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On Second Order q -Difference Equations Satisfied by Al-Salam–Carlitz I-Sobolev Type Polynomials of Higher Order

Author

Listed:
  • Carlos Hermoso

    (Departamento de Física y Matemáticas, Universidad de Alcalá, Ctra. Madrid-Barcelona, Km. 33,600, Facultad de Ciencias, 28805 Alcalá de Henares, Madrid, Spain
    These authors contributed equally to this work.)

  • Edmundo J. Huertas

    (Departamento de Física y Matemáticas, Universidad de Alcalá, Ctra. Madrid-Barcelona, Km. 33,600, Facultad de Ciencias, 28805 Alcalá de Henares, Madrid, Spain
    These authors contributed equally to this work.)

  • Alberto Lastra

    (Departamento de Física y Matemáticas, Universidad de Alcalá, Ctra. Madrid-Barcelona, Km. 33,600, Facultad de Ciencias, 28805 Alcalá de Henares, Madrid, Spain
    These authors contributed equally to this work.)

  • Anier Soria-Lorente

    (Departamento de Tecnología, Universidad de Granma, Km. 17,5 de la Carretera de Bayamo-Manzanillo, Bayamo 85100, Cuba
    These authors contributed equally to this work.)

Abstract

This contribution deals with the sequence { U n ( a ) ( x ; q , j ) } n ≥ 0 of monic polynomials in x , orthogonal with respect to a Sobolev-type inner product related to the Al-Salam–Carlitz I orthogonal polynomials, and involving an arbitrary number j of q -derivatives on the two boundaries of the corresponding orthogonality interval, for some fixed real number q ∈ ( 0 , 1 ) . We provide several versions of the corresponding connection formulas, ladder operators, and several versions of the second order q -difference equations satisfied by polynomials in this sequence. As a novel contribution to the literature, we provide certain three term recurrence formula with rational coefficients satisfied by U n ( a ) ( x ; q , j ) , which paves the way to establish an appealing generalization of the so-called J -fractions to the framework of Sobolev-type orthogonality.

Suggested Citation

  • Carlos Hermoso & Edmundo J. Huertas & Alberto Lastra & Anier Soria-Lorente, 2020. "On Second Order q -Difference Equations Satisfied by Al-Salam–Carlitz I-Sobolev Type Polynomials of Higher Order," Mathematics, MDPI, vol. 8(8), pages 1-21, August.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:8:p:1300-:d:395302
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