Linearization of the product of orthogonal polynomials of a discrete variable

Saïd Belmehdi; Stanisław Lewanowicz; André Ronveaux

Applicationes Mathematicae (1997)

  • Volume: 24, Issue: 4, page 445-455
  • ISSN: 1233-7234

Abstract

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Let P k be any sequence of classical orthogonal polynomials of a discrete variable. We give explicitly a recurrence relation (in k) for the coefficients in P i P j = ∑ k c ( i , j , k ) P k , in terms of the coefficients σ and τ of the Pearson equation satisfied by the weight function ϱ, and the coefficients of the three-term recurrence relation and of two structure relations obeyed by P k .

How to cite

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Belmehdi, Saïd, Lewanowicz, Stanisław, and Ronveaux, André. "Linearization of the product of orthogonal polynomials of a discrete variable." Applicationes Mathematicae 24.4 (1997): 445-455. <http://eudml.org/doc/219184>.

@article{Belmehdi1997,
abstract = {Let $P_k$ be any sequence of classical orthogonal polynomials of a discrete variable. We give explicitly a recurrence relation (in k) for the coefficients in $P_iP_j=\sum _kc(i,j,k)P_k$, in terms of the coefficients σ and τ of the Pearson equation satisfied by the weight function ϱ, and the coefficients of the three-term recurrence relation and of two structure relations obeyed by $P_k$.},
author = {Belmehdi, Saïd, Lewanowicz, Stanisław, Ronveaux, André},
journal = {Applicationes Mathematicae},
keywords = {linearization coefficients; classical orthogonal polynomials of a discrete variable; recurrence relations},
language = {eng},
number = {4},
pages = {445-455},
title = {Linearization of the product of orthogonal polynomials of a discrete variable},
url = {http://eudml.org/doc/219184},
volume = {24},
year = {1997},
}

TY - JOUR
AU - Belmehdi, Saïd
AU - Lewanowicz, Stanisław
AU - Ronveaux, André
TI - Linearization of the product of orthogonal polynomials of a discrete variable
JO - Applicationes Mathematicae
PY - 1997
VL - 24
IS - 4
SP - 445
EP - 455
AB - Let $P_k$ be any sequence of classical orthogonal polynomials of a discrete variable. We give explicitly a recurrence relation (in k) for the coefficients in $P_iP_j=\sum _kc(i,j,k)P_k$, in terms of the coefficients σ and τ of the Pearson equation satisfied by the weight function ϱ, and the coefficients of the three-term recurrence relation and of two structure relations obeyed by $P_k$.
LA - eng
KW - linearization coefficients; classical orthogonal polynomials of a discrete variable; recurrence relations
UR - http://eudml.org/doc/219184
ER -

References

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  1. R. Askey, Orthogonal Polynomials and Special Functions, Regional Conf. Ser. Appl. Math. 21, SIAM, Philadelphia, 1975. 
  2. R. Askey and G. Gasper, Convolution structures for Laguerre polynomials, J. Anal. Math. 31 (1977), 48-68. Zbl0347.33006
  3. B. W. Char, K. O. Geddes, G. H. Gonnet, B. L. Leong, M. B. Monagan and S. M. Watt, Maple V Language Reference Manual, Springer, New York, 1991. 
  4. T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978. Zbl0389.33008
  5. A. G. Garcia, F. Marcellán and L. Salto, A distributional study of discrete classical orthogonal polynomials, J. Comput. Appl. Math. 57 (1995), 147-162. Zbl0853.33009
  6. R. Koekoek and R. F. Swarttouw, The Askey scheme of hypergeometric orthogonal polynomials and its q-analogue, Fac. Techn. Math. Informatics, Delft Univ. of Technology, Rep. 94-05, Delft, 1994. 
  7. J. Letessier, A. Ronveaux and G. Valent, Fourth order difference equation for the associated Meixner and Charlier polynomials, J. Comput. Appl. Math. 71 (1996), 331-341. Zbl0856.33008
  8. S. Lewanowicz, Recurrence relations for the connection coefficients of orthogonal polynomials of a discrete variable, ibid. 76 (1996), 213-229. Zbl0877.33003
  9. A. F. Nikiforov, S. K. Suslov and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer, Berlin, 1991. Zbl0743.33001
  10. A. Ronveaux, S. Belmehdi, E. Godoy and A. Zarzo, Recurrence relations approach for connection coefficients-Applications to classical discrete orthogonal polynomials, in: Symmetries and Integrability of Difference Equations, D. Levi, L. Vinet and P. Winternitz (eds.), Centre de Recherches Mathématiques, CRM Proc. and Lecture Notes Ser. 9, Amer. Math. Soc., Providence, 1996, 321-337. Zbl0862.33006

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