$q$-analogs of sinc sums and integrals

TL;DR

This paper explores $q$-analogues of sum-integral relations involving sinc functions and binomial coefficients, extending known $q$-hypergeometric series to multibasic fractional generalizations beyond traditional hypergeometric functions.

Contribution

It introduces novel multibasic fractional $q$-analogues of sum-equals-integral relations that are not based on $q$-hypergeometric functions.

Findings

Established new $q$-analogues for sinc sums and integrals.

Extended the framework to multibasic fractional generalizations.

Provided mathematical relations beyond existing $q$-hypergeometric series.

Abstract

$q$-analogs of sum equals integral relations $\sum_{n\in\mathbb{Z}}f(n)=\int_{-\infty}^\infty f(x)dx$ for sinc functions and binomial coefficients are studied. Such analogs are already known in the context of $q$-hypergeometric series. This paper deals with multibasic `fractional' generalizations that are not $q$-hypergeometric functions.