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

2008 ◽  
Vol 199 (2) ◽  
pp. 723-737 ◽  
Author(s):  
Junesang Choi ◽  
P.J. Anderson ◽  
H.M. Srivastava
Keyword(s):  
Zeta Function ◽  
Hurwitz Zeta Function ◽  
Euler Polynomials

scholarly journals Fourier expansion and integral representation generalized Apostol-type Frobenius–Euler polynomials

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Alejandro Urieles ◽  
William Ramírez ◽  
María José Ortega ◽  
Daniel Bedoya
Keyword(s):  
Fourier Series ◽  
Integral Representation ◽  
Zeta Function ◽  
Fourier Expansion ◽  
Series Representation ◽  
Hurwitz Zeta Function ◽  
Euler Polynomials ◽  
Genocchi Polynomials ◽  
Lerch Zeta Function ◽  
Rational Arguments

Abstract The main purpose of this paper is to investigate the Fourier series representation of the generalized Apostol-type Frobenius–Euler polynomials, and using the above-mentioned series we find its integral representation. At the same time applying the Fourier series representation of the Apostol Frobenius–Genocchi and Apostol Genocchi polynomials, we obtain its integral representation. Furthermore, using the Hurwitz–Lerch zeta function we introduce the formula in rational arguments of the generalized Apostol-type Frobenius–Euler polynomials in terms of the Hurwitz zeta function. Finally, we show the representation of rational arguments of the Apostol Frobenius Euler polynomials and the Apostol Frobenius–Genocchi polynomials.

scholarly journals Some formulas for Apostol-Euler polynomials associated with Hurwitz zeta function at rational arguments

2009 ◽  
Vol 3 (2) ◽  
pp. 336-346 ◽  
Author(s):  
Qiu-Ming Luo
Keyword(s):  
Zeta Function ◽  
Higher Order ◽  
Hurwitz Zeta Function ◽  
Euler Polynomials ◽  
Special Cases ◽  
Lerch Zeta Function ◽  
Series Representations ◽  
Rational Arguments

We give some explicit relationships between the Apostol-Euler polynomials and generalized Hurwitz-Lerch Zeta function and obtain some series representations of the Apostol-Euler polynomials of higher order in terms of the generalized Hurwitz-Lerch Zeta function. Several interesting special cases are also shown.

scholarly journals On theq-Extension of Apostol-Euler Numbers and Polynomials

2008 ◽  
Vol 2008 ◽  
pp. 1-10 ◽  
Author(s):  
Young-Hee Kim ◽  
Wonjoo Kim ◽  
Lee-Chae Jang
Keyword(s):  
Zeta Function ◽  
Generating Functions ◽  
Hurwitz Zeta Function ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
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.

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 Asymptotics to all orders of the Hurwitz zeta function

2018 ◽  
Vol 465 (1) ◽  
pp. 423-458 ◽  
Author(s):  
Arran Fernandez ◽  
Athanassios S. Fokas
Keyword(s):  
Zeta Function ◽  
Hurwitz Zeta Function

scholarly journals Relations among the Riemann Zeta and Hurwitz Zeta Functions, as Well as Their Products

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 754 ◽  
Author(s):  
A. C. L. Ashton ◽  
A. S. Fokas
Keyword(s):  
Zeta Function ◽  
Zeta Functions ◽  
Formal Proof ◽  
Hurwitz Zeta Function ◽  
Mean Square ◽  
Novel Approach ◽  
Starting Point ◽  
Lindelöf Hypothesis ◽  
Riemann Zeta ◽  
The Mean

In this paper, several relations are obtained among the Riemann zeta and Hurwitz zeta functions, as well as their products. A particular case of these relations give rise to a simple re-derivation of the important results of Katsurada and Matsumoto on the mean square of the Hurwitz zeta function. Also, a relation derived here provides the starting point of a novel approach which, in a series of companion papers, yields a formal proof of the Lindelöf hypothesis. Some of the above relations motivate the need for analysing the large α behaviour of the modified Hurwitz zeta function ζ 1 ( s , α ) , s ∈ C , α ∈ ( 0 , ∞ ) , which is also presented here.

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