On the generalized higher-order q-Bernoulli numbers and polynomials

TL;DR

This paper explores new equations involving p-adic q-integrals and systematically studies extended Carlitz q-Bernoulli numbers and polynomials within p-adic number fields, revealing their properties and relationships.

Contribution

It introduces a systematic study of generalized higher-order q-Bernoulli numbers and polynomials derived from p-adic q-integrals, extending existing theories.

Findings

Derived new equations for p-adic q-integrals on Zp

Established properties of extended Carlitz q-Bernoulli numbers

Analyzed relationships among q-Bernoulli polynomials

Abstract

In this paper we give some interesting equation of p-adic q-integrals on Zp. From those p-adic q-integrals, we present a systemic study of some families of extended Carlitz q-Bernoulli numbers and polynomials in p-adic number field.