# On Positivities of Certain q-Special Functions

## TL;DR

This paper uses Bochner's theorem to establish positivity of specific q-special functions, leading to new inequalities for the Jacobi theta and q-Gamma functions.

## Contribution

It introduces a novel application of Bochner's theorem to prove positivity of q-exponentials and derives new inequalities for important q-special functions.

## Key findings

- Proved certain q-exponentials are positive definite functions.
- Derived new inequalities for the Jacobi theta function.
- Established properties of the q-Gamma function.

## Abstract

In this work we shall apply the Bochner's theorem to prove certain combinations of Euler's q-exponentials are positive definite functions. Then we apply this positivity to prove curious inequalities for the Jacobi theta function $\vartheta_{4}$ and q-Gamma function $\Gamma_{q}$.