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 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 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 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 An extension ofq-zeta function

2004 ◽  
Vol 2004 (49) ◽  
pp. 2649-2651 ◽  
Author(s):  
T. Kim ◽  
L. C. Jang ◽  
S. H. Rim
Keyword(s):  
Integral Representation ◽  
Zeta Function ◽  
Hurwitz Zeta Function

We will define the extension ofq-Hurwitz zeta function due to Kim and Rim (2000) and study its properties. Finally, we lead to a useful new integral representation for theq-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.

scholarly journals Real zeros of Hurwitz–Lerch zeta and Hurwitz–Lerch type of Euler–Zagier double zeta functions

2015 ◽  
Vol 160 (1) ◽  
pp. 39-50 ◽  
Author(s):  
TAKASHI NAKAMURA
Keyword(s):  
Zeta Function ◽  
Zeta Functions ◽  
Hurwitz Zeta Function ◽  
Real Zeros ◽  
Double Zeta ◽  
Lerch Zeta Function

AbstractLet 0 < a ⩽ 1, s, z ∈ ${\mathbb{C}}$ and 0 < |z| ⩽ 1. Then the Hurwitz–Lerch zeta function is defined by Φ(s, a, z) ≔ ∑∞n = 0zn(n + a)− s when σ ≔ ℜ(s) > 1. In this paper, we show that the Hurwitz zeta function ζ(σ, a) ≔ Φ(σ, a, 1) does not vanish for all 0 < σ < 1 if and only if a ⩾ 1/2. Moreover, we prove that Φ(σ, a, z) ≠ 0 for all 0 < σ < 1 and 0 < a ⩽ 1 when z ≠ 1. Real zeros of Hurwitz–Lerch type of Euler–Zagier double zeta functions are studied as well.

scholarly journals A New Family of Zeta Type Functions Involving the Hurwitz Zeta Function and the Alternating Hurwitz Zeta Function

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 233
Author(s):  
Daeyeoul Kim ◽  
Yilmaz Simsek
Keyword(s):  
Zeta Function ◽  
Generating Function ◽  
Bernoulli Polynomials ◽  
Hurwitz Zeta Function ◽  
New Class ◽  
Mellin Transformation ◽  
New Family ◽  
Euler Numbers And Polynomials ◽  
Lerch Zeta Function ◽  
Combinatorial Sums

In this paper, we further study the generating function involving a variety of special numbers and ploynomials constructed by the second author. Applying the Mellin transformation to this generating function, we define a new class of zeta type functions, which is related to the interpolation functions of the Apostol–Bernoulli polynomials, the Bernoulli polynomials, and the Euler polynomials. This new class of zeta type functions is related to the Hurwitz zeta function, the alternating Hurwitz zeta function, and the Lerch zeta function. Furthermore, by using these functions, we derive some identities and combinatorial sums involving the Bernoulli numbers and polynomials and the Euler numbers and polynomials.

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