Algebraic identities on q-harmonic numbers and q-binomial coefficients

TL;DR

This paper introduces a general algebraic identity that leads to new formulas involving q-binomial coefficients and q-harmonic numbers, recovering known identities and establishing a q-analog of Euler's formula with potential applications in q-supercongruences.

Contribution

It presents a novel algebraic identity that unifies and extends existing results on q-harmonic numbers and q-binomial coefficients, including known identities and new q-analog formulas.

Findings

Derived new formulas involving q-binomial coefficients and q-harmonic numbers

Recovered known identities, including Zheng's q-Apéry number identity

Established a q-analog of Euler's formula

Abstract

The aim of this paper is to present a general algebraic identity. Applying this identity, we provide several formulas involving the q-binomial coefficients and the q-harmonic numbers. We also recover some known identities including an algebraic identity of D. Y. Zheng on q-Ap\'{e}ry numbers and we establish the q-analog of Euler's formula. The proposed results may have important applications in the theory of q-supercongruences.