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 F₁ hypergeometric series, and it is used to extend these formulas to products of ₄ φ ₃ basic hypergeometric series. The ₄ φ 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 φ ₃ series and expansion formulas for basic hypergeometric series are also derived.