Multiple Hurwitz Zeta Function Research Articles

A symbolic approach to multiple Hurwitz zeta values at non-positive integers

Abstract In this article, we give another method to calculate the values of multiple Hurwitz zeta function at non-positive integers by means of Raabe’s formula and the Bernoulli numbers and we simplify these values by symbolic computation techniques.

Derivation of computational formulas for certain class of finite sums: Approach to generating functions arising from p$$ p $$‐adic integrals and special functions

The aim of this paper is to construct generating functions for certain families of special finite sums by using the Newton–Mercator series, hypergeometric functions, and ‐adic integral. By using these generating functions with their functional and partial derivative equations, many novel computational formulas involving the special finite sums of (inverse) binomial coefficients, the Bernoulli type polynomials and numbers, Euler polynomials and numbers, the Stirling numbers, the (alternating) harmonic numbers, the Leibnitz polynomials, and others are derived. We also develop a computation algorithm for these finite sums and provide some of their special values. By using these finite sums and combinatorial numbers, we find some formulas involving multiple alternating zeta functions, the Bernoulli polynomials of higher order and the Euler polynomials of higher order. We then obtain a decomposition from these formulas, which are related to the multiple Hurwitz zeta functions.

On multiple Hurwitz zeta function of Mordell–Tornheim type

We introduce a certain multiple Hurwitz zeta function as a generalization of the Mordell–Tornheim multiple zeta function, and study its analytic properties. In particular, we evaluate the values of the function and its first and second derivatives at non-positive integers.

Absolute series for higher Euler constants

We give families of series for generalized Euler constants of multiple Hurwitz zeta functions in terms of absolute zeta functions of algebraic groups, extending recent work of Kurokawa and Tanaka. Our series also converge p-adically to give p-adic limit formulas for higher p-adic Euler constants.

Limit formulas for multiple Hurwitz zeta functions

We prove the limit formula for the multiple Hurwitz zeta function using absolute zeta functions. These formulas contain Euler's formula for the Euler constant proved in 1776.

On the sum relation of multiple Hurwitz zeta functions

In this paper we shall define the special-valued multiple Hurwitz zeta functions, namely the multiple t-values t(α) and define similarly the multiple star t-values as t⋆(α). Then we consider the sum of all such multiple (star) t-values of fixed depth and weight with even argument and prove that such a sum can be evaluated when the evaluations of t({2m}n) and t*({2m}n) are clear. We give the evaluations of them in terms of the classical Euler numbers through their generating functions.

Multiple Lerch Zeta Functions and an Idea of Ramanujan

In this article, we derive meromorphic continuation of multiple Lerch zeta functions by generalizing an elegant identity of Ramanujan. Further, we describe the set of all possible singularities of these functions. Finally, for the multiple Hurwitz zeta functions, we list the exact set of singularities.

Analytic continuation of multiple Hurwitz zeta functions

We obtain the analytic continuation of multiple Hurwitz zeta functions by using a simple and elementary translation formula. We also locate the polar hyperplanes for these functions and express the residues, along these hyperplanes, as coefficients of certain infinite matrices.

Q-Hypergeometric representations of the multiple Hurwitz Zeta function

The basic hypergeometric series started essentially by Euler back in (1748) that emphasis on generating functions of partitions. Later, Gauss (1813) and Cauchy (1852) found several transformations and summations formulas related to basic hypergeometric series. In this paper, the main goal is to introduce some new representations for the q-analogue of the multiple Hurwitz Zeta function are derived.

A Combinatorial Identity of Multiple Zeta Values with Even Arguments

Let $\zeta(s_1,s_2,\cdots,s_k;\alpha)$ be the multiple Hurwitz zeta function. Given two positive integers $k$ and $n$ with $k\leq n$, let $E(2n, k;\alpha)$ be the sum of all multiple zeta values with even arguments whose weight is $2n$ and whose depth is $k$. In this note we present some generating series for the numbers $E(2n,k;\alpha)$.

Multiple zeta functions and double wrapping in planar [formula omitted] SYM

Using the FiNLIE solution of the AdS/CFT Y-system, we compute the anomalous dimension of the Konishi operator in planar N=4 SYM up to eight loops, i.e. up to the leading double wrapping order. At this order a non-reducible Euler–Zagier sum, ζ1,2,8, appears for the first time. We find that at all orders in perturbation, every spectral-dependent quantity of the Y-system is expressed through multiple Hurwitz zeta functions, hence we provide a Mathematica package to manipulate these functions, including the particular case of Euler–Zagier sums. Furthermore, we conjecture that only Euler–Zagier sums can appear in the answer for the anomalous dimension at any order in perturbation theory.We also resum the leading transcendentality terms of the anomalous dimension at all orders, obtaining a simple result in terms of Bessel functions. Finally, we demonstrate that exact Bethe equations should be related to an absence of poles condition that becomes especially non-trivial at double wrapping.

The Multiple Hurwitz Zeta Function and the Multiple Hurwitz-Euler Eta Function

Almost eleven decades ago, Barnes introduced and made a systematic investigation on the multiple Gamma functions $\Gamma_n$. In about the middle of 1980s, these multiple Gamma functions were revived in the study of the determinants of Laplacians on the $n$-dimensional unit sphere ${\bf S}^n$ by using the multiple Hurwitz zeta functions $\zeta_n(s,a)$. In this paper, we first aim at presenting a generalized Hurwitz formula for $\zeta_n(s,a)$ together with its various special cases. Secondly, we give analytic continuations of multiple Hurwitz-Euler eta function $\eta_n(s,a)$ in two different ways. As a by-product of our second investigation, a relationship between $\eta_n(-\ell,a)$ $(\ell \in \mathbb{N}_0)$ and the generalized Euler polynomials $E_\ell^{(n)}(n-a)$ is also presented.

A note on sums of products of Bernoulli numbers

In this work we obtain a new approach to closed expressions for sums of products of Bernoulli numbers by using the relation of values at non-positive integers of the important representation of the multiple Hurwitz zeta function in terms of the Hurwitz zeta function.

ASYMPTOTIC EXPANSIONS FOR PERIOD DEFORMATIONS OF MULTIPLE HURWITZ ZETA FUNCTIONS AND THEIR ASSOCIATED FUNCTIONS

We study the asymptotic behavior of multiple Hurwitz zeta functions and generalized multiple gamma and sine functions as some period parameters tend to 0. We obtain the perfect asymptotic expansions and get some applications in refining a previous paper by the author.

On theq-Extension of Apostol-Euler Numbers and Polynomials

Recently, Choi et al. (2008) have studied theq-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of ordernand multiple Hurwitz zeta function. In this paper, we define Apostol's typeq-Euler numbersEn,q,ξandq-Euler polynomialsEn,q,ξ(x). We obtain the generating functions ofEn,q,ξandEn,q,ξ(x), respectively. We also have the distribution relation for Apostol's typeq-Euler polynomials. Finally, we obtainq-zeta function associated with Apostol's typeq-Euler numbers and Hurwitz's typeq-zeta function associated with Apostol's typeq-Euler polynomials for negative integers.

Some q-extensions of the Apostol–Bernoulli and the Apostol–Euler polynomials of order n, and the multiple Hurwitz zeta function

In this paper, we first investigate several further interesting properties of the multiple Hurwitz–Lerch Zeta function Φ n ( z, s, a) which was introduced recently by Choi et al. [J. Choi, D.S. Jang, H.M. Srivastava, A generalization of the Hurwitz–Lerch Zeta function, Integral Transform. Spec. Funct., 19 (2008)]. We then introduce and investigate some q-extensions of the multiple Hurwitz–Lerch Zeta function Φ n ( z, s, a), the Apostol–Bernoulli polynomials B k ( n ) ( x ; λ ) of order n, and the Apostol–Euler polynomials E k ( n ) ( x ; λ ) of order n. Relevant connections of the results presented here with those obtained in earlier works are also indicated precisely.

PERIOD DEFORMATIONS OF MULTIPLE HURWITZ ZETA FUNCTIONS

We study continuous period deformations of the multiple Hurwitz zeta functions and their derivatives. Moreover we investigate period deformations of the generalized multiple gamma and sine functions and give applications.

The Multiple Hurwitz Zeta Function and a Generalization of Lerch's Formula

We investigate the multiple Hurwitz zeta function $\zeta_n(s_1, \ldots, s_n;a)$, in particular those values at non-positive integers. Then, as an application, we give a generalization of Lerch's formula.

Absolute tensor products, Kummer's formula and functional equations for multiple Hurwitz zeta functions

Absolute tensor products, Kummer's formula and functional equations for multiple Hurwitz zeta functions

Zeta functions and normalized multiple sine functions

By using normalized multiple sine functions we show expressions for special values of zeta functions and L-functions containing ζ(3), ζ(5), etc. Our result reveals the importance of division values of normalized multiple sine functions. Properties of multiple Hurwitz zeta functions are crucial for the proof.