On certain $q$-multiple sums

TL;DR

This paper introduces a general method for deriving $q$-multiple sum identities, extending previous results and providing new classes of such sums related to divisor functions and harmonic series.

Contribution

The paper develops a unified approach to $q$-multiple sum identities, generalizing prior results and duality relations, and introduces new classes of $q$-multiple sums.

Findings

Generalized $q$-series identities related to divisor functions

Extended duality relations for finite multiple harmonic $q$-series

New classes of $q$-multiple sums

Abstract

We present outlines of a general method to reach certain kinds of $q$-multiple sum identities. Throughout our exposition, we shall give generalizations to the results given by Dilcher, Prodinger, Fu and Lascoux, Zeng, and Guo and Zhang concerning $q$-series identities related to divisor functions. Our exposition shall also provide a generalization of the duality relation for finite multiple harmonic $q$-series given by Bradley. Utilizing these generalizations, we will also arrive at some new interesting classes of $q$-multiple sums.