# General Multi-sum Transformations and Some Implications

## TL;DR

This paper introduces two broad transformations for hypergeometric multi-sums involving arbitrary sequences, enabling their reduction to simpler forms and revealing new summation formulas and product representations.

## Contribution

It presents novel general transformations for hypergeometric multi-sums of arbitrary depth, expanding the toolkit for analyzing complex q-series.

## Key findings

- Derived summation formulas for q-orthogonal polynomials
- Expressed multi-sums as infinite products
- Reduced complex multi-sums to simpler forms

## Abstract

We give two general transformations that allows certain quite general basic hypergeometric multi-sums of arbitrary depth (sums that involve an arbitrary sequence $\{g(k)\}$), to be reduced to an infinite $q$-product times a single basic hypergeometric sum. Various applications are given, including summation formulae for some $q$ orthogonal polynomials, and various multi-sums that are expressible as infinite products.