmic polynomials", establishes an explicit formula, an identity, and two recurrence relations by virtue of the Fa a di Bruno formula and two identities of the Bell polynomials of the second kind in terms of the Stirling numbers of the fi rst and second kinds, and constructs some determinantal inequalities, product inequalities, logarithmic convexity for multi-order logarithmic numbers and polynomials by virtue of some properties of completely monotonic functions.