Short proofs for $q$-Raabe formula and integrals for Jacobi theta functions

TL;DR

This paper provides concise proofs for the $q$-analogue of Raabe's integral formula and evaluates an integral involving the first Jacobi theta function, addressing open questions and reproducing known identities.

Contribution

It introduces short proofs for the $q$-Raabe formula and evaluates a theta function integral, advancing understanding of $q$-analogues and theta function identities.

Findings

Answer to Mez ext{"o}'s question on $q$-Raabe formula

Evaluation of an integral involving the first theta function

Reproduction of known identities for Mez ext{"o}

Abstract

We shall answer a question of Mez\H{o} on the $q$-analogue of the Raabe's integral formula for $0<q<1$ and we shall evaluate an integral involving the first theta function. Moreover, we will reproduce short proofs for some identities of Mez\H{o}.