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.     

Generalized higher order Bernoulli number pairs and generalized Stirling number pairs

From a delta series f(t) and its compositional inverse g(t), Hsu defined the generalized Stirling number pair . In this paper, we further define from f(t) and g(t) the generalized higher order Bernoulli number pair . Making use of the Bell polynomials, the potential polynomials as well as the Lagrange inversion formula, we give some explicit expressions and recurrences of the generalized higher order Bernoulli numbers, present the relations between the generalized higher order Bernoulli numbers of both kinds and the corresponding generalized Stirling numbers of both kinds, and study the relations between any two generalized higher order Bernoulli numbers. Moreover, we apply the general results to some special number pairs and obtain series of combinatorial identities. It can be found that the introduction of generalized Bernoulli number pair and generalized Stirling number pair provides a unified approach to lots of sequences in mathematics, and as a consequence, many known results are special cases of ours.     

Arithmetical properties of hypergeometric Bernoulli numbers

In a recent paper, Byrnes et al. (2014) have developed some recurrence relations for the hypergeometric zeta functions. Moreover, the authors made two conjectures for arithmetical properties of the denominators of the reduced fraction of the hypergeometric Bernoulli numbers. In this paper, we prove these conjectures using some recurrence relations. Furthermore, we assert that the above properties hold for both Carlitz and Howard numbers.     

Sums of products of Cauchy numbers

In this paper, we consider a kind of sums involving Cauchy numbers, which have not been studied in the literature. By means of the method of coefficients, we give some properties of the sums. We further derive some recurrence relations and establish a series of identities involving the sums, Stirling numbers, generalized Bernoulli numbers, generalized Euler numbers, Lah numbers, and harmonic numbers. In particular, we generalize some relations between two kinds of Cauchy numbers and some identities for Cauchy numbers and Stirling numbers.     

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.     

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.     

Higher-order convolutions for Bernoulli and Euler polynomials

We prove convolution identities of arbitrary orders for Bernoulli and Euler polynomials, i.e., sums of products of a fixed but arbitrary number of these polynomials. They differ from the more usual convolutions found in the literature by not having multinomial coefficients as factors. This generalizes a special type of convolution identity for Bernoulli numbers which was first discovered by Yu. Matiyasevich.     

高阶Bernoulli多项式和高阶Euler多项式的关系

利用发生函数的方法，讨论了高阶Bernoulli数和高阶Euler数，高阶Bernoulli多项式和高阶Euler多项式之间的关系，得到了经典Bernoulli数和Euler数，经典Bernoulli多项式和Euler多项式之间的新型关系。     

高阶退化Bernoulli数和多项式

本文研究了高阶退化Berrioulli数和多项式的两个显明公式，得到了一个包含高阶Bemoulli数和Stirling数的恒等式，并推广了F．H．Howard，S．Shirai和K．I．Sato的结果。     

New convolutions for complete and elementary symmetric functions

In this paper, we give new relationships between complete and elementary symmetric functions. These results can be used to discover and prove some identities involving r-Whitney numbers, Jacobi–Stirling numbers, Bernoulli numbers and other numbers that are specializations of complete and elementary symmetric functions.     

随机变量序列的大数定律及常返性定理

对一类有界独立或相依的随机变量序列|ξn|,获得了它的伯努利大数定律、波雷尔强大数定律及常返性定理.作为应用,得出了Loève专著[1]中的推广的伯努利大数定律、常返性定理,改进了[1]中的推广的波雷尔强大数定律.     

广义Bernoulli数和广义高阶Bernoulli数

定义了广义Bernoulli数和广义高阶Bernoulli数，建立了它们的递推公式和有关性质，从而推广了Bernoulli数和高阶Bernoulli数。     

Padé approximation and Apostol-Bernoulli and Apostol-Euler polynomials

Using the Padé approximation of the exponential function, we obtain recurrence relations between Apostol-Bernoulli and between Apostol-Euler polynomials. As applications, we derive some new lacunary recurrence relations for Bernoulli and Euler polynomials with gap of length 4 and lacunary relations for Bernoulli and Euler numbers with gap of length 6.     

高阶Bernoulli数与Stirling数的几个恒等式

利用第一、二类高阶Bernoulli数和二类Stirling数S1(n,k),S2(n,k)的定义.研究了二类高阶Bernoulli数母函数的幂级数展开,揭示了二类高阶Bernoulli数之间以及与第一类Stirling数S1(n,k)、第二类Stirling数S2(n,k)之间的内在联系,得到了几个关于二类高阶Bernoulli数和第一类Stirling数S1(n,k)、第二类Stirling数S2(n,k)之间有趣的恒等式.     

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.     

Reciprocal relations of Bernoulli and Euler numbers/polynomials

By means of the symmetric summation theorem on polynomial differences due to Chu and Magli [Summation formulae on reciprocal sequences. European J Combin. 2007;28(3):921–930], we examine Bernoulli and Euler polynomials of higher order. Several reciprocal relations on Bernoulli and Euler numbers and polynomials are established, including some recent ones obtained by Agoh Shortened recurrence relations for generalized Bernoulli numbers and polynomials. J Number Theory. 2017;176:149–173.     

Congruences for degenerate number sequences

The degenerate Stirling numbers and degenerate Eulerian polynomials are intimately connected to the arithmetic of generalized factorials. In this article, we show that these numbers and similar sequences may in fact be expressed as p-adic integrals of generalized factorials. As an application of this identification we deduce systems of congruences which are analogues and generalizations of the Kummer congruences for the ordinary Bernoulli numbers.     

Zeta functions interpolating the convolution of the Bernoulli polynomials

We prove several relations on multiple Hurwitz–Riemann zeta functions. Using analytic continuation of these multiple Hurwitz–Riemann zeta functions, we quote at negative integers Euler's nonlinear relation for generalized Bernoulli polynomials and numbers. As an application, we give a general convolution identity for Bernoulli numbers.     

Symmetry p-adic invariant integral on ℤ p for Bernoulli and Euler polynomials

The main purpose of this paper is to investigate several further interesting properties of symmetry for the p-adic invariant integrals on ? _p . From these symmetry, we can derive many interesting recurrence identities for Bernoulli and Euler polynomials. Finally we introduce the new concept of symmetry of fermionic p-adic invariant integral on ? _p . By using this symmetry of fermionic p-adic invariant integral on ? _p , we will give some relations of symmetry between the power sum polynomials and Euler numbers. The relation between the q-Bernoulli polynomials and q-Dedekind type sums which discussed in Y. Simsek (q-Dedekind type sums related to q-zeta function and basic L-series, J. Math. Anal. Appl. 318 (2006), pp. 333–351) can be also derived by using the properties of symmetry of fermionic p-adic integral on ? _p .     

A 2-adic formula for Bernoulli numbers of the second kind and for the Nörlund numbers

We give a formula expressing Bernoulli numbers of the second kind as 2-adically convergent sums of traces of algebraic integers. We use this formula to prove and explain the formulas and conjectures of Adelberg concerning the initial 2-adic digits of these numbers. We also give analogous results for the Nörlund numbers.