On a theta product of Jacobi and its applications to $q$-gamma products

Abstract

We give a new proof for a product formula of Jacobi which turns out to be equivalent to a $q$-trigonometric product which was stated without proof by Gosper. We apply this formula to derive a $q$-analogue for the Gauss multiplication formula for the gamma function. Furthermore, we give explicit formulas for short products of $q$-gamma functions.

13 pages, published in J. Math. Anal. Appl