A limit đ=-1 for the big đ-Jacobi polynomials
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Bibliographic record
Abstract
We study a new family of âclassicalâ orthogonal polynomials, here called big <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="negative 1"> <mml:semantics> <mml:mrow> <mml:mo> â </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Jacobi polynomials, which satisfy (apart from a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -term recurrence relation) an eigenvalue problem with differential operators of Dunkl type. These polynomials can be obtained from the big <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Jacobi polynomials in the limit <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q right-arrow negative 1"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo stretchy="false"> â </mml:mo> <mml:mo> â </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">q \to -1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . An explicit expression of these polynomials in terms of Gaussâ hypergeometric functions is found. The big <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="negative 1"> <mml:semantics> <mml:mrow> <mml:mo> â </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Jacobi polynomials are orthogonal on the union of two symmetric intervals of the real axis. We show that the big <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="negative 1"> <mml:semantics> <mml:mrow> <mml:mo> â </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Jacobi polynomials can be obtained from the (terminating) Bannai-Ito polynomials when the orthogonality support is extended to an infinite number of points. We further indicate that these polynomials provide a nontrivial realization of the Askey-Wilson algebra for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q right-arrow negative 1"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo stretchy="false"> â </mml:mo> <mml:mo> â </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">q \to -1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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