PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2000 | 27 | 2 | 187-196
Tytuł artykułu

Linearization of Arbitrary products of classical orthogonal polynomials

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Rocznik
Tom
27
Numer
2
Strony
187-196
Opis fizyczny
Daty
wydano
2000
otrzymano
1998-09-30
poprawiono
1999-05-18
Twórcy
  • Unité de Recherche en Physique Théorique (URPT), Institut de Mathématique et de Sciences Physiques (IMSP), Université Nationale du Benin, BP 613 Porto-Novo, Benin
  • UFR de Mathématique, Université des Sciences et Technologies, Lille I, F-59655 Villeneuve d'Ascq, France
  • Facultés Universitaires Notre-Dame de la Paix, rue de Bruxelles, 61, B-5000 Namur, Belgium
Bibliografia
  • [1] R. Askey, Orthogonal Polynomials and SpecialFunctions, Regional Conf. Ser. Appl. Math. 21, SIAM, 1975, 39-46.
  • [2] S. Belmehdi, S. Lewanowicz and A. Ronveaux, Linearizationof product of orthogonal polynomials of a discrete variable, Appl. Math. (Warsaw) 24 (1997), 445-455.
  • [3] T. S. Chihara, An Introduction toOrthogonal Polynomials, Gordon and Breach, New York, 1978.
  • [4] E. Feldheim, Quelques nouvelles relations pour les polynômes d'Hermite, J. London Math. Soc. 13 (1938), 22-29.
  • [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.
  • [6] E. W. Hobson, The Theory of Spherical andEllipsoidal Harmonics, Chelsea, New York, 1965.
  • [7] R. Hylleraas, Linearization of products ofJacobi polynomials, Math. Scand. 10 (1962), 189-200.
  • [8] R. Lasser, Linearization of the product ofassociated Legendre polynomials, SIAM J. Math. Anal. 14 (1983), 403-408.
  • [9] S. Lewanowicz, Second-order recurrence relationfor the linearization coefficients of the classical polynomials, J.Comput. Appl. Math. 69 (1994), 159-170.
  • [10] S. Lewanowicz and A. Ronveaux, Linearization of powers of classicalorthogonal polynomial of a discrete variable, J. Math. Phys. Sci. (Madras), in print.
  • [11] A. Nikiforov et V. Ouvarov,
  • [12] Élémentsde la Théorie des Fonctions Spéciales, Mir, Moscow, 1976.
  • [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.
  • [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] 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.
  • [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.
  • [17] A. Ronveaux, M. N. Hounkonnou and S. Belmehdi, Recurrence relations between linearization coefficients oforthogonal polynomials, Report Laboratoire de PhysiqueMathématique FUNDP, Namur, 1993.
  • [18] A. Ronveaux, M. N. Hounkonnou and S. Belmehdi, Generalized linearization problems, J. Phys. A. 28 (1995), 4423-4430.
  • [19] M. E. Rose, Elementary Theory of AngularMomentum, Wiley, New York, 1957.
  • [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] R. Szwarc, Linearization and connectioncoefficients of orthogonal polynomials, Monatsh. Math. 113 (1992), 319-29.
  • [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.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-zmv27i2p187bwm
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.