scholarly journals Integral Representations of q-analogues of the Hurwitz Zeta Function

2006 ◽  
Vol 149 (2) ◽  
pp. 141-154 ◽  
Author(s):  
Masato Wakayama ◽  
Yoshinori Yamasaki
Keyword(s):  
Zeta Function ◽  
Integral Representations ◽  
Hurwitz Zeta Function

scholarly journals Hurwitz Zeta Function of Two Variables and Associated Properties

2020 ◽  
pp. 297-315
Author(s):  
M. A. Pathan ◽  
Maged G. Bin-Saad ◽  
J. A. Younis
Keyword(s):  
Zeta Function ◽  
Integral Representations ◽  
Hurwitz Zeta Function ◽  
Summation Formulas ◽  
Lerch Zeta Function ◽  
Function Of Two Variables

The main objective of this work is to introduce a new generalization of Hurwitz-Lerch zeta function of two variables. Also, we investigate several interesting properties such as integral representations, operational connections and summation formulas.

Application of the Hurwitz Zeta Function to the Evaluation of Certain Integrals

1993 ◽  
Vol 36 (3) ◽  
pp. 373-384 ◽  
Author(s):  
Zhang Nan Yue ◽  
Kenneth S. Williams
Keyword(s):  
Zeta Function ◽  
Complex Plane ◽  
Integral Representations ◽  
Simple Pole ◽  
Hurwitz Zeta Function ◽  
Improper Integrals

AbstractThe Hurwitz zeta function ζ(s, a) is defined by the seriesfor 0 < a ≤ 1 and σ = Re(s) > 1, and can be continued analytically to the whole complex plane except for a simple pole at s = 1 with residue 1. The integral functions C(s, a) and S(s, a) are defined in terms of the Hurwitz zeta function as follows:Using integral representations of C(s, a) and S(s, a), we evaluate explicitly a class of improper integrals. For example if 0 < a < 1 we show that

scholarly journals Contributions to the theory of the Hurwitz zeta-function

2007 ◽  
Vol Volume 30 ◽  
Author(s):  
S Kanemitsu ◽  
Y Tanigawa ◽  
H Tsukada ◽  
M Yoshimoto
Keyword(s):  
Zeta Function ◽  
Closed Form ◽  
Historical Perspective ◽  
Integral Representations ◽  
Hurwitz Zeta Function ◽  
Psi Function ◽  
Principal Solution ◽  
International Audience ◽  
The Difference ◽  
Almost All

International audience We give various contributions to the theory of Hurwitz zeta-function. An elementary part is the argument relating to the partial sum of the defining Dirichlet series for it; how much can we retrieve the whole from the part. We also give the sixth proof of the far-reaching Ramanujan -- Yoshimoto formula, which is a closed form for the important sum $\sum^\infty_{m=2} \frac{\zeta(m,\alpha)}{m+\lambda} z^{m+\lambda}$. This proof, incorporating the structure of the Hurwitz zeta-function as the principal solution of the difference equation, seems one of the most natural ones. The formula may be applied to deduce almost all formulas in H.~M.~Srivastava and J.~Choi. The same is applied to obtain closed form for the integral of the Euler psi function and give Espisona-Moll results. %In this paper we shall give various contributions to the theory of the Hurwitz zeta-function. In \S1 we shall continue our previous study and give integral representations (for the derivatives as well) which give another basis of the theory of gamma and related functions. In \S2 we shall give the sixth proof of the Ramanujan formula with two examples which supersede those results presented in the book of Srivastava and Choi. In \S3 we shall give two more proofs of the closed formula for the integral of the psi-function, thus recovering the recent result of Episona and Moll. Finally, in \S4 we shall give another proof of the functional equation. Hereby we put all existing literature in the hierarchical and historical perspective.

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 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.

scholarly journals A (p,v)-Extension of Hurwitz-Lerch Zeta Function and its Properties

2018 ◽  
Author(s):  
Gauhar Rahman ◽  
KS Nisar ◽  
Shahid Mubeen
Keyword(s):  
Zeta Function ◽  
Beta Function ◽  
Integral Representations ◽  
Basic Properties ◽  
Mellin Transformation ◽  
Special Cases ◽  
Lerch Zeta Function ◽  
Derivative Formulas ◽  
Generating Relations

In this paper, we define a (p,v)-extension of Hurwitz-Lerch Zeta function by considering an extension of beta function defined by Parmar et al. [J. Classical Anal. 11 (2017) 81&ndash;106]. We obtain its basic properties which include integral representations, Mellin transformation, derivative formulas and certain generating relations. Also, we establish the special cases of the main results.

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