# On q-analogues of quadratic Euler sums

## TL;DR

This paper introduces generalized q-analogues of Euler sums, deriving new identities and relations to q-polylogarithms, and evaluates q-series with q-harmonic numbers, bridging to classical Euler sums as q approaches 1.

## Contribution

It develops a comprehensive framework for q-analogues of Euler sums, including new identities and evaluation methods using Jackson q-integrals and stuffle products.

## Key findings

- Derived new identities for q-analogues of Euler sums.
- Established relations between quadratic and linear sums and q-polylogarithms.
- Evaluated q-series involving q-harmonic numbers.

## Abstract

In this paper we define the generalized q-analogues of Euler sums and present a new family of identities for q-analogues of Euler sums by using the method of Jackson q-integral rep- resentations of series. We then apply it to obtain a family of identities relating quadratic Euler sums to linear sums and q-polylogarithms. Furthermore, we also use certain stuffle products to evaluate several q-series with q-harmonic numbers. Some interesting new results and illustrative examples are considered. Finally, we can obtain some explicit relations for the classical Euler sums when q approaches to 1.