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On Polynomials Orthogonal with Respect to an Inner Product Involving Higher-Order Differences: The Meixner Case

Author

Listed:
  • Roberto S. Costas-Santos

    (Facultad de Ciencias Técnicas, Universidad de Granma, Km. 17.5 de la carretera de Bayano-Manzanillo, Bayamo 85100, Cuba)

  • Anier Soria-Lorente

    (Facultad de Ciencias Técnicas, Universidad de Granma, Km. 17.5 de la carretera de Bayano-Manzanillo, Bayamo 85100, Cuba)

  • Jean-Marie Vilaire

    (Institut des Sciences, des Technologies et des Études Avancées d’Haïti, #10, Rue Mercier-Laham, Delmas 60, Musseau, Port-au-Prince 15953, Haiti)

Abstract

In this contribution we consider sequences of monic polynomials orthogonal with respect to the Sobolev-type inner product f , g = ⟨ u M , f g ⟩ + λ T j f ( α ) T j g ( α ) , where u M is the Meixner linear operator, λ ∈ R + , j ∈ N , α ≤ 0 , and T is the forward difference operator Δ or the backward difference operator ∇. Moreover, we derive an explicit representation for these polynomials. The ladder operators associated with these polynomials are obtained, and the linear difference equation of the second order is also given. In addition, for these polynomials, we derive a ( 2 j + 3 ) -term recurrence relation. Finally, we find the Mehler–Heine type formula for the particular case α = 0 .

Suggested Citation

  • Roberto S. Costas-Santos & Anier Soria-Lorente & Jean-Marie Vilaire, 2022. "On Polynomials Orthogonal with Respect to an Inner Product Involving Higher-Order Differences: The Meixner Case," Mathematics, MDPI, vol. 10(11), pages 1-16, June.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:11:p:1952-:d:832703
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