Modified Clebsch-Gordan-type expansions for products of discrete hypergeometric polynomials

Journal of Computational and Applied Mathematics

Volumn 89, Issue 1, 1998, Pages 171-197

  Álvarez Nodarse, Renato ^a,b   Yanez R J ^b   Dehesa, Jesús S ^b  

^a UNIVERSIDAD CARLOS III DE MADRID   (Spain)

^b UNIVERSITY OF GRANADA   (Spain)

Author keywords

Connection and linearization problems; Discrete inversion formulas; Discrete polynomials; Second order difference equations

Indexed keywords

COMPUTATIONAL METHODS; DIFFERENCE EQUATIONS; INVERSE PROBLEMS; LINEARIZATION; SET THEORY;

DISCRETE INVERSION FORMULAS; DISCRETE POLYNOMIALS; SECOND ORDER DIFFERENCE EQUATIONS;

POLYNOMIALS;

EID: 0032498749     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0427(97)00244-6     Document Type: Article

References (55)

1. [1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1964.
2. [2] R. Álvarez-Nodarse, Polinomios hipergeométricos y q-polinomios, Monografías de la Academia de Ciencias de Zaragoza, Spain, 1998, to appear.
3. 2s. J. Phys. A 29 (1996) 7165-7175.
4. [4] R. Álvarez-Nodarse, J.S. Dehesa, Linearization formulas between orthogonal q-polynomials, in preparation.
5. [5] I. Area, E. Godoy, A. Ronveaux, A. Zarzo, Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: discrete case. J. Comput. Appl. Math. (1997) submitted.
6. [6] P.L. Artes, J.S. Dehesa, A. Martinez-Finkelshtein, J. Sánchez-Ruiz, Linearization and connection coefficients for hypergeometric type polynomials. Preprint 1997, submitted.
7. [7] T. Asahi, Spectral theory of difference equations, Prog. Theor. Phys. (Suppl.) 36 (1966) 55-96.
8. [8] R. Askey, Orthogonal Polynomials and Special Functions, Regional Conf. in Applied Mathematics, vol. 21, SIAM. Philadelphia, PA, 1975.
9. [9] R. Askey, Orthogonal expansions with positive coefficients, Proc. Amer. Math. Soc. 26 (1965) 1191-1194.
10. [10] R. Askey, G. Gasper, Convolution structures for Laguerre polynomials, J. d'Analyse Math. 31 (1977) 48-68.
11. [11] S. Belmehdi, S. Lewanowicz, A. Ronveaux, Linearization of products of orthogonal polynomials of a discrete variable. Applicationes Mathematicae 24 (1997) 445-455.
12. [12] N. Bogdanova, V. Gadjokov, Use of orthogonal polynomials in calibration problems, Comput. Phys. Commun. 24 (1981) 225-229.
13. [13] C. Dunkl, D.E. Ramirez, Krawtchouk polynomials and the symmetrization of hypergroups, SIAM J. Math. Anal. 179 (1974) 351-366.
14. [14] C.F. Dunkl, Spherical functions on compact groups and applications to special functions, Symp. Math. 22 (1977) 145-161.
15. [15] A.R.M. Edmonds, Angular Momentum in Quantum Mechanics, Princeton Univ. Press, Princeton, 1957.
16. [16] J.L. Fields, J. Wimp, Expansion of hypergeometric functions in hypergeometric functions, Math. Comput. 15 (1961) 390-395.
17. [17] G. Gasper, Projection formulas for orthogonal polynomials of a discrete variable, J. Math. Anal. Appl. 45 (1974) 176-198.
18. [18] G. Gasper, Positivity and special functions, in: R. Askey (Ed.), Theory and Applications of Special Functions, Academic Press, New York, 1975, pp. 375-433
19. [19] E. Godoy, A. Ronveaux, A. Zarzo, I. Area, Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: continuous case, J. Comput. Appl. Math. 84 (1997) 257-275.
20. [20] E. Godoy, A. Ronveaux, A. Zarzo, I. Area, On the limit relations between classical continuous and discrete orthogonal polynomials, Preprint (1997).
21. [21] A. Jirari, Second-order Sturm-Liuoville Difference Equations and Orthogonal Polynomials, Memoirs AMS, vol. 113, Amer. Math. Soc., Providence. RI, 1995.
22. [22] E.R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975.
23. [23] M.N. Hounkonnou, S. Belmedi, A. Ronveaux, Linearization of arbitrary products of classical orthogonal polynomials, Preprint (1997), submitted.
24. [24] H.F. King, M. Dupuis, Numerical integration using Rys polynomials, J. Comput. Phys. 21 (1976) 144-165.
25. [25] W. Koepf, D. Schmersau, Representation of orthogonal polynomials, Preprint ZIB-Berlin SC 97-06 (1997).
26. [26] P.A. Lee, An integral representation and some summation formulas for the Hahn polynomials, SIAM J. Appl. Math. 19 (1970) 266-272.
27. [27] P.A. Lee, Explicit formula for the multiple generating function of product of generalized Laguerre polynomials, J. Phys. A 30 (1997) L183-L186.
28. [28] P. Lesky, Über Polynomlösungen von Differentialgleichungen und Differenzengleichungen zweiter Ordnung, Anzeiger der österreichischen Akad. Wissenschaften, math.-naturwiss, Klaese 121 (1985) 29-33.
29. [29] S. Lewanowicz, Recurrence relations for hypergcometric functions of unit argument, Math. Comput. 45 (1985) 521-535, Errata 48 (1987) 853.
30. [30] S. Lewanowicz, Recurrence relations for the connection coefficients of orthogonal polynomials of a discrete variable, J. Comput. Appl. Math. 76 (1996) 213-229.
31. [31] S. Lewanowicz, A. Ronveaux, Linearization of powers of an orthogonal polynomial of a discrete variable, Preprint (1996).
32. [32] S. Lewanowicz, Second-order recurrence relations for the linearization coefficients of the classical orthogonal polynomials, J. Comput. Appl. Math. 69 (1996) 159-160.
33. [33] S. Lewanowicz, Recurrence relations for the connection coefficients of orthogonal polynomials of a discrete variable on the lattice x(s) = exp(2ws), Preprint (1997).
34. [34] S. Lewanowicz, A general approach to the connection and linearization problems for the classical orthogonal polynomials, Preprint (1997).
35. [35] S. Lewanowicz, The hypergeometric functions approach to the connection problem for the classical orthogonal polynomials, Preprint (1997) and Private communication.
36. [36] Y.L. Luke, The Special Functions and their Approximations, Academic Press, New York, 1969.
37. [37] C. Markett, Linearization of the product of symmetric orthogonal polynomials, Constr. Approx. 10 (1994) 317-338.
38. [38] A. Messina, E. Paladino, An operator approach to the construction of generating function for products of associated Laguerre polynomials, J. Phys. A 29 (1996) L263-L270.
39. [39] R.E. Mickens, Exact difference schemes for the wave equations with spherical symmetry, J. Difference Equations Appl. 2(3) (1997) 560-566.
40. [40] A.F. Nikiforov, V.B. Uvarov, Special Functions in Mathematical Physics, Birkhauser, Basel, 1988.
41. [41] A.F. Nikiforov, S.K. Suslov, V.B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer, Berlin, 1991.
42. [42] A.F. Nikiforov, V.B. Uvarov, Polynomial solutions of hypergeometric type difference equations and their classification, Integral Transforms and Special Functions, 1 (1993) 223-249.
43. [43] A.W. Niukkanen, Clebsch-Gordan-type linearization relations for the products of Laguerre and hydrogen-like functions, J. Phys. A 18 (1985) 1399-1437.
44. [44] M. Petkovšek, Hypergeometric solutions of difference equations with polynomial coefficients, J. Symbol. Comput. 14 (1992) 243-264.
45. [45] A. Ronveaux, I. Area, E. Godoy, A. Zarzo, Lectures on recursive approach to connection and linearization coefficients between polynomials, in: Proc. Int. Workshop on Special Functions and Differential Equations, Madras, India, January 1997, pp. 13-24.
46. [46] A. Ronveaux, E. Godoy, A. Zarzo, First associated and co-recursive of classical discrete polynomials, Difference equation and connection problems, Preprint (1995), submitted.
47. [47] A. Ronveaux, A. Zarzo, E. Godoy, Recurrence relation for connection coefficients between two families of orthogonal polynomials, J. Comput. Appl. Math. 62 (1995) 67-73.
48. [48] A. Ronveaux, S. Belmehdi, E. Godoy, A. Zarzo, Recurrence relation for connection coefficients, Applications to classical discrete orthogonal polynomials, in: D. Levi, L. Vinet, P. Witerwitz (Eds.), Proc. Workshop on Symmetries and Integrability of Difference Equations, C.R.M. Proc. Lecture Notes Series, vol. 9, Providence, RI, 1996, pp. 321-337.
49. [49] E. Rosato, M. Vigilante, N. de Cesare, E. Perillo, G. Spadaccini, Weighted fit by orthogonal polynomials, Comput. Phys. Commun. 22 (1981) 311-315.
50. [50] J. Sánchez-Ruiz, J.S. Dehesa, Expansions in series of orthogonal hypergeometric polynomials, J. Comput. Appl. Math. 89 (1998) to appear.
51. [51] R. Szwarc, Linearization and connection coefficients of orthogonal polynomials, Mh. Math. 113 (1992) 319-329.
52. [52] R. Szwarc, Connection coefficients of orthogonal polynomials, Can. Math. Bull. 35 (4) (1992) 548-556.
53. [53] N.Ja. Vilenkin, A.U. Klimyk, Representations of Lie Groups and Special Functions, vol. II, Kluwer Academic Publishers, Dordrecht, 1992.
54. [54] J. Wimp, Recursion formulae for hypergeometric functions, Math. Comput. 22 (1968) 363-373.
55. [55] A. Zarzo, I. Area, E. Godoy, A. Ronveaux, Results on some inversion problems for classical continuous and discrete orthogonal polynomials, J. Phys. A 30 (1997) L35-L40.