Differential Equation for Classical-Type Orthogonal Polynomials

Ronveaux, A. ; Marcellan, F.

Canadian mathematical bulletin, Tome 32 (1989) no. 4, pp. 404-411

Résumé

The second order differential equation of Littlejohn-Shore for Laguerre type orthogonal polynomials is generalized in two ways. First the positive Dirac mass can be situated at any point and secondly the weight can be any classical weight modified by an arbitrary number of Dirac distributions.

Détail

DOI : 10.4153/CMB-1989-058-5

Mots-clés : 33A65, 42C05

Bibliographie

[1] 1. Cachafeiro, A. and F. Marcellan, Polinomios ortogonales y medidas singulares sobre curvas, in Actas Jornadas Matematicas Hispano Lusas Badajoz, (1986) p. 128-139. Google Scholar

[2] 2. Chihara, T. S., Orthogonal polynomials and measures with end point masses, Rocky Mountain J. of Math. 15, 3, (1985), p. 705–719. Google Scholar

[3] 3. Gautschi, W., A survey of Gauss Christoffel quadrature formulae, Christoffel Symposium, E. Butzer, Birkhàuser Verlag (1981), p. 73–145. Google Scholar

[4] 4. Hendriksen, E. and H. Van Rossum, Semiclassical orthogonal polynomials, Lect. Notes in Math., Springer Verlag (1985) Vol. 1171, Cl. Brezinski et coll. éd., p. 354–361. Google Scholar

[5] 5. Koornwinder, T. H., Orthogonal polynomials with weight function (1 — x)a(\ + x)@ + M6(x + 1) + N6(x - 1), Canad. Math. Bull. 27 (2) (1984), p. 205–214. Google Scholar

[6] 6. Laguerre, E. N., Sur la réduction en fractions continues d'une fraction qui satisfait à une équation différentielle donnée, J. Math, pures et appl., 1 (1885), p. 135-165. Google Scholar

[7] 7. Littlejohn, L. L. and Shore, S. D., Non classical orthogonal polynomials as solutions to second order differential equations, Canad. Math. Bull. 25 (3), (1982), p. 291–295. Google Scholar

[8] 8. Maroni, P., Une caractérisation des polynômes orthogonaux semi-classiques, C.R. Acad. Sci. Paris, 301, série 1, (1985), p. 269–272. Google Scholar

[9] 9. Nevai, P., Orthogonal polynomials. Memoirs, Amer. Math. Soc, 213, Amer. Math. Soc, Providence, R.I., (1979). Google Scholar

[10] 10. Nikiforov, A. et V. Ouvarov, Eléments de la théorie des fonctions spéciales, Mir, Moscou, (1978). Google Scholar

[11] 11. Nikiforov, A. et Ouvarov, V., Fonctions spéciales de la Physique mathématique, Mir, Moscou, (1983). Google Scholar

[12] 12. Ronveaux, A., Sur l'équation différentielle du second ordre satisfaite par une classe de polynômes othogonaux semi-classiques, C.R. Acad. Sci. Paris, 305, Série 1, (1987) p. 163–166. Google Scholar

[13] 13. Ronveaux, A. and Thiry, G., Differential equations of some orthogonal families in Reduce. J. of Symbolic Comp., in print. Google Scholar

[14] 14. Shohat, J., A differential equation for orthogonal polynomials, Duke Math. J. 5 (1939), p. 401–417. Google Scholar

[15] 15. Uvarov, V. B., Relation between polynomials orthogonal with different weights, Dokl. Akad. Nauk, USSR, 126, p. 33–36 (Russian), (1959). Google Scholar

[16] 16. Szegô, G., Orthogonal polynomials, fourth edition, Amer. Math. Soc, Providence, R.I., (1975). Google Scholar