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arXiv · 2410.00246

Bilateral discrete and continuous orthogonality relations in the $q^{-1}$-symmetric Askey scheme

Howard S. Cohl·Hans Volkmer
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Abstract

In the $q^{-1}$-symmetric Askey scheme, namely the $q^{-1}$-Askey--Wilson, continuous dual $q^{-1}$-Hahn, $q^{-1}$-Al-Salam--Chihara, continuous big $q^{-1}$-Hermite and continuous $q^{-1}$-Hermite polynomials, we compute bilateral discrete and continuous orthogonality relations. We also derive a $q$-beta integral which comes from the continuous orthogonality relation for the $q^{-1}$-Askey--Wilson polynomials. In the $q\to 1^{-}$ limit, this $q$-beta integral corresponds to a beta integral of Ramanujan-type which we present and provide two proofs for.

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Howard S. Cohl, Hans Volkmer. 2024-09-30. Bilateral discrete and continuous orthogonality relations in the $q^{-1}$-symmetric Askey scheme. https://arxiv.org/abs/2410.00246

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