Positivity of the Poisson Kernel for the Continuous q-Ultraspherical Polynomials

Rogers’ [Proc. London Math. Soc., 24 (1893), pp. 117–179] bilinear generating function for the continuous q-Hermite polynomials \[ {{( {t^2 ;q} )_∞ }} {{| {( {tei(θ + ψ ) ;q} )_∞ ( {tei(θ - ψ ) ;q} )_∞ } |^2 }} = ∑n = 0^∞ {{{H_n ( {cos θ |q} )H_n ( {cos ψ |q} )}} {{(q;q)_n }}} t^n , \] where (q;q)ₙ = (1 - q)(1 - q² ) ⋯ (1 - qⁿ ) and (a;q)_∞ = (1 - a)(1 - aq)(1 - aq² ) ⋯, is extended to the continuous q-ultraspherical polynomials Cₙ (x;β |q) and used to give conditions for the positivity of the Poisson kernel for these polynomials. Related bilinear generating functions are also considered.