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 \phi _2 $ series with an arbitrary argument is obtained as a sum of two balanced ${}_5 \phi _4 $ series. This is then extended to a very well-poised ${}_5 \phi _4 $ series with arbitrary argument. These results are used to derive some generating functions for the q-Wilson polynomials $p_n (x;a,b,c,d;q)$ when $ad = bc$ and an expression for the Poisson kernel $K_t (x,y;a,b,c,{{bc} / {a;}}q)$ as a sum of three sums of very well-poised ${}_10 \phi _9 $ series which clearly demonstrates its positivity for $0 \leqq t < 1$, $0 \leqq q < 1$ in the continuous q-Jacobi case when $a = q^{{\alpha / 2} + {1 / 4}} $, $b = q^{{\alpha / 2} + {3 / 4}}$, $c = q^{{\beta / 2} + {1 / 4}} $ and $\alpha $, $\beta > - 1$. Additional quadratic transformation formulas are derived, along with q-analogues of Watson’s and Whipple’s summation formulas.