Product Formulas of Watson, Bailey and Bateman Types and Positivity of the Poisson Kernel for q-Racah Polynomials

A new method is introduced for proving certain important formulas due to Watson, Bailey and Bateman for products of ${}_2 F_1 $ hypergeometric series, and it is used to extend these formulas to products of ${}_4 \phi _3 $ basic hypergeometric series. The ${}_4 \phi _3 $ analogue of Watson’s product formula is used to give conditions under which the Poisson kernels for q-Racah polynomials, q-Hahn polynomials and little q-Jacobi polynomials are positive. A transformation formula for a certain ${}_4 \phi _3 $ series and expansion formulas for basic hypergeometric series are also derived.