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

Carlos Hermoso, Edmundo J. Huertas, Alberto Lastra and Anier Soria-Lorente
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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
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
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
Anier Soria-Lorente: Departamento de Tecnología, Universidad de Granma, Km. 17,5 de la Carretera de Bayamo-Manzanillo, Bayamo 85100, Cuba

Mathematics, 2020, vol. 8, issue 8, 1-21

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.

Keywords: Al-Salam–Carlitz I polynomials; Al-Salam–Carlitz I-Sobolev type polynomials; second order linear q -difference equations; structure relations; recurrence relations; basic hypergeometric series (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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