Note on q-extensions of Euler numbers and polynomials of higher order

TL;DR

This paper introduces a new q-extension of Euler numbers and polynomials, deriving identities, constructing q-Euler zeta functions, and exploring their properties and sums, expanding the mathematical understanding of q-analogs.

Contribution

It presents a novel q-extension of Euler numbers and polynomials, distinct from previous work, and develops related zeta functions and sum formulas.

Findings

Derived identities for the new q-Euler numbers and polynomials

Constructed q-Euler zeta functions interpolating at negative integers

Established formulas for sums and products of q-Euler numbers and polynomials

Abstract

In [14] Ozden-Simsek-Cangul constructed generating functions of higher-order twisted $(h,q)$-extension of Euler polynomials and numbers, by using $p$-adic q-deformed fermionic integral on $\Bbb Z_p$. By applying their generating functions, they derived the complete sums of products of the twisted $(h,q)$-extension of Euler polynomials and numbers, see[13, 14]. In this paper we cosider the new $q$-extension of Euler numbers and polynomials to be different which is treated by Ozden-Simsek-Cangul. From our $q$-Euler numbers and polynomials we derive some interesting identities and we construct $q$-Euler zeta functions which interpolate the new $q$-Euler numbers and polynomials at a negative integer. Furthermore we study Barnes' type $q$-Euler zeta functions. Finally we will derive the new formula for " sums products of $q$-Euler numbers and polynomials" by using fermionic $p$-adic $q$-integral on $\Bbb Z_p$.