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Telescoping partial fractions decompositions, the little q-Jacobi functions of complex order, and the nonterminating q-Saalschütz sum

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Abstract

We use telescoping partial fractions decompositions to give new proofs of the orthogonality property and the normalization relation for the little q-Jacobi polynomials, and the q-Saalschütz sum. In [20], we followed the development [19] of Schur functions for partitions with complex parts, and we showed that there exist natural little q-Jacobi functions of complex order which satisfy extensions of the orthogonality property and normalization relation of the little q-Jacobi polynomials, and that these two results follow from and together imply the nonterminating form of the q-Saalschütz sum. Writing the q-Pochhammer symbol of complex order as a ratio of infinite products in the usual way, we obtain new telescoping partial fractions decomposition proofs of our results [20] for the little q-Jacobi functions of complex order. We give several new proofs of the q-Saalschütz sum and its nonterminating form.

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

    Kevin W. J. Kadell

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  1. Kevin W. J. Kadell
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Correspondence to Kevin W. J. Kadell.

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For our friends Dick and Liz

2000 Mathematics Subject Classification Primary—42C05; Secondary—33C45, 33C47

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Kadell, K.W.J. Telescoping partial fractions decompositions, the little q-Jacobi functions of complex order, and the nonterminating q-Saalschütz sum. Ramanujan J 13, 449–469 (2007). https://doi.org/10.1007/s11139-006-0261-1

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