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Periodica Mathematica Hungarica

, Volume 72, Issue 2, pp 200–217 | Cite as

Appell polynomial sequences with respect to some differential operators

  • P. Maroni
  • Teresa A. Mesquita
Article
  • 82 Downloads

Abstract

We present a study of a specific kind of lowering operator, herein called \(\Lambda \), which is defined as a finite sum of lowering operators and might be presented by various configurations. We characterize the polynomial sequences fulfilling an Appell relation with respect to \(\Lambda \), and considering a concrete cubic decomposition of a simple Appell sequence, we prove that the polynomial component sequences are \(\Lambda \)-Appell, with \(\Lambda \) defined as previously, although by a three term sum. Ultimately, we prove the non-existence of orthogonal polynomial sequences which are also \(\Lambda \)-Appell, when \(\Lambda \) is the lowering operator \(\Lambda =a_{0}D+a_{1}DxD+a_{2}\left( Dx\right) ^2D\), where \(a_{0}\), \(a_{1}\) and \(a_{2}\) are constants and \(a_{2} \ne 0\). The case where \(a_{2}=0\) and \(a_{1} \ne 0\) is also naturally recaptured.

Keywords

Orthogonal polynomials Appell sequences Stirling numbers  Cubic decomposition Laguerre polynomials 

Mathematics Subject Classification

Primary 42C05 Secondary 44A55 16R60 33C45 11B73 

Notes

Acknowledgments

We would like to thank the referee’s remarks and suggestions which enhanced the final presentation of this paper. T. A. Mesquita was partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds (FEDER), under the partnership agreement PT2020.

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2016

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis LionsCNRS, UMR 7598ParisFrance
  2. 2.Laboratoire Jacques-Louis LionsUPMC Univ Paris 06, UMR 7598ParisFrance
  3. 3.Instituto Politécnico de Viana do Castelo & Centro de Matemática da Universidade do PortoViana do CasteloPortugal

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