scholarly journals On the Barnes' Type Related to Multiple Genocchi Polynomials on

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
J. Y. Kang ◽  
H. Y. Lee ◽  
N. S. Jung ◽  
C. S. Ryoo
Keyword(s):  
Zeta Function ◽  
Genocchi Numbers ◽  
Genocchi Polynomials ◽  
Invariant Integral

Using fermionic -adic invariant integral on , we construct the Barnes' type multiple Genocchi numbers and polynomials. From those numbers and polynomials, we derive the twisted Barnes' type multiple Genocchi numbers and polynomials. Moreover, we will find the Barnes' type multiple Genocchi zeta function.

scholarly journals On Genocchi Numbers and Polynomials

2008 ◽  
Vol 2008 ◽  
pp. 1-7 ◽  
Author(s):  
Seog-Hoon Rim ◽  
Kyoung Ho Park ◽  
Eun Jung Moon
Keyword(s):  
Zeta Function ◽  
Genocchi Numbers ◽  
Genocchi Polynomials

The main purpose of this paper is to study the distribution of Genocchi polynomials. Finally, we construct the Genocchi zeta function which interpolates Genocchi polynomials at negative integers.

scholarly journals Some Properties of Multiple Generalizedq-Genocchi Polynomials with Weight and Weak Weight

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
J. Y. Kang
Keyword(s):  
Zeta Function ◽  
Genocchi Numbers ◽  
Genocchi Polynomials ◽  
New Type

The present paper deals with the variousq-Genocchi numbers and polynomials. We define a new type of multiple generalizedq-Genocchi numbers and polynomials with weightαand weak weightβby applying the method ofp-adicq-integral. We will find a link between their numbers and polynomials with weightαand weak weightβ. Also we will obtain the interesting properties of their numbers and polynomials with weightαand weak weightβ. Moreover, we construct a Hurwitz-type zeta function which interpolates multiple generalizedq-Genocchi polynomials with weightαand weak weightβand find some combinatorial relations.

scholarly journals q-Genocchi Numbers and Polynomials Associated with Fermionicp-Adic Invariant Integrals onℤp

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
Leechae Jang ◽  
Taekyun Kim
Keyword(s):  
Higher Order ◽  
Genocchi Numbers ◽  
Genocchi Polynomials ◽  
Invariant Integrals ◽  
Invariant Integral

The main purpose of this paper is to present a systemic study of some families of multiple Genocchi numbers and polynomials. In particular, by using the fermionicp-adic invariant integral onℤp, we constructp-adic Genocchi numbers and polynomials of higher order. Finally, we derive the following interesting formula:Gn+k,q(k)(x)=2kk!(n+kk)∑l=0∞∑d0+d1+⋯+dk=k−1,di∈ℕ(−1)l(l+x)n, whereGn+k,q(k)(x)are theq-Genocchi polynomials of orderk.

scholarly journals A New Class of Laguerre-based Generalized Apostol Polynomials

2016 ◽  
Vol 57 (1) ◽  
pp. 67-89 ◽  
Author(s):  
N.U. Khan ◽  
T. Usman
Keyword(s):  
Generating Functions ◽  
General Symmetry ◽  
Genocchi Numbers ◽  
New Class ◽  
Genocchi Polynomials ◽  
Summation Formulae

Abstract In this paper, we introduce a unified family of Laguerre-based Apostol Bernoulli, Euler and Genocchi polynomials and derive some implicit summation formulae and general symmetry identities arising from different analytical means and applying generating functions. The result extend some known summations and identities of generalized Bernoulli, Euler and Genocchi numbers and 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 A note on the generalized q-Genocchi measures with weight

2011 ◽  
Vol 31 (1) ◽  
pp. 17 ◽  
Author(s):  
Hassan Jolany ◽  
Serkan Araci ◽  
Mehmet Acikgoz ◽  
Jong-Jin Seo
Keyword(s):  
Integral Representation ◽  
Generating Function ◽  
Genocchi Numbers ◽  
Genocchi Polynomials ◽  
Insight Into

In this paper we investigate special generalized q-Genocchi measures. We introduce q-Genocchi measures with weight alpha. The present paper deals with q-extension of Genocchi measure. Some earlier results of T. Kim in terms of q-Genocchi polynomials can be deduced. We apply the method of generating function, which are exploited to derive further classes of q-Genocchi polynomials and develop q-Genocchi measures. To be more precise, we present the integral representation of p-adic q-Genocchi measure with weight alpha which yields a deeper insight into the effectiveness of this type of generalizations. Generalized q-Genocchi numbers with weight alpha possess a number of interesting properties which we state in this paper.

scholarly journals On Multi Poly-Genocchi Polynomials with Parameters a, b and c

2020 ◽  
Vol 13 (3) ◽  
pp. 444-458
Author(s):  
Roberto Bagsarsa Corcino ◽  
Mark Laurente ◽  
Mary Ann Ritzell Vega
Keyword(s):  
Recurrence Relations ◽  
Euler Polynomials ◽  
Explicit Formulas ◽  
Multiple Parameters ◽  
Genocchi Numbers ◽  
Genocchi Polynomials

Most identities of Genocchi numbers and polynomials are related to the well-knownBenoulli and Euler polynomials. In this paper, multi poly-Genocchi polynomials withparameters a, b and c are dened by means of multiple parameters polylogarithm. Several properties of these polynomials are established including some recurrence relations and explicit formulas.

scholarly journals A Family of Generalized Legendre-Based Apostol-Type Polynomials

Axioms ◽  
2022 ◽  
Vol 11 (1) ◽  
pp. 29
Author(s):  
Talha Usman ◽  
Nabiullah Khan ◽  
Mohd Aman ◽  
Junesang Choi
Keyword(s):  
Integral Formula ◽  
Umbral Calculus ◽  
Maclaurin Series ◽  
General Symmetry ◽  
Genocchi Numbers ◽  
Genocchi Polynomials ◽  
Research Fields ◽  
Summation Formulas ◽  
Potential Applications ◽  
Addition Formulas

Numerous polynomials, their extensions, and variations have been thoroughly explored, owing to their potential applications in a wide variety of research fields. The purpose of this work is to provide a unified family of Legendre-based generalized Apostol-Bernoulli, Apostol-Euler, and Apostol-Genocchi polynomials, with appropriate constraints for the Maclaurin series. Then we look at the formulae and identities that are involved, including an integral formula, differential formulas, addition formulas, implicit summation formulas, and general symmetry identities. We also provide an explicit representation for these new polynomials. Due to the generality of the findings given here, various formulae and identities for relatively simple polynomials and numbers, such as generalized Bernoulli, Euler, and Genocchi numbers and polynomials, are indicated to be deducible. Furthermore, we employ the umbral calculus theory to offer some additional formulae for these new polynomials.

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