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<p>A unified algebraic interpretation of both finite families of orthogonal polynomials and biorthogonal rational functions of <i>q</i>-Hahn type is provided. The approach relies on the meta <i>q</i>-Hahn algebra and its finite-dimensional bidiagonal representations. The functions of <i>q</i>-Hahn type are identified as overlaps (up to global factors) between bases solving ordinary or generalized eigenvalue problems in the representation of the meta <i>q</i>-Hahn algebra. Moreover, (bi)orthogonality relations, recurrence relations, difference equations and some contiguity relations satisfied by these functions are recovered algebraically using the actions of the generators of the meta <i>q</i>-Hahn algebra on various bases.</p>

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Meta algebras and biorthogonal rational functions: the q-Hahn case

  • 期刊论文

  • 发表日期：

    2025年11月6日

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  • Pierre-Antoine Bernard,
  • Abderahmane Bouziane,
  • Samuel Pellerin,
  • Simone Têtu,
  • Satoshi Tsujimoto,
  • Luc Vinet,
  • Meri Zaimi,
  • Alexei Zhedanov

摘要

A unified algebraic interpretation of both finite families of orthogonal polynomials and biorthogonal rational functions of q-Hahn type is provided. The approach relies on the meta q-Hahn algebra and its finite-dimensional bidiagonal representations. The functions of q-Hahn type are identified as overlaps (up to global factors) between bases solving ordinary or generalized eigenvalue problems in the representation of the meta q-Hahn algebra. Moreover, (bi)orthogonality relations, recurrence relations, difference equations and some contiguity relations satisfied by these functions are recovered algebraically using the actions of the generators of the meta q-Hahn algebra on various bases.