Some identities of q-Euler polynomials arising from q-umbral calculus

TL;DR

This paper introduces new q-umbral calculus techniques to derive identities involving q-Euler polynomials and numbers, connecting them with q-Bernoulli entities and expanding the theoretical framework of q-extensions.

Contribution

It develops a novel q-umbral calculus method to study and derive identities for q-Euler polynomials and numbers, enhancing the understanding of their properties and relations.

Findings

New identities relating q-Euler and q-Bernoulli polynomials

Development of a q-umbral calculus approach

Connections between q-Euler polynomials and higher-order variants

Abstract

Recently, Araci-Acikgoz-Sen derived some interesting identities on weighted q-Euler polynomials and higher-order q-Euler polynomials from the applications of umbral calculus (See [1]). In this paper, we develop the new method of q-umbral calculus due to Roman and we study new q-extension of Euler numbers and polynomials which are derived from q-umbral calculus. Finally, we give some interesting identities on our q-Euler polynomials related to the q-Bernoulli numbers and polynomials of Hegazi and Mansour.