[Paper Review] Note on the sums of powers of consecutive $q$-integers

Yılmaz Şimşek, D. Kim+2 more|arXiv (Cornell University)|Mar 11, 2005

Advanced Mathematical Identities15 references3 citations

TL;DR

This paper constructs the q-analogue of Barnes' Bernoulli numbers and polynomials of degree 2 for positive even integers, providing a q-analogue of the sums of powers of consecutive q-integers. It resolves part of Schlosser's open question by deriving a general formula for S_{m,n}(q), linking q-Bernoulli polynomials to q-sums of powers via generating functions and Mellin transforms.

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.

Motivation & Objective

• To construct the q-analogue of Barnes' Bernoulli numbers and polynomials of degree 2 for positive even integers.
• To resolve a part of Schlosser's open question regarding q-analogues of sums of powers of consecutive integers.
• To derive a general formula for S_{m,n}(q), the q-analogue of sums of powers of consecutive q-integers.
• To define and study q-zeta functions using Mellin transforms of q-generating functions.

Proposed method

• Defining a generating function F_{k,q}^*(t) for q-Bernoulli numbers β_{n,k,q}^* via a sum over q-integers and exponential generating terms.
• Deriving a closed-form expression for β_{n,k,q}^* using binomial coefficients, q-Pochhammer-like terms, and rational functions in q.
• Introducing a second generating function F_{k,q}^*(t;k) for q-Bernoulli polynomials β_{n,k,q}^*(k), with shifted indices and q-exponential terms.
• Establishing a general formula S_{n,k-1}(q) = (β_{n,k,q}^*(k) - β_{n,k,q}^*) / n, linking q-sums to q-Bernoulli polynomials.
• Applying Mellin transformation to the generating functions to define q-zeta functions ζ_{k,q}^*(s) and ζ_{k,q}^*(s;k) for Re(s) > 2.
• Using the Cauchy residue theorem to relate special values of the q-zeta function to q-Bernoulli numbers: ζ_{k,q}^*(1-n) = -β_{n,k,q}^*/n.

Experimental results

Research questions

• RQ1 Can a q-analogue of Barnes' Bernoulli numbers and polynomials of degree 2 be constructed for positive even integers?
• RQ2 Does a general q-analogue formula exist for sums of powers of consecutive q-integers S_{m,n}(q)?
• RQ3 How can q-zeta functions be defined via Mellin transforms of q-generating functions?
• RQ4 What is the relationship between the special values of q-zeta functions and q-Bernoulli numbers?
• RQ5 How do the q-Bernoulli numbers and polynomials reduce to classical Bernoulli numbers in the limit q → 1?

Key findings

• The q-Bernoulli numbers β_{n,k,q}^* are explicitly given by a closed-form formula involving binomial coefficients and rational functions in q, valid for even n.
• The q-Bernoulli polynomials β_{n,k,q}^*(k) are derived in closed form, involving weighted sums over m with q-exponential terms and denominators of the form (1 - q^{m - (n-1)/2 - 2}) and (1 - q^{m - (n-1)/2}).
• A general formula is established: S_{n,k-1}(q) = (β_{n,k,q}^*(k) - β_{n,k,q}^*) / n, providing a q-analogue of the classical sum of powers.
• The q-zeta function ζ_{k,q}^*(s) is defined via Mellin transform of F_{k,q}^*(-t), with ζ_{k,q}^*(1-n) = -β_{n,k,q}^*/n for positive integers n.
• The limit lim_{q→1} F_{k,q}^*(t) = -F_2(t; -1; -1, -1), showing consistency with the classical Barnes double zeta function.
• The constructed q-Bernoulli numbers and polynomials reduce to classical Bernoulli numbers in the limit q → 1, confirming q-analogue consistency.