scholarly journals Identities of Symmetry for Higher-Order Generalizedq-Euler Polynomials

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
D. V. Dolgy ◽  
D. S. Kim ◽  
T. G. Kim ◽  
J. J. Seo
Keyword(s):  
Higher Order ◽  
Euler Polynomials ◽  
Power Sums

We investigate the properties of symmetry in two variables related to multiple Eulerq-l-function which interpolates higher-orderq-Euler polynomials at negative integers. From our investigation, we can derive many interesting identities of symmetry in two variables related to generalized higher-orderq-Euler polynomials and alternating generalizedq-power sums.

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 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 note on the higher-order Frobenius-Euler polynomials and Sheffer sequences

2013 ◽  
Vol 2013 (1) ◽  
Author(s):  
Dae San Kim ◽  
Taekyun Kim ◽  
Sang-Hun Lee ◽  
Seog-Hoon Rim
Keyword(s):  
Higher Order ◽  
Euler Polynomials ◽  
Sheffer Sequences

On modified higher-order q-Euler polynomials under S_5

2015 ◽  
Vol 9 ◽  
pp. 4171-4178
Author(s):  
Taekyun Kim ◽  
Jong Jin Seo
Keyword(s):  
Higher Order ◽  
Euler Polynomials

scholarly journals Sums of Products Involving Power Sums of φ(n) Integers

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Jitender Singh
Keyword(s):  
Closed Form ◽  
Complex Variable ◽  
Bernoulli Polynomials ◽  
Rational Numbers ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Power Sums ◽  
Power Sum ◽  
Mobius Function ◽  
Sums Of Products

A sequence of rational numbers as a generalization of the sequence of Bernoulli numbers is introduced. Sums of products involving the terms of this generalized sequence are then obtained using an application of Faà di Bruno's formula. These sums of products are analogous to the higher order Bernoulli numbers and are used to develop the closed form expressions for the sums of products involving the power sums Ψk(x,n):=∑d|n‍μ(d)dkSkx/d,  n∈ℤ+ which are defined via the Möbius function μ and the usual power sum Sk(x) of a real or complex variable x. The power sum Sk(x) is expressible in terms of the well-known Bernoulli polynomials by Sk(x):=(Bk+1(x+1)-Bk+1(1))/(k+1).

Some Explicit Formulas for the Frobenius-Euler Polynomials of Higher Order

2017 ◽  
Vol 11 (2) ◽  
pp. 621-626 ◽  
Author(s):  
H. M. Srivastava ◽  
Mohamed Amine Boutiche ◽  
Mourad Rahmani
Keyword(s):  
Higher Order ◽  
Euler Polynomials ◽  
Explicit Formulas
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