New summation relations for the Stieltjes constants

2006 ◽  
Vol 462 (2073) ◽  
pp. 2563-2573 ◽  
Author(s):  
Mark W Coffey
Keyword(s):  
Zeta Function ◽  
Hypergeometric Function ◽  
Laurent Expansion ◽  
Hurwitz Zeta Function ◽  
Stieltjes Constants ◽  
Positive Integers

The Stieltjes constants have been of interest for over a century, yet their detailed behaviour remains under investigation. These constants appear in the Laurent expansion of the Hurwitz zeta function about . We obtain novel single and double summatory relations for , including single summation relations for and , where a and b are real and p and q are positive integers. In addition, we obtain new integration formulae for the Hurwitz zeta function and a new expression for the Stieltjes constants . Portions of the presentation show an intertwining of the theory of the hypergeometric function with that of the Hurwitz zeta function.

On multiple Hurwitz zeta function of Mordell–Tornheim type

2021 ◽  
pp. 1-34
Author(s):  
Kazuhiro Onodera
Keyword(s):  
Zeta Function ◽  
Hurwitz Zeta Function ◽  
Multiple Zeta Function ◽  
Multiple Zeta ◽  
Second Derivatives ◽  
Positive Integers

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.

scholarly journals A procedure for generating infinite series identities

2004 ◽  
Vol 2004 (67) ◽  
pp. 3653-3662
Author(s):  
Anthony A. Ruffa
Keyword(s):  
Zeta Function ◽  
Hypergeometric Function ◽  
Infinite Series ◽  
Special Functions ◽  
Hurwitz Zeta Function ◽  
Gauss Hypergeometric Function ◽  
Polygamma Function ◽  
Polygamma Functions ◽  
Infinite Sums ◽  
Lerch Transcendent

A procedure for generating infinite series identities makes use of the generalized method of exhaustion by analytically evaluating the inner series of the resulting double summation. Identities are generated involving both elementary and special functions. Infinite sums of special functions include those of the gamma and polygamma functions, the Hurwitz Zeta function, the polygamma function, the Gauss hypergeometric function, and the Lerch transcendent. The procedure can be automated withMathematica(or equivalent software).

scholarly journals A Combinatorial Identity of Multiple Zeta Values with Even Arguments

10.37236/3923 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Shifeng Ding ◽  
Lihua Feng ◽  
Weijun Liu
Keyword(s):  
Zeta Function ◽  
Hurwitz Zeta Function ◽  
Combinatorial Identity ◽  
Multiple Zeta Values ◽  
Multiple Zeta ◽  
Generating Series ◽  
Zeta Values ◽  
Positive Integers

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 $E(2n,k;\alpha)$.

scholarly journals A note on a certain class of functions related to Hurwitz zeta function and Lambert transform

2000 ◽  
Vol 31 (1) ◽  
pp. 49-56
Author(s):  
R. K. Raina ◽  
T. S. Nahar
Keyword(s):  
Zeta Function ◽  
Hurwitz Zeta Function ◽  
Multiple Series ◽  
Generating Relations ◽  
Class Of Functions

In this paper we obtain multiple-series generating relations involving a class of function $ \theta_{(p_n)}^{(\mu_n)}(s,a;x_1,\ldots,x_n)$ which are connected to the Hurwitz zeta function. Also, a new generalization of Lambert transform is introduced, and its relationship with the above class of functions further depicted.

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