Abstract
Aq-integral representation of Rogers'q-ultraspherical polynomialsC n (x;β∥q) is obtained by using Sears' summation formula for balanced non-terminating3 φ 2 series. It is then used to give a simple derivation of the Gasper-Rahman formula for the Poisson kernel ofC n (x;β∥q). As another application it is shown how this representation can be directly used to give an asymptotic expansion of theq-ultraspherical polynomials.
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Communicated by Edward B. Saff.
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Rahman, M., Verma, A. A q-integral representation of Rogers' q-ultraspherical polynomials and some applications. Constr. Approx 2, 1–10 (1986). https://doi.org/10.1007/BF01893413
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DOI: https://doi.org/10.1007/BF01893413