scholarly journals Series Representations at Special Values of Generalized Hurwitz-Lerch Zeta Function

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
S. Gaboury ◽  
A. Bayad
Keyword(s):  
Zeta Function ◽  
Higher Order ◽  
Euler Polynomials ◽  
Expansion Formula ◽  
Special Values ◽  
Lerch Zeta Function ◽  
Series Representations ◽  
Rational Arguments

By making use of some explicit relationships between the Apostol-Bernoulli, Apostol-Euler, Apostol-Genocchi, and Apostol-Frobenius-Euler polynomials of higher order and the generalized Hurwitz-Lerch zeta function as well as a new expansion formula for the generalized Hurwitz-Lerch zeta function obtained recently by Gaboury and Bayad , in this paper we present some series representations for these polynomials at rational arguments. These results provide extensions of those obtained by Apostol (1951) and by Srivastava (2000).

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 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.

SPECIAL VALUES OF SOME EXTENSIONS OF ZETA FUNCTION VIA BETA DISTRIBUTIONS

2012 ◽  
Vol 15 (02) ◽  
pp. 1250008
Author(s):  
NORIYOSHI SAKUMA ◽  
RYOICHI SUZUKI
Keyword(s):  
Zeta Function ◽  
Poisson Distribution ◽  
Special Values ◽  
Beta Distributions ◽  
Lerch Zeta Function ◽  
Basel Problem

In this paper, we show the Basel problem via the beta distributions, which include the free Poisson distribution and positive arcsine law. This is a generalization of Ref. 2 by Fujita. We also obtain special values of an extension of generalized Hurwitz–Lerch zeta function, which was introduced by Gang, Jain and Kalla.

scholarly journals Integral Representation and Explicit Formula at Rational Arguments for Apostol–Tangent Polynomials

Symmetry ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 35
Author(s):  
Cristina B. Corcino ◽  
Roberto B. Corcino ◽  
Baby Ann A. Damgo ◽  
Joy Ann A. Cañete
Keyword(s):  
Fourier Series ◽  
Integral Representation ◽  
Zeta Function ◽  
Explicit Formula ◽  
Summation Formula ◽  
Fourier Series Expansion ◽  
Residue Theorem ◽  
Lerch Zeta Function ◽  
Complex Integral ◽  
Rational Arguments

The Fourier series expansion of Apostol–tangent polynomials is derived using the Cauchy residue theorem and a complex integral over a contour. This Fourier series and the Hurwitz–Lerch zeta function are utilized to obtain the explicit formula at rational arguments of these polynomials. Using the Lipschitz summation formula, an integral representation of Apostol–tangent polynomials is also obtained.

scholarly journals NOTE ON q-DEDEKIND-TYPE SUMS RELATED TO q-EULER POLYNOMIALS

2011 ◽  
Vol 54 (1) ◽  
pp. 121-125 ◽  
Author(s):  
TAEKYUN KIM
Keyword(s):  
Continuous Function ◽  
Zeta Function ◽  
Higher Order ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Euler Numbers And Polynomials ◽  
Euler Polynomials And Numbers

AbstractRecently, q-Dedekind-type sums related to q-zeta function and basic L-series are studied by Simsek in [13] (Y. Simsek, q-Dedekind type sums related to q-zeta function and basic L-series, J. Math. Anal. Appl. 318 (2006), 333–351) and Dedekind-type sums related to Euler numbers and polynomials are introduced in the previous paper [11] (T. Kim, Note on Dedekind type DC sums, Adv. Stud. Contem. Math. 18 (2009), 249–260). It is the purpose of this paper to construct a p-adic continuous function for an odd prime to contain a p-adic q-analogue of the higher order Dedekind the type sums related to q-Euler polynomials and numbers by using an invariant p-adic q-integrals.

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