Linearization of Arbitrary products of classical orthogonal polynomials

Mahouton Hounkonnou; Said Belmehdi; André Ronveaux

Applicationes Mathematicae (2000)

  • Volume: 27, Issue: 2, page 187-196
  • ISSN: 1233-7234

Abstract

top
A procedure is proposed in order to expand w = ∏ j = 1 N P i j ( x ) = ∑ k = 0 M L k P k ( x ) where P i ( x ) belongs to aclassical orthogonal polynomial sequence (Jacobi, Bessel, Laguerre and Hermite) ( M = ∑ j = 1 N i j ). We first derive a linear differential equation of order 2 N satisfied by w, fromwhich we deduce a recurrence relation in k for the linearizationcoefficients L k . We develop in detail the two cases [ P i ( x ) ] N , P i ( x ) P j ( x ) P k ( x ) and give the recurrencerelation in some cases (N=3,4), when the polynomials P i ( x ) are monic Hermite orthogonal polynomials.

How to cite

top

Hounkonnou, Mahouton, Belmehdi, Said, and Ronveaux, André. "Linearization of Arbitrary products of classical orthogonal polynomials." Applicationes Mathematicae 27.2 (2000): 187-196. <http://eudml.org/doc/219266>.

@article{Hounkonnou2000,
abstract = {A procedure is proposed in order to expand $w=\prod ^N_\{j=1\} P_\{i_j\}(x)=\sum ^M_\{k=0\} L_ k P_ k(x)$ where $P_i(x)$ belongs to aclassical orthogonal polynomial sequence (Jacobi, Bessel, Laguerre and Hermite) ($M=\sum ^N_\{j=1\} i_j$). We first derive a linear differential equation of order $2^N$ satisfied by w, fromwhich we deduce a recurrence relation in k for the linearizationcoefficients $L_k$. We develop in detail the two cases $[P_i(x)]^N$, $P_ i(x)P_ j(x)P_ k(x)$ and give the recurrencerelation in some cases (N=3,4), when the polynomials $P_i(x)$are monic Hermite orthogonal polynomials.},
author = {Hounkonnou, Mahouton, Belmehdi, Said, Ronveaux, André},
journal = {Applicationes Mathematicae},
keywords = {classical orthogonal polynomials; Hermite orthogonal polynomials; linearization coefficients; recurrence relations; differential equations},
language = {eng},
number = {2},
pages = {187-196},
title = {Linearization of Arbitrary products of classical orthogonal polynomials},
url = {http://eudml.org/doc/219266},
volume = {27},
year = {2000},
}

TY - JOUR
AU - Hounkonnou, Mahouton
AU - Belmehdi, Said
AU - Ronveaux, André
TI - Linearization of Arbitrary products of classical orthogonal polynomials
JO - Applicationes Mathematicae
PY - 2000
VL - 27
IS - 2
SP - 187
EP - 196
AB - A procedure is proposed in order to expand $w=\prod ^N_{j=1} P_{i_j}(x)=\sum ^M_{k=0} L_ k P_ k(x)$ where $P_i(x)$ belongs to aclassical orthogonal polynomial sequence (Jacobi, Bessel, Laguerre and Hermite) ($M=\sum ^N_{j=1} i_j$). We first derive a linear differential equation of order $2^N$ satisfied by w, fromwhich we deduce a recurrence relation in k for the linearizationcoefficients $L_k$. We develop in detail the two cases $[P_i(x)]^N$, $P_ i(x)P_ j(x)P_ k(x)$ and give the recurrencerelation in some cases (N=3,4), when the polynomials $P_i(x)$are monic Hermite orthogonal polynomials.
LA - eng
KW - classical orthogonal polynomials; Hermite orthogonal polynomials; linearization coefficients; recurrence relations; differential equations
UR - http://eudml.org/doc/219266
ER -

References

top
  1. [1] R. Askey, Orthogonal Polynomials and SpecialFunctions, Regional Conf. Ser. Appl. Math. 21, SIAM, 1975, 39-46. 
  2. [2] S. Belmehdi, S. Lewanowicz and A. Ronveaux, Linearizationof product of orthogonal polynomials of a discrete variable, Appl. Math. (Warsaw) 24 (1997), 445-455. Zbl0891.33006
  3. [3] T. S. Chihara, An Introduction toOrthogonal Polynomials, Gordon and Breach, New York, 1978. 
  4. [4] E. Feldheim, Quelques nouvelles relations pour les polynômes d'Hermite, J. London Math. Soc. 13 (1938), 22-29. Zbl64.0354.05
  5. [5] E. Godoy, I. Area, A. Ronveaux and A. Zarzo, Minimalrecurrence relations for connection coefficients between classical orthogonalpolynomials: Continuous case, J. Comput. Appl. Math. 84 (1997), 257-275. Zbl0909.65008
  6. [6] E. W. Hobson, The Theory of Spherical andEllipsoidal Harmonics, Chelsea, New York, 1965. 
  7. [7] R. Hylleraas, Linearization of products ofJacobi polynomials, Math. Scand. 10 (1962), 189-200. Zbl0109.29603
  8. [8] R. Lasser, Linearization of the product ofassociated Legendre polynomials, SIAM J. Math. Anal. 14 (1983), 403-408. Zbl0509.33007
  9. [9] S. Lewanowicz, Second-order recurrence relationfor the linearization coefficients of the classical polynomials, J.Comput. Appl. Math. 69 (1994), 159-170. Zbl0885.33003
  10. [10] S. Lewanowicz and A. Ronveaux, Linearization of powers of classicalorthogonal polynomial of a discrete variable, J. Math. Phys. Sci. (Madras), in print. 
  11. [11] A. Nikiforov et V. Ouvarov, 
  12. [12] Élémentsde la Théorie des Fonctions Spéciales, Mir, Moscow, 1976. 
  13. [13] A. Ronveaux, Orthogonal polynomials: Connection andlinearization coefficients, in: Proc. International Workshop onOrthogonal Polynomials in Mathematical Physics in honour of Professor André Ronveaux (Leganes, Universidad Carlos III, Madrid, 1996), M. Alfaro et al. (eds.), 131-142. Zbl0928.33009
  14. [14] A. Ronveaux, Some 4th order differentialequations related to classical orthogonal polynomials, in: Sobre polynomios orthogonales y applicationes(Vigo, 1988), A. Cachafeiro and E. Godoy (eds.), Esc. Tec. Super. Ing. Ind. Vigo, 1989, 159-169. 
  15. [15] A. Ronveaux, S. Belmehdi, E. Godoy and A. Zarzo, Recurrence relations approach for connection coefficients. Applications toclassical discrete orthogonal polynomials, in: CRM Proc. Lecture Notes 9, Amer. Math. Soc., 1996, 319-335. Zbl0862.33006
  16. [16] A. Ronveaux, E. Godoy and A. Zarzo, Recurrencerelations for connection coefficients between two families of orthogonalpolynomials, J. Comput. Appl. Math. 62 (1995), 67-73. Zbl0876.65005
  17. [17] A. Ronveaux, M. N. Hounkonnou and S. Belmehdi, Recurrence relations between linearization coefficients oforthogonal polynomials, Report Laboratoire de PhysiqueMathématique FUNDP, Namur, 1993. 
  18. [18] A. Ronveaux, M. N. Hounkonnou and S. Belmehdi, Generalized linearization problems, J. Phys. A. 28 (1995), 4423-4430. Zbl0867.33003
  19. [19] M. E. Rose, Elementary Theory of AngularMomentum, Wiley, New York, 1957. 
  20. [20] I. A. Šapkrarev, Über lineare Differentialgleichungenmit der Eigenschaft dass k-te Potenzen der Integrale einer linearenDifferentiagleichung zweiter Ordnung ihre Integrale sind, Mat. Vesnik(4) 19 (1967), 67-70. 
  21. [21] R. Szwarc, Linearization and connectioncoefficients of orthogonal polynomials, Monatsh. Math. 113 (1992), 319-29. Zbl0766.33008
  22. [22] A. Zarzo, I. Area, E. Godoy and A. Ronveaux, Resultsfor some inversion problems for classical continuous and discrete orthogonalpolynomials, J. Phys. A. 30 (1997), L35-L40. 

NotesEmbed ?

top

You must be logged in to post comments.