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

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.