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The Ramanujan Journal

, Volume 39, Issue 3, pp 497–531 | Cite as

Coefficients of multiplication formulas for classical orthogonal polynomials

  • D. D. Tcheutia
  • M. Foupouagnigni
  • W. KoepfEmail author
  • P. Njionou Sadjang
Article

Abstract

In this paper using both analytic and algorithmic approaches, we derive the coefficients \(D_m(n,a)\) of the multiplication formula
$$\begin{aligned} p_n(ax)=\sum _{m=0}^nD_m(n,a)p_m(x) \end{aligned}$$
or the translation formula
$$\begin{aligned} p_n(x+a)=\sum _{m=0}^nD_m(n,a)p_m(x), \end{aligned}$$
where \(\{p_n\}_{n\ge 0}\) is an orthogonal polynomial set, including the classical continuous orthogonal polynomials, the classical discrete orthogonal polynomials, the \(q\)-classical orthogonal polynomials, as well as the classical orthogonal polynomials on a quadratic lattice and a \(q\)-quadratic lattice. We give a representation of the coefficients \(D_m(n,a)\) as a single, double or triple sum whereas in many cases we get simple representations.

Keywords

Orthogonal polynomials Hypergeometric representation  \(q\)-Hypergeometric representation Inversion coefficients  Multiplication coefficients Translation coefficients 

Mathematics Subject Classification

33C45 33D45 33D15 33F10 68W30 

Notes

Acknowledgments

We would like to thank the three anonymous referees of this paper for very carefully reading the manuscript, and also for their valuable comments and suggestions which improved the paper significantly.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • D. D. Tcheutia
    • 1
    • 2
  • M. Foupouagnigni
    • 1
    • 2
  • W. Koepf
    • 3
    Email author
  • P. Njionou Sadjang
    • 1
    • 2
  1. 1.Department of Mathematics, Higher Teachers’ Training CollegeUniversity of Yaounde IYaoundéCameroon
  2. 2.African Institute for Mathematical SciencesLimbéCameroon
  3. 3.Institute of MathematicsUniversity of KasselKasselGermany

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