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 $\Gamma_{q}$.
Positivity and Fourier integrals over regular hexagon
q-Special functions, a tutorial
q-Special functions, an overview
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