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.     

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.     

Rational series for multiple zeta and log gamma functions

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We give series expansions for the Barnes multiple zeta functions in terms of rational functions whose numerators are complex-order Bernoulli polynomials, and whose denominators are linear. We also derive corresponding rational expansions for Dirichlet L-functions and multiple log gamma functions in terms of higher order Bernoulli polynomials. These expansions naturally express many of the well-known properties of these functions. As corollaries many special values of these transcendental functions are expressed as series of higher order Bernoulli numbers.

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The Multiple Gamma Function and Its Application to Computation of Series

The multiple gamma function Γ_n, defined by a recurrence-functional equation as a generalization of the Euler gamma function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Katz and Sarnak, interest in the multiple gamma function has been revived. This paper discusses some theoretical aspects of the Γ_n function and their applications to summation of series and infinite products.This work was supported by NFS grant CCR-0204003.2000 Mathematics Subject Classification: Primary—33E20, 33F99, 11M35, 11B73     

Another discrete Fourier transform pairs associated with the Lipschitz-Lerch zeta function

It is demonstrated that the alternating Lipschitz-Lerch zeta function and the alternating Hurwitz zeta function constitute a discrete Fourier transform pair. This discrete transform pair makes it possible to deduce, as special cases and consequences, many (mainly new) transformation relations involving the values at rational arguments of alternating variants of various zeta functions, such as the Lerch and Hurwitz zeta functions and Legendre chi function.     

利用渐近展开法求一类多重级数的闭形式

对于慢收敛多重级数Ik=∑ from (n1,n2,…,nk=1) to ∞((-)1~nlnn/n|n=n1+n2+…+nk,利用渐近展开方法给出闭形式.     

On asymptotics of the q-exponential and q-gamma functions

In this work we present a derivation for the complete asymptotic expansions of Euler?s q-exponential function and Jackson?s q-gamma function via Mellin transform. These formulas are valid everywhere, uniformly on any compact subset of the complex plane.     

Analysis of the Bernstein basis functions: an approach to combinatorial sums involving binomial coefficients and Catalan numbers

We give some alternative forms of the generating functions for the Bernstein basis functions. Using these forms,we derive a collection of functional equations for the generating functions. By applying these equations, we prove some identities for the Bernstein basis functions. Integrating these identities, we derive a variety of identities and formulas, some old and some new, for combinatorial sums involving binomial coefficients, Pascal's rule, Vandermonde's type of convolution, the Bernoulli polynomials, and the Catalan numbers. Copyright © 2014 John Wiley & Sons, Ltd.     

Notes on generalization of the Bernoulli type polynomials

Recently, Srivastava et al. introduced a new generalization of the Bernoulli, Euler and Genocchi polynomials (see [H.M. Srivastava, M. Garg, S. Choudhary, Russian J. Math. Phys. 17 (2010) 251-261] and [H.M. Srivastava, M. Garg, S. Choudhary, Taiwanese J. Math. 15 (2011) 283-305]). They established several interesting properties of these general polynomials, the generalized Hurwitz-Lerch zeta functions and also in series involving the familiar Gaussian hypergeometric function. By the same motivation of Srivastava’s et al. [11] and [12], we introduce and derive multiplication formula and some identities related to the generalized Bernoulli type polynomials of higher order associated with positive real parameters a, b and c. We also establish multiple alternating sums in terms of these polynomials. Moreover, by differentiating the generating function of these polynomials, we give a interpolation function of these polynomials.     

Some Expansions in Basic Fourier Series and Related Topics

We consider explicit expansions of some elementary and q-functions in basic Fourier series introduced recently by Bustoz and Suslov. Natural q-extensions of the Bernoulli and Euler polynomials, numbers, and the Riemann zeta function are discussed as a by-product.     

Computation of the Mordell-Tornheim zeta values

In this paper the authors present several algorithmic formulas which are potentially useful in computing the following Mordell-Tornheim zeta values:

for the special cases

   and

Some interesting (known or new) consequences and illustrative examples are also considered.

     

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.     

Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials

The main object of this paper is to give analogous definitions of Apostol type (see [T.M. Apostol, On the Lerch Zeta function, Pacific J. Math. 1 (1951) 161-167] and [H.M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambridge Philos. Soc. 129 (2000) 77-84]) for the so-called Apostol-Bernoulli numbers and polynomials of higher order. We establish their elementary properties, derive several explicit representations for them in terms of the Gaussian hypergeometric function and the Hurwitz (or generalized) Zeta function, and deduce their special cases and applications which are shown here to lead to the corresponding results for the classical Bernoulli numbers and polynomials of higher order.     

Carlitz’s q-Bernoulli and q-Euler numbers and polynomials and a class of generalized q-Hurwitz zeta functions

In this paper, we systematically recover the identities for the q-eta numbers η_k and the q-eta polynomials η_k(x), presented by Carlitz [L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. 15 (1948) 987–1000], which we define here via generating series rather than via the difference equations of Carlitz. Following a method developed by Kaneko et al. [M. Kaneko, N. Kurokawa, M. Wakayama, A variation of Euler’s approach to the Riemann zeta function, Kyushu J. Math. 57 (2003) 175–192] for a canonical q-extension of the Riemann zeta function, we investigate a similarly constructed q-extension of the Hurwitz zeta function. The details of this investigation disclose some interesting connections among q-eta polynomials, Carlitz’s q-Bernoulli polynomials -polynomials, and the q-Bernoulli polynomials that emerge from the q-extension of the Hurwitz zeta function discussed here.     

A family of p-adic twisted interpolation functions associated with the modified Bernoulli numbers

The main purpose of this paper is to construct a family of modified p-adic twisted functions, which interpolate the modified twisted q-Bernoulli polynomials and the generalized twisted q-Bernoulli numbers at negative integers. We also give some applications and examples related to these functions and numbers.     

Series representations for some mathematical constants

The authors apply a classical series identity involving the psi (or digamma) function with a view to deriving series representations for a number of known mathematical constants. Several closely-related consequences and results are also considered.     

The sharp Jackson inequality in the spaceL p on the sphere

The sharp Jackson inequality in the spaceL _p, 1≤p<2, on the unit Euclidean sphereS ⁿ⁻¹ ,n≥3, is proved. Forn=2, it was established by N. I. Chernykh. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 50–62, July, 1999.     

Probabilistic properties of number fields

In general a bound on number theoretic invariants under the Generalized Riemann Hypothesis (GRH) for the Dedekind zeta function of a number field K   is much stronger than an unconditional one. In this article, we consider three invariants; the residue of _ζK(s)${\zeta }_K(s)$ at s=1$s=1$, the logarithmic derivative of Artin L-function attached to K   at s=1$s=1$, and the smallest prime which does not split completely in K. We obtain bounds on them just as good as the bounds under GRH except for a density zero set of number fields.