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<p>The properties of the Wilson rational functions <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\( _{10}\phi _9\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mn>10</mn> <mrow /> </mmultiscripts> <msub> <mi>ϕ</mi> <mn>9</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> with three different normalizations are described. For one normalization, it satisfies an <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(R_{II}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mrow> <mi mathvariant="italic">II</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> recurrence relation, whereas for the two other ones, they satisfy a generalized eigenvalue problem. The so-called Wilson rational algebra is introduced, which encodes algebraically the spectral properties of these special functions. Finally, different limits are considered, leading up to functions proportional to <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\( _{4}\phi _3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mrow /> <mn>4</mn> <mrow /> </mmultiscripts> <msub> <mi>ϕ</mi> <mn>3</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. For one of these, the spectral algebra simplifies to yield the meta <i>q</i>-Racah algebra.</p>

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Algebras behind the bispectrality of the Wilson rational functions and their \( _4\phi _3\) limits

  • 期刊论文

  • 发表日期：

    2025年5月26日

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  • Nicolas Crampé,
  • Satoshi Tsujimoto,
  • Luc Vinet,
  • Alexei Zhedanov

摘要

The properties of the Wilson rational functions \( _{10}\phi _9\) 10 ϕ 9 with three different normalizations are described. For one normalization, it satisfies an \(R_{II}\) R II recurrence relation, whereas for the two other ones, they satisfy a generalized eigenvalue problem. The so-called Wilson rational algebra is introduced, which encodes algebraically the spectral properties of these special functions. Finally, different limits are considered, leading up to functions proportional to \( _{4}\phi _3\) 4 ϕ 3 . For one of these, the spectral algebra simplifies to yield the meta q-Racah algebra.