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Around q-Appell polynomial sequences

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Abstract

First we show that the quadratic decomposition of the Appell polynomials with respect to the q-divided difference operator is supplied by two other Appell sequences with respect to a new operator \(\mathcal{M}_{q;q^{-\varepsilon}}\), where ε represents a complex parameter different from any negative even integer number. While seeking all the orthogonal polynomial sequences invariant under the action of \(\mathcal{M}_{\sqrt{q};q^{-\varepsilon/2}}\) (the \(\mathcal{M}_{\sqrt{q};q^{-\varepsilon/2}}\)-Appell), only the Wall q-polynomials with parameter q ε/2+1 are achieved, up to a linear transformation. This brings a new characterization of these polynomial sequences.

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Authors and Affiliations

  1. Inst. Sup. de Engenharia de Coimbra & CMUP, Rua Pedro Nunes-Quinta da Nora, 3030-199, Coimbra, Portugal

    Ana F. Loureiro

  2. Lab. Jacques-Louis Lions—CNRS, Université Pierre et Marie Curie, Bôite courrier 187, 75252, Paris cedex 05, France

    Pascal Maroni

Authors
  1. Ana F. Loureiro
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  2. Pascal Maroni
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Correspondence to Ana F. Loureiro.

Additional information

The first author was supported in part by FCT-Portugal through the Centro de Matemática da Universidade do Porto.

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Loureiro, A.F., Maroni, P. Around q-Appell polynomial sequences. Ramanujan J 26, 311–321 (2011). https://doi.org/10.1007/s11139-011-9336-8

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  • DOI: https://doi.org/10.1007/s11139-011-9336-8

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