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 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 Some Identities on the -Genocchi Polynomials of Higher-Order and -Stirling Numbers by the Fermionic -Adic Integral on

2010 ◽  
Vol 2010 ◽  
pp. 1-14 ◽  
Author(s):  
Seog-Hoon Rim ◽  
Jeong-Hee Jin ◽  
Eun-Jung Moon ◽  
Sun-Jung Lee
Keyword(s):  
Stirling Numbers ◽  
Higher Order ◽  
Genocchi Numbers ◽  
Genocchi Polynomials

A systemic study of some families of -Genocchi numbers and families of polynomials of Nörlund type is presented by using the multivariate fermionic -adic integral on . The study of these higher-order -Genocchi numbers and polynomials yields an interesting -analog of identities for Stirling numbers.

scholarly journals Combinatorial Identities with Generalized Higher-order Genocchi Sequences

2019 ◽  
Vol 12 (2) ◽  
pp. 605-621
Author(s):  
Tian Hao ◽  
Wuyungaowa Bao
Keyword(s):  
Probabilistic Method ◽  
Fibonacci Numbers ◽  
Stirling Numbers ◽  
Higher Order ◽  
Harmonic Numbers ◽  
Bell Numbers ◽  
Genocchi Numbers ◽  
Genocchi Polynomials ◽  
Cauchy Numbers ◽  
The Moment

In this paper, we make use of the probabilistic method to calculate the moment representation of generalized higher-order Genocchi polynomials. We obtain the moment expression of the generalized higher-order Genocchi numbers with a and b parameters. Some characteriza tions and identities of generalized higher-order Genocchi polynomials are given by the proof of the moment expression. As far as properties given by predecessors are concerned, we prove them by the probabilistic method. Finally, new identities of relationships involving generalized higher-order Genocchi numbers and harmonic numbers, derangement numbers, Fibonacci numbers, Bell numbers, Bernoulli numbers, Euler numbers, Cauchy numbers and Stirling numbers of the second kind are established. 

scholarly journals A Note on (h,q)-Genocchi Polynomials and Numbers of Higher Order

2010 ◽  
Vol 2010 ◽  
pp. 1-7 ◽  
Author(s):  
Lee-Chae Jang ◽  
Kyung-Won Hwang ◽  
Young-Hee Kim
Keyword(s):  
Higher Order ◽  
Genocchi Polynomials

scholarly journals Some Identities of Ordinary and Degenerate Bernoulli Numbers and Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 847 ◽  
Author(s):  
Dmitry V. Dolgy ◽  
Dae San Kim ◽  
Jongkyum Kwon ◽  
Taekyun Kim
Keyword(s):  
Random Variables ◽  
Infinite Family ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Integer Power ◽  
Bernoulli Numbers And Polynomials ◽  
Power Sums ◽  
Power Sum ◽  
Invariant Integrals

In this paper, we investigate some identities on Bernoulli numbers and polynomials and those on degenerate Bernoulli numbers and polynomials arising from certain p-adic invariant integrals on Z p . In particular, we derive various expressions for the polynomials associated with integer power sums, called integer power sum polynomials and also for their degenerate versions. Further, we compute the expectations of an infinite family of random variables which involve the degenerate Stirling polynomials of the second and some value of higher-order Bernoulli polynomials.

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 On the unified family of generalized Apostol-type polynomials of higher order and multiple power sums

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 929-935 ◽  
Author(s):  
Veli Kurt
Keyword(s):  
Recurrence Relations ◽  
Higher Order ◽  
Euler Polynomials ◽  
Multiplication Formula ◽  
Power Sums ◽  
Genocchi Polynomials ◽  
Alternating Sums ◽  
Multiple Power

In last last decade, many mathematicians studied the unification of the Bernoulli and Euler polynomials. Firstly Karande B. K. and Thakare N. K. in [6] introduced and generalized the multiplication formula. Ozden et. al. in [14] defined the unified Apostol-Bernoulli, Euler and Genocchi polynomials and proved some relations. M. A. Ozarslan in [13] proved the explicit relations, symmetry identities and multiplication formula. El-Desouky et. al. in ([3], [4]) defined a new unified family of the generalized Apostol-Euler, Apostol-Bernoulli and Apostol-Genocchi polynomials and gave some relations for the unification of multiparameter Apostol-type polynomials and numbers. In this study, we give some symmetry identities and recurrence relations for the unified Apostol-type polynomials related to multiple alternating sums.

scholarly journals A Note on Type 2 w-Daehee Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 697
Author(s):  
Minyoung Ma ◽  
Dongkyu Lim
Keyword(s):  
Order Type ◽  
Higher Order ◽  
Special Polynomials ◽  
Invariant Integral ◽  
Symmetric Identities

In the paper, by virtue of the p-adic invariant integral on Z p , the authors consider a type 2 w-Daehee polynomials and present some properties and identities of these polynomials related with well-known special polynomials. In addition, we present some symmetric identities involving the higher order type 2 w-Daehee polynomials. These identities extend and generalize some known results.

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.

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