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The Little q-Jacobi Functions of Complex Order

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Theory and Applications of Special Functions

Part of the book series: Developments in Mathematics ((DEVM,volume 13))

Abstract

We use Ismail's argument and an elementary combinatorial identity to prove the q-binomial theorem, the symmetry of the Rogers-Fine function, Ramanujan's 1ψ1 sum, and Heine's q-Gauss sum and give many other proofs of these results. We prove a special case of Heine's 2ϕ1 transformation and write Ramanujan's 1ψ1 sum as the nonterminating q-Chu-Vandermonde sum. We show that the q-SaalschÜtz and q-Chu-Vandermonde sums are equivalent to the evaluations of certain moments and to the orthogonality of the little q-Jacobi polynomials; hence the q-Chu-Vandermonde sum implies the q-Saalschütz sum. We extend the little q-Jacobi polynomials naturally to the little q-Jacobi functions of complex order. We show that the nonterminating q-Saalschütz and q-Chu-Vandermonde sums are equivalent to the evaluations of certain moments and, using the Liouville-Ismail argument, to two orthogonality relations. We show that the nonterminating q-Chu-Vandermonde sum implies the nonterminating q-Saalschütz sum.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Arizona State University, Tempe, AZ, 85287-1804

    Kevin W. J. Kadell

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Editors and Affiliations

  1. University of Central Florida, USA

    Mourad E.H. Ismail

  2. Technische Universiteit Delft, The Netherlands

    Erik Koelink

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Kadell, K.W.J. (2005). The Little q-Jacobi Functions of Complex Order. In: Ismail, M.E., Koelink, E. (eds) Theory and Applications of Special Functions. Developments in Mathematics, vol 13. Springer, Boston, MA. https://doi.org/10.1007/0-387-24233-3_13

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