Positivity of the Poisson Kernel for the Continuous q-Jacobi Polynomials and Some Quadratic Transformation Formulas for Basic Hypergeometric Series

A nonterminating extension of the Sears–Carlitz quadratic transformation formula for a well-posed ₃ φ ₂ series with an arbitrary argument is obtained as a sum of two balanced ₅ φ ₄ series. This is then extended to a very well-poised ₅ φ ₄ series with arbitrary argument. These results are used to derive some generating functions for the q-Wilson polynomials pₙ (x;a,b,c,d;q) when $ad = bc$ and an expression for the Poisson kernel Kₜ (x,y;a,b,c,bc / a;q) as a sum of three sums of very well-poised ₁0 φ ₉ series which clearly demonstrates its positivity for 0 t < 1, 0 q < 1 in the continuous q-Jacobi case when a = q^α / 2 + 1 / 4, b = q^α / 2 + 3 / 4, c = q^β / 2 + 1 / 4 and α, β > - 1. Additional quadratic transformation formulas are derived, along with q-analogues of Watson’s and Whipple’s summation formulas.