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.     

Curious congruences related to the Bell polynomials

In this paper we give some congruences on the r-derangement polynomials (defined below), Lah polynomials and some versions of Bell numbers and polynomials.     

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.     

Symmetric identities on Bernoulli polynomials

In this paper, we obtain a generalization of an identity due to Carlitz on Bernoulli polynomials. Then we use this generalized formula to derive two symmetric identities which reduce to some known identities on Bernoulli polynomials and Bernoulli numbers, including the Miki identity.     

Identities for the Riemann zeta function

In this paper, we obtain several expansions for ζ(s) involving a sequence of polynomials in s, denoted by α _k (s). These polynomials can be regarded as a generalization of Stirling numbers of the first kind and our identities extend some series expansions for the zeta function that are known for integer values of s. The expansions also give a different approach to the analytic continuation of the Riemann zeta function.     

Arakawa-Kaneko L-functions and generalized poly-Bernoulli polynomials

We introduce and investigate generalized poly-Bernoulli numbers and polynomials. We state and prove several properties satisfied by these polynomials. The generalized poly-Bernoulli numbers are algebraic numbers. We introduce and study the Arakawa-Kaneko L-functions. The non-positive integer values of the complex variable s of these L-functions are expressed rationally in terms of generalized poly-Bernoulli numbers and polynomials. Furthermore, we prove difference and Raabe?s type formulae for these L-functions.     

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.     

Poly-p-Bernoulli polynomials and generalized Arakawa–Kaneko zeta function

In this paper, we first obtain several properties of poly-p-Bernoulli polynomials. In particular, we achieve some new results for poly-Bernoulli polynomials. We next define a generalization of the Arakawa–Kaneko zeta function associated with poly-p-Bernoulli polynomials, investigate some its particular values, and give asymptotic and series expansions.

     

Constructing function fields¶with many rational places via the Carlitz module

We consider certain decomposition fields in extensions of ?q (Z) by the Carlitz module and give formulas for their genera and numbers of rational places, suitable for automatic computations. By extensive calculations we found some function fields which have more rational places than the known examples of the respective genus. Received: 7 September 2001/ Revised version: 19 October 2001     

Some properties on the integral of the product of several Euler polynomials

Abstract

In this paper, we study the formula for a product of two Euler poly-nomials. From this study, we derive some formulae for the integral of the product of two or more Euler polynomials.     

Construction a new generating function of Bernstein type polynomials

Main purpose of this paper is to reconstruct generating function of the Bernstein type polynomials. Some properties of this generating functions are given. By applying this generating function, not only derivative of these polynomials but also recurrence relations of these polynomials are found. Interpolation function of these polynomials is also constructed by Mellin transformation. This function interpolates these polynomials at negative integers which are given explicitly. Moreover, relations between these polynomials, the Stirling numbers of the second kind and Bernoulli polynomials of higher order are given. Furthermore some remarks associated with the Bezier curves are given.     

Hyperharmonic series involving Hurwitz zeta function

We show that the sum of the series formed by the so-called hyperharmonic numbers can be expressed in terms of the Riemann zeta function. These results enable us to reformulate Euler's formula involving the Hurwitz zeta function. In additon, we improve Conway and Guy's formula for hyperharmonic numbers.     

On arithmetic numbers

An integer n is said to be arithmetic if the arithmetic mean of its divisors is an integer. In this paper, using properties of the factorization of values of cyclotomic polynomials, we characterize arithmetic numbers. As an application, we give an interesting characterization of Mersenne numbers.     

Matrices related to the Bell polynomials

In this paper, we study the matrices related to the partial exponential Bell polynomials and those related to the Bell polynomials with respect to Ω. As a result, the factorizations of these matrices are obtained, which give unified approaches to the factorizations of many lower triangular matrices. Moreover, some combinatorial identities are also derived from the corresponding matrix representations.     

Logarithmic behavior of some combinatorial sequences

Two general methods for establishing the logarithmic behavior of recursively defined sequences of real numbers are presented. One is the interlacing method, and the other one is based on calculus. Both methods are used to prove logarithmic behavior of some combinatorially relevant sequences, such as Motzkin and Schröder numbers, sequences of values of some classic orthogonal polynomials and many others. The calculus method extends also to numbers indexed by two or more parameters.     

The Möbius inversion formula for Fourier series applied to Bernoulli and Euler polynomials

Hurwitz found the Fourier expansion of the Bernoulli polynomials over a century ago. In general, Fourier analysis can be fruitfully employed to obtain properties of the Bernoulli polynomials and related functions in a simple manner. In addition, applying the technique of Möbius inversion from analytic number theory to Fourier expansions, we derive identities involving Bernoulli polynomials, Bernoulli numbers, and the Möbius function; this includes formulas for the Bernoulli polynomials at rational arguments. Finally, we show some asymptotic properties concerning the Bernoulli and Euler polynomials.     

An <Emphasis Type="Italic">ABC</Emphasis> inequality for Mahler’s measure

We prove an upper bound for the Mahler measure of the Wronskian of a collection of N linearly independent polynomials with complex coefficients. If the coefficients of the polynomials are algebraic numbers we obtain an inequality for the absolute Weil heights of the roots of the polynomials. This later inequality is analogous to the abc inequality for polynomials, and also has applications to Diophantine problems. Research supported in part by the National Science Foundation (DMS-06-03282) and the Erwin Schr?dinger Institute. Author’s address: Department of Mathematics, University of Texas, Austin, Texas 78712, USA     

Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind

Recently, the authors introduced some generalizations of the Apostol-Bernoulli polynomials and the Apostol-Euler polynomials (see [Q.-M. Luo, H.M. Srivastava, J. Math. Anal. Appl. 308 (2005) 290-302] and [Q.-M. Luo, Taiwanese J. Math. 10 (2006) 917-925]). The main object of this paper is to investigate an analogous generalization of the Genocchi polynomials of higher order, that is, the so-called Apostol-Genocchi polynomials of higher order. For these generalized Apostol-Genocchi polynomials, we establish several elementary properties, provide some explicit relationships with the Apostol-Bernoulli polynomials and the Apostol-Euler polynomials, and derive various explicit series representations in terms of the Gaussian hypergeometric function and the Hurwitz (or generalized) zeta function. We also deduce their special cases and applications which are shown here to lead to the corresponding results for the Genocchi and Euler polynomials of higher order. By introducing an analogue of the Stirling numbers of the second kind, that is, the so-called λ-Stirling numbers of the second kind, we derive some basic properties and formulas and consider some interesting applications to the family of the Apostol type polynomials. Furthermore, we also correct an error in a previous paper [Q.-M. Luo, H.M. Srivastava, Comput. Math. Appl. 51 (2006) 631-642] and pose two open problems on the subject of our investigation.     

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.     

A multiplication theorem for the Lerch zeta function and explicit representations of the Bernoulli and Euler polynomials

A multiplication theorem for the Lerch zeta function ?(s,a,ξ) is obtained, from which, when evaluating at s=−n for integers n?0, explicit representations for the Bernoulli and Euler polynomials are derived in terms of two arrays of polynomials related to the classical Stirling and Eulerian numbers. As consequences, explicit formulas for some special values of the Bernoulli and Euler polynomials are given.