Some properties and an application of multivariate exponential polynomials

In the paper, the authors introduce a notion “multivariate exponential polynomials” which generalize exponential numbers and polynomials, establish explicit formulas, inversion formulas, and recurrence relations for multivariate exponential polynomials in terms of the Stirling numbers of the first and second kinds with the help of the Faà di Bruno formula, two identities for the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds, construct some determinantal inequalities and product inequalities for multivariate exponential polynomials with the aid of some properties of completely monotonic functions and other known results, derive the logarithmic convexity and logarithmic concavity for multivariate exponential polynomials, and finally find an application of multivariate exponential polynomials to white noise distribution theory by confirming that multivariate exponential polynomials satisfy conditions for sequences required in white noise distribution theory.     

一些包含Chebyshev多项式和Stirling数的恒等式

利用初等方法研究Chebyshev多项式的性质,建立了广义第二类Chebyshev多项式的一个显明公式,并得到了一些包含第一类Chebyshev多项式,第一类Stirling数和Lucas数的恒等式.     

An explicit formula for generalized potential polynomials and its applications

We define the generalized potential polynomials associated to an independent variable, and prove an explicit formula involving the generalized potential polynomials and the exponential Bell polynomials. We use this formula to describe closed type formulas for the higher order Bernoulli, Eulerian, Euler, Genocchi, Apostol-Bernoulli, Apostol-Euler polynomials and the polynomials involving the Stirling numbers of the second kind. As further applications, we derive several known identities involving the Bernoulli numbers and polynomials and Euler polynomials, and new relations for the higher order tangent numbers, the higher order Bernoulli numbers of the second kind, the numbers , the higher order Bernoulli numbers and polynomials and the higher order Euler polynomials and their coefficients.     

A closed formula for the generating function of p-Bernoulli numbers

Abstract

In this paper, using geometric polynomials, we obtain a generating function of p-Bernoulli numbers in terms of harmonic numbers. As consequences of this generating function, we derive closed formulas for the finite summation of Bernoulli and harmonic numbers involving Stirling numbers of the second kind. We also give a relationship between the p-Bernoulli numbers and the generalized Bernoulli polynomials.     

Enumeration of labelled threshold graphs and a theorem of frobenius involving eulerian polynomials

We enumerate labelled threshold graphs by the number of vertices, the number of isolated vertices, and the number of distinct vertex-degrees and we give the exact asymptotics for the number of labelled threshold graphs withn vertices. We obtain the appropriate generating function and point out a combinatorial interpretation relating its coefficients to the Stirling numbers of the second kind. We use these results to derive a new proof of a theorem of Frobenius expressing the Eulerian polynomials in terms of the Stirling numbers.     

On multivariate logarithmic polynomials and their properties

In the paper, the author introduces a new notion “multivariate logarithmic polynomial”, establishes two recurrence relations, an explicit formula, and an identity for multivariate logarithmic polynomials by virtue of the Faà di Bruno formula and two identities for the Bell polynomials of the second kind in terms of the Stirling numbers of the first and second kinds, and constructs some determinantal inequalities, some product inequalities, and logarithmic convexity for multivariate logarithmic polynomials by virtue of some properties of completely monotonic functions.     

Degenerate Bernoulli polynomials, generalized factorial sums, and their applications

We prove a general symmetric identity involving the degenerate Bernoulli polynomials and sums of generalized falling factorials, which unifies several known identities for Bernoulli and degenerate Bernoulli numbers and polynomials. We use this identity to describe some combinatorial relations between these polynomials and generalized factorial sums. As further applications we derive several identities, recurrences, and congruences involving the Bernoulli numbers, degenerate Bernoulli numbers, generalized factorial sums, Stirling numbers of the first kind, Bernoulli numbers of higher order, and Bernoulli numbers of the second kind.     

Explicit Formulas Associated with Some Families of Generalized Bernoulli and Euler Polynomials

In this paper, we propose and derive several new explicit formulas of the generalized Bernoulli and Euler polynomials in terms of the generalized Stirling numbers of the second kind. A study of some families of the modified generalized Euler polynomials yields an interesting algorithm for calculating the generalized Euler polynomials.     

Reciprocity for multirestricted Stirling numbers

Multirestricted Stirling numbers of the second kind count the number of partitions of a given set into a given number of parts, each part being restricted to at most a fixed number of elements. Multirestricted numbers of the first kind are then defined as elements of the matrix inverse to the matrix of corresponding multirestricted numbers of the second kind. The anomalous sign behavior of these latter numbers makes them impervious to combinatorial analysis. In answer to a conjecture that has remained open for several years, we derive a reciprocity law for multirestricted Stirling numbers using algebraic techniques based on polynomial recursions. As corollaries, we obtain new recurrence relations for multirestricted numbers, and a new algebraic derivation of the reciprocity law for Stirling numbers.     

联系Bernoulli数和第二类Stirling数的一个恒等式

利用指数型生成函数建立起联系Bernoulli数和第二类Stirling数的一个有趣的恒等式．     

Computation methods for combinatorial sums and Euler‐type numbers related to new families of numbers

The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order. We can show that these numbers are related to the well‐known numbers and polynomials such as the Stirling numbers of the second kind and the central factorial numbers, the array polynomials, the rook numbers and polynomials, the Bernstein basis functions and others. In order to derive our new identities and relations for these numbers, we use a technique including the generating functions and functional equations. Finally, we give not only a computational algorithm for these numbers but also some numerical values of these numbers and the Euler numbers of negative order with tables. We also give some combinatorial interpretations of our new numbers. Copyright © 2016 John Wiley & Sons, Ltd.     

Srivastava–Pintér theorems for 2D‐Appell polynomials and their applications

Recently, Srivastava and Pintér proved addition theorems for the generalized Bernoulli and Euler polynomials. Luo and Srivastava obtained the anologous results for the generalized Apostol–Bernoulli polynomials and the generalized Apostol–Euler polynomials. Finally, Tremblay et al. gave analogues of the Srivastava–Pintér addition theorem for general family of Bernoulli polynomials. In this paper, we obtain Srivastava–Pintér type theorems for 2D‐Appell Polynomials. We also give the representation of 2D‐Appell Polynomials in terms of the Stirling numbers of the second kind and 1D‐Appell polynomials. Furthermore, we introduce the unified 2D‐Apostol polynomials. In particular, we obtain some relations between that family of polynomials and the generalized Hurwitz–Lerch zeta function as well as the Gauss hypergeometric function. Finally, we present some applications of Srivastava–Pintér type theorems for 2D‐Appell Polynomials. Copyright © 2013 John Wiley & Sons, Ltd.     

Explicit formulas and recurrence relations for higher order Eulerian polynomials

In the paper, the authors present explicit formulas, nonlinear ordinary differential equations, and recurrence relations for Eulerian polynomials, higher order Eulerian polynomials, and their generating functions in terms of the Stirling numbers of the second kind.     

Riordan阵的差分性质及其应用

本文考虑了Riordan阵的差分性质, 并给出一些涉及经典组合序列的差分恒等式, 包括广义Stirling数, 第一类和第二类Stirling数, 第一类和第二类B型Stirling数以及Gegenbauer-Humbert型多项式.     

Generalized Bessel numbers and some combinatorial settings

Stirling numbers and Bessel numbers have a long history, and both have been generalized in a variety of directions. Here, we present a second level generalization that has both as special cases. This generalization often preserves the inverse relation between the first and second kind, and has simple combinatorial interpretations. We also frame the discussion in terms of the exponential Riordan group. Then the inverse relation is just the group inverse, and factoring inside the group leads to many results connecting the various Stirling and Bessel numbers.     

On some expansions for the Euler Gamma function and the Riemann Zeta function

In the present paper we introduce some expansions which use the falling factorials for the Euler Gamma function and the Riemann Zeta function. In the proofs we use the Faá di Bruno formula, Bell polynomials, potential polynomials, Mittag-Leffler polynomials, derivative polynomials and special numbers (Eulerian numbers and Stirling numbers of both kinds). We investigate the rate of convergence of the series and give some numerical examples.     

Genocchi polynomials associated with the Umbral algebra

The aim of this paper is to study on the Genocchi polynomials of higher order on P, the algebra of polynomials in the single variable x over the field C of characteristic zero and P^∗, the vector spaces of all linear functional on P. By using the action of a linear functional L on a polynomial p(x) Sheffer sequences and Appell sequences, we obtain some fundamental properties of the Genocchi polynomials. Furthermore, we give relations between, the first and second kind Stirling numbers, Euler polynomials of higher order and Genocchi polynomials of higher order.     

Some inequalities for the Bell numbers

In this paper, we present derivatives of the generating functions for the Bell numbers by induction and by the Faà di Bruno formula, recover an explicit formula in terms of the Stirling numbers of the second kind, find the (logarithmically) absolute and complete monotonicity of the generating functions, and construct some inequalities for the Bell numbers. From these inequalities, we derive the logarithmic convexity of the sequence of the Bell numbers.     

Interpolation and Combinatorial Functions

Using some basic results about polynomial interpolation, divided differences, and Newton polynomial sequences we develop a theory of generalized binomial coefficients that permits the unified study of the usual binomial coefficients, the Stirling numbers of the second kind, the q-Gaussian coefficients, and other combinatorial functions. We obtain a large number of combinatorial identities as special cases of general formulas. For example, Leibniz's rule for divided differences becomes a Chu-Vandermonde convolution formula for each particular family of generalized binomial coefficients.     

Peters type polynomials and numbers and their generating functions: Approach with p‐adic integral method

The aim of this paper is to introduce and investigate some of the primary generalizations and unifications of the Peters polynomials and numbers by means of convenient generating functions and p‐adic integrals method. Various fundamental properties of these polynomials and numbers involving some explicit series and integral representations in terms of the generalized Stirling numbers, generalized harmonic sums, and some well‐known special numbers and polynomials are presented. By using p‐adic integrals, we construct generating functions for Peters type polynomials and numbers (Apostol‐type Peters numbers and polynomials). By using these functions with their partial derivative eqautions and functional equations, we derive many properties, relations, explicit formulas, and identities including the Apostol‐Bernoulli polynomials, the Apostol‐Euler polynomials, the Boole polynomials, the Bernoulli polynomials, and numbers of the second kind, generalized harmonic sums. A brief revealing and historical information for the Peters type polynomials are given. Some of the formulas given in this article are given critiques and comments between previously well‐known formulas. Finally, two open problems for interpolation functions for Apostol‐type Peters numbers and polynomials are revealed.