Unification of q-exponential function and related q-numbers and polynomials

TL;DR

This paper introduces a unified framework for q-exponential functions and related q-numbers and polynomials, extending classical results through a generalized parameter, and investigates their properties and conditions.

Contribution

It unifies various forms of q-exponential functions into a single framework using an additional parameter, connecting to classical q-Bernoulli and Euler polynomials.

Findings

Unified q-exponential functions with a new parameter

Derived conditions linking to classical q-Bernoulli numbers

Extended properties of generalized Bernoulli and Euler polynomials

Abstract

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 study some conditions on this parameter to related this to some classical results for q-Bernoulli numbers and polynomials.