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In this paper we construct $q$-Genocchi numbers and polynomials. By using these numbers and polynomials, we investigate the $q$-analogue of alternating sums of powers of consecutive integers due to Euler.

Number Theory · Mathematics 2007-05-23 Taekyun Kim

In this paper, we investigate some properties of several Sheffer sequences of several polynomials arising from umbral calculus. From our investigation, we can derive many interesting identities of several polynomials

Number Theory · Mathematics 2013-02-21 Taekyun Kim , Dae San Kim , Seog-Hoon Rim , Dmitry V. Dolgy

By using p-adic q-integrals, we study the q-Bernoulli numbers and polynomials of higher order.

Number Theory · Mathematics 2015-06-26 Taekyun Kim

A new $q$-analogue of Appell polynomial sequences and their generalizations are introduced and their main characterizations are proved. As consequences new $q$-analogue of Bernoulli and Euler polynomials and numbers is introduced, their…

Classical Analysis and ODEs · Mathematics 2018-01-29 P. Njionou Sadjang

In the present paper, we investigate special generalized q-Euler numbers and polynomials. Some earlier results of T. Kim in terms of q-Euler polynomials with weight alpha can be deduced. For presentation of our formulas we apply the method…

Number Theory · Mathematics 2018-07-23 Serkan Araci , Mehmet Acikgoz , Hassan Jolany

The purpose of this paper is to investigate some properties of q-Euler numbers and polynomials with weight 0. From those q-Euler numbers with weight 0, we derive some identities on the q-Euler numbers and polynomials with weight 0.

Number Theory · Mathematics 2011-10-11 T. Kim

The purpose of this paper is to present a systemic study of some families of multiple q-Genocchi and euler numbers by using multivariate q-Volkenborn integral. From the studies of those q-Genocchi numbers and polynomials of higher order we…

Number Theory · Mathematics 2009-11-13 Taekyun Kim

The main purpose of this paper is to introduce and investigate a class of generalized Bernoulli polynomials and Euler polynomials based on the generating function. we unify all forms of q-exponential functions by one more parameter. we…

Complex Variables · Mathematics 2018-10-24 N. I. Mahmudov , Mohammad Momenzadeh

Recently, Kim-Kim Introduced some interesting identities of symmetry for q-Bernoulli polynomials under symmetry group of degree n. In this paper, we study the degenerate q-Euler polynomials and derive some identities of symmetry for these…

Number Theory · Mathematics 2016-08-02 D. V. Dolgy , T. Kim , L. -C. Jang , H. -I Kwon

In this paper we obtain several new identities for Bernoulli and Euler polynomials; some of them extend Miki's and Matiyasevich's identities. Our new method involves differences and derivatives of polynomials.

Number Theory · Mathematics 2007-05-23 Hao Pan , Zhi-Wei Sun

In the paper, using the extended fermionic $p$-adic integral on $\mathbb{Z}_p$, the authors find some applications of the umbral calculus. From these applications, the authors derive some identities on the weighted Euler numbers and…

Number Theory · Mathematics 2018-01-12 Feng Qi , Serkan Araci , Mehmet Acikgoz

In this paper, we study some properties of the q-Appell polynomials, including the recurrence relations and the q-difference equations which extend some known calssical (q=1) results. We also provide the recurrence relations and the…

Classical Analysis and ODEs · Mathematics 2014-03-04 Nazim I. Mahmudov

In this article, we introduce combinatorial models for poly-Bernoulli polynomials and poly-Euler numbers of both kinds. As their applications, we provide combinatorial proofs of some identities involving poly-Bernoulli polynomials.

Combinatorics · Mathematics 2022-07-04 Beáta Bényi , Toshiki Matsusaka

We describe applications of the classical umbral calculus to bilinear generating functions for polynomial sequences, identities for Bernoulli and related numbers, and Kummer congruences.

Combinatorics · Mathematics 2013-04-02 Ira M. Gessel

The aim of this paper is to give a new approach to modified $q$-Bernstein polynomials for functions of several variables. By using these polynomials, the recurrence formulas and some new interesting identities related to the second Stirling…

Number Theory · Mathematics 2019-07-04 Serkan Araci , Mehmet Acikgoz , Hassan Jolany , Armen Bagdasaryan

Certain generalization of Euler numbers was defined in 1935 by Lehmer using cubic roots of unity, as a natural generalization of Bernoulli and Euler numbers. In this paper, we define a new polynomial related to the higher-order generalized…

Combinatorics · Mathematics 2025-07-24 Wei-Wei Qi

We establish some new combinatorial identities involving Euler polynomials and balancing (Lucas-balancing) polynomials. The derivations use elementary techniques and are based on functional equations for the respective generating functions.…

Number Theory · Mathematics 2020-09-22 Robert Frontczak , Taras Goy

In this paper we give new identities involving q-Euler polynomials of higher order.

Number Theory · Mathematics 2015-05-14 Taekyun Kim , Y. H. Kim

In the present article, we study Bell based Euler polynomial of order {\alpha} and investigate some useful correlation formula, summation formula and derivative formula. Also, we introduce some relation of string number of the second kind.…

Number Theory · Mathematics 2021-04-20 Nabiullah Khan , Saddam Husain

In this paper we study the genralized q-Euler numbers and polynomials. From our results, we derive some interesting congruences related tothe generalized q-Euler numbers.

Number Theory · Mathematics 2009-10-15 Kyoung-Ho Park , Young-Hee Kim , Taekyun Kim