In the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$q^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>q</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>-symmetric Askey scheme, namely the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$q^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>q</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>-Askey–Wilson, continuous dual <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$q^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>q</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>-Hahn, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$q^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>q</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>-Al-Salam–Chihara, continuous big <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$q^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>q</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>-Hermite and continuous <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$q^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>q</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>-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 <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$q^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>q</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>-Askey–Wilson polynomials. In the <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$q\rightarrow 1^{-}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo stretchy="false">→</mo> <msup> <mn>1</mn> <mo>-</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> limit, this q-beta integral corresponds to a beta integral of Ramanujan-type which we present and provide two proofs for.

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

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2025年11月29日

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Howard S. Cohl,
Hans Volkmer

摘要

In the $q^{-1}$ $q^{- 1}$ -symmetric Askey scheme, namely the $q^{-1}$ $q^{- 1}$ -Askey–Wilson, continuous dual $q^{-1}$ $q^{- 1}$ -Hahn, $q^{-1}$ $q^{- 1}$ -Al-Salam–Chihara, continuous big $q^{-1}$ $q^{- 1}$ -Hermite and continuous $q^{-1}$ $q^{- 1}$ -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 $q^{-1}$ $q^{- 1}$ -Askey–Wilson polynomials. In the $q\rightarrow 1^{-}$ $q \to 1^{-}$ limit, this q-beta integral corresponds to a beta integral of Ramanujan-type which we present and provide two proofs for.

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

摘要