scholarly journals Some identities on the Bernoulli, Euler and Genocchi polynomials via power sums and alternate power sums

2009 ◽  
Vol 309 (10) ◽  
pp. 3346-3363 ◽  
Author(s):  
Hongmei Liu ◽  
Weiping Wang
Keyword(s):  
Power Sums ◽  
Genocchi 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 Symmetric Identities for Carlitz’s Type Higher-Order (p,q)-Genocchi Polynomials

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1670
Author(s):  
Ahyun Kim ◽  
Cheon Seoung Ryoo
Keyword(s):  
Higher Order ◽  
Power Sums ◽  
Genocchi Polynomials ◽  
Symmetric Identities

In this paper, we study Carlitz’s type higher-order (p,q)-Genocchi polynomials. To be specific, we define Carlitz’s type higher-order (p,q)-Genocchi polynomials and Carlitz’s type higher-order (h,p,q)-Genocchi polynomials. This paper also explores properties including distribution relation and symmetric identities. In addition, we find alternating (p,q)-power sums. We identify symmetric identities using Carlitz’s type higher-order (h,p,q)-Genocchi polynomials and alternating (p,q)-power sums.

scholarly journals Degenerate poly-Bell polynomials and numbers

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Hye Kyung Kim
Keyword(s):  
Combinatorial Identities ◽  
Bell Polynomials ◽  
Genocchi Polynomials ◽  
Special Polynomials ◽  
Explicit Representations

AbstractNumerous mathematicians have studied ‘poly’ as one of the generalizations to special polynomials, such as Bernoulli, Euler, Cauchy, and Genocchi polynomials. In relation to this, in this paper, we introduce the degenerate poly-Bell polynomials emanating from the degenerate polyexponential functions which are called the poly-Bell polynomials when $\lambda \rightarrow 0$ λ → 0 . Specifically, we demonstrate that they are reduced to the degenerate Bell polynomials if $k = 1$ k = 1 . We also provide explicit representations and combinatorial identities for these polynomials, including Dobinski-like formulas, recurrence relationships, etc.

A concise proof of a theorem on products of power sums

1978 ◽  
Vol 6 (1) ◽  
pp. 11-17 ◽  
Author(s):  
P. S. Dwyer ◽  
N. N. Mikhail ◽  
D. S. Tracy
Keyword(s):  
Power Sums

scholarly journals IDENTITIES OF SYMMETRY FOR GENERALIZED TWISTED BERNOULLI POLYNOMIALS TWISTED BY RAMIFIED ROOTS OF UNITY

2014 ◽  
Vol 60 (1) ◽  
pp. 19-36
Author(s):  
Dae San Kim
Keyword(s):  
Generating Function ◽  
Bernoulli Polynomials ◽  
Roots Of Unity ◽  
Integral Expression ◽  
Power Sums ◽  
Exponential Generating Function

Abstract We derive eight identities of symmetry in three variables related to generalized twisted Bernoulli polynomials and generalized twisted power sums, both of which are twisted by ramified roots of unity. All of these are new, since there have been results only about identities of symmetry in two variables. The derivations of identities are based on the p-adic integral expression of the generating function for the generalized twisted Bernoulli polynomials and the quotient of p-adic integrals that can be expressed as the exponential generating function for the generalized twisted power sums.

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 Symmetry for the Generalized Bernoulli Numbers and Polynomials

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
Taekyun Kim ◽  
Seog-Hoon Rim ◽  
Byungje Lee
Keyword(s):  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
Power Sums ◽  
Invariant Integral ◽  
Generalized Bernoulli Polynomials ◽  
Symmetric Properties

By the properties ofp-adic invariant integral onℤp, we establish various identities concerning the generalized Bernoulli numbers and polynomials. From the symmetric properties ofp-adic invariant integral onℤp, we give some interesting relationship between the power sums and the generalized Bernoulli polynomials.

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