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 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 New results for Srivastava’s λ-generalized Hurwitz-Lerch Zeta function

Filomat ◽  
2017 ◽  
Vol 31 (8) ◽  
pp. 2219-2229 ◽  
Author(s):  
Min-Jie Luo ◽  
R.K. Raina
Keyword(s):  
Integral Representation ◽  
Zeta Function ◽  
Series Representation ◽  
Slight Modification ◽  
Summation Formula ◽  
Integral Transforms ◽  
Lerch Zeta Function ◽  
The Relationship ◽  
Appl Math

In view of the relationship with the Kr?tzel function, we derive a new series representation for the ?-generalized Hurwitz-Lerch Zeta function introduced by H.M. Srivastava [Appl. Math. Inf. Sci. 8 (2014) 1485-1500] and determine the monotonicity of its coeficients. An integral representation of the Mathieu (a;?)-series is rederived by applying the Abel?s summation formula (which provides a slight modification of the result given by Pog?ny [Integral Transforms Spec. Funct. 16 (8) (2005) 685-689]) and this modified form of the result is then used to obtain a new integral representation for Srivastava?s ?-generalized Hurwitz-Lerch Zeta function. Finally, by making use of the various results presented in this paper, we establish two sets of two-sided inequalities for Srivastava?s ?-generalized Hurwitz-Lerch Zeta function.

scholarly journals Further Extension of the Generalized Hurwitz-Lerch Zeta Function of Two Variables

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 48
Author(s):  
Kottakkaran Sooppy Nisar
Keyword(s):  
Zeta Function ◽  
Hypergeometric Function ◽  
Summation Formula ◽  
Integral Representations ◽  
Generalized Hypergeometric Function ◽  
Special Cases ◽  
Lerch Zeta Function ◽  
Function Of Two Variables

The main aim of this paper is to provide a new generalization of Hurwitz-Lerch Zeta function of two variables. We also investigate several interesting properties such as integral representations, summation formula, and a connection with the generalized hypergeometric function. To strengthen the main results we also consider some important special cases.

scholarly journals A new derivation of $\zeta(-r)=-\frac{B_{r+1}}{r+1}$

2019 ◽  
Author(s):  
Sumit Kumar Jha
Keyword(s):  
Integral Representation ◽  
Zeta Function ◽  
Explicit Formula ◽  
Riemann Zeta Function ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Master Theorem

In this note, we give a new derivation for the fact that $\zeta(-r)=-\frac{B_{r+1}}{r+1}$ where $\zeta(s)$ represents the Riemann zeta function, and $B_{r}$ represents the Bernoulli numbers. Our proof uses the well-known explicit formula for the Bernoulli numbers in terms of the Stirling numbers of the second kind, and the Ramanujan's master theorem to obtain an integral representation for the Riemann zeta function.

scholarly journals On the Evaluation of Riemann Zeta Functions for Even Integers

2021 ◽  
Author(s):  
Ribhu Paul
Keyword(s):  
Fourier Series ◽  
Zeta Function ◽  
Closed Form ◽  
Series Expansion ◽  
Recursion Relation ◽  
Zeta Functions ◽  
Fourier Series Expansion ◽  
Recursive Method ◽  
Riemann Zeta

A recursive method for obtaining the Zeta function for even integers is obtained starting from the Fourier series expansion of the function f(x) = x. Repeating the method after term by term integration yields the final, simplified closed form that happens to be a recursion relation. Using the obtained recursion relation, one can successively evaluate the values of zeta functions for even integers.

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 An extension of the generalized Hurwitz-Lerch Zeta function of two variables

Filomat ◽  
2017 ◽  
Vol 31 (1) ◽  
pp. 91-96 ◽  
Author(s):  
Junesang Choi ◽  
Rakesh Parmar
Keyword(s):  
Zeta Function ◽  
Zeta Functions ◽  
Summation Formula ◽  
Integral Representations ◽  
Contour Integral ◽  
Functions Of Two Variables ◽  
Special Cases ◽  
Lerch Zeta Function ◽  
Function Of Two Variables

The main object of this paper is to introduce a new extension of the generalized Hurwitz-Lerch Zeta functions of two variables. We then systematically investigate such its several interesting properties and related formulas as (for example) various integral representations, which provide certain new and known extensions of earlier corresponding results, a summation formula and Mellin-Barnes type contour integral representations. We also consider some important special cases.

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