Note on the sums of powers of consecutive $q$-integers

TL;DR

This paper develops a $q$-analogue of Barnes's Bernoulli numbers and polynomials for even degrees, addressing part of Schlosser's question, and explores the $q$-analogue of sums of powers of consecutive integers.

Contribution

It introduces a new $q$-analogue of Barnes's Bernoulli numbers and polynomials for degree 2, providing partial answers to Schlosser's question.

Findings

Constructed $q$-analogue of Barnes's Bernoulli numbers for degree 2

Addressed the $q$-analogue of sums of powers of consecutive integers

Partially answered Schlosser's question for even integers

Abstract

In this paper we construct the $q$-analogue of Barnes's Bernoulli numbers and polynomials of degree 2, for positive even integers, which is an answer to a part of Schlosser's question. For positive odd integers, Schlosser's question is still open. Finally, we will treat the $q$-analogue of the sums of powers of consecutive integers.