An algebraic description of the bispectrality of the biorthogonal rational functions of Hahn type
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Bibliographic record
Abstract
The biorthogonal rational functions of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Subscript 3 Baseline upper F 2"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi/> <mml:mn>3</mml:mn> </mml:msub> </mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{_3}F_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> type on the uniform grid provide the simplest example of rational functions with bispectrality properties that are similar to those of classical orthogonal polynomials. These properties are described by three difference operators <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X comma upper Y comma upper Z"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">X,Y,Z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which are tridiagonal with respect to three distinct bases of the relevant finite-dimensional space. The pairwise commutators of the operators <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X comma upper Y comma upper Z"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">X,Y,Z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> generate a quadratic algebra which is akin to the algebras of Askey–Wilson type attached to hypergeometric polynomials.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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