On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials

Journal of Physics A: Mathematical and General

Volumn 37, Issue 3, 2004, Pages 657-675

  Doha, E H ^a  

^a CAIRO UNIVERSITY   (Egypt)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0742271179     PISSN: 03054470     EISSN: None     Source Type: Journal    
DOI: 10.1088/0305-4470/37/3/010     Document Type: Article

References (58)

1. Abramowitz M and Stegun I A (ed) 1970 Handbook of Mathematical Functions (Appl. Maths. Series vol 55) (New York: National Bureau of Standards)
2. 2s J. Phys. A: Math. Gen. 29 7165-75
3. Álvarez-Nodarse R, Yáñez R J and Dehesa J S 1998 Modified Clebsch-Gordan-type expansions for products of discrete hypergeometric polynomials J. Comput. Appl. Math. 89 171-97
4. Andrews G E, Askey R and Roy R 1999 Special Functions (Cambridge: Cambridge University Press)
5. Area I, Godoy E, Ronveaux A and Zarzo A 1998 Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: discrete case J. Comput. Appl. Math. 89 309-25
6. Askey R and Gasper G 1971 Jacobi polynomials expansions of Jacobi polynomials with non-negative coefficients Proc. Camb. Phil Soc. 70 243-55
7. Ben-yu G 1998a Gegenbauer approximation and its applications to differential equations on the whole line J. Math. Anal. Appl. 226 180-206
8. Ben-yu G 1998b Spectral Methods and Their Applications (London: World Scientific)
9. Canuto C, Hussaini M Y, Quarteroni A and Zang T A 1988 Spectral Methods in Fluid Dynamics (Berlin: Springer)
10. Coutsias E A, Hagstrom T and Torres D 1996 An efficient spectral method for ordinary differential equations with rational function coefficients Math. Comput. 65 611-35
11. Doha E H 1990 An accurate solution of parabolic equations by expansion in ultraspherical polynomials Comput. Math. Appl. 19 75-88
12. Doha E H 1991 The coefficients of differentiated expansions and derivatives of ultraspherical polynomials J. Comput. Math. Appl. 21 115-22
13. Doha E H 1994 The first and second kinds Chebyshev coefficients of the moments of the general-order derivative of an infinitely differentiable function Int. J. Comput. Math. 51 21-35
14. Doha E H 1998 The ultraspherical coefficients of the moments of a general-order derivative of an infinitely differentiable function J. Comput. Appl. Math. 89 53-72
15. Doha E H 2000 The coefficients of differentiated expansions of double and triple ultraspherical polynomials Ann. Univ. Sci. Budepest Sect. Comput. 19 57-73
16. Doha E H 2002a On the coefficients of differentiated expansions and derivatives of Jacobi polynomials J. Phys. A: Math. Gen. 35 3467-78
17. Doha E H 2002b On the coefficients of integrated expansions and integrals of ultraspherical polynomials and their applications for solving differential equations J. Comput. Appl. Math. 139 275-98
18. Doha E H 2003a Explicit formulae for the coefficients of integrated expansions for Jacobi polynomials and their integrals Integral Transforms Spec. Funct. 14 69-86
19. Doha E H 2003b On the connection coefficients and recurrence relations arising from expansions in series of Laguerre polynomials J. Phys. A: Math. Gen. 36 5449-62
20. Doha E H 2003c On the connection coefficients and recurrence relations arising from expansions in series of Hermite polynomials Integral Transforms Spec. Funct. at press
21. Doha E H and Abd-Elhameed W M 2002 Efficient spectral Galerkin-algorithms for direct solution of second-order equations using ultraspherical polynomials SIAM J. Sci. Comput. 24 548-71
22. Doha E H and Al-Kholi F M R 2001 An efficient double Legendre spectral method for parabolic and elliptic partial differential equations Int. J. Comput. Math. 78 413-32
23. Doha E H and El-Soubhy S I 1995 On the Legendre coefficients of the moments of the general-order derivative of an infinitely differentiable function Int. J. Comput. Math. 56 107-22
24. Doha E H and Helal M A 1997 An accurate double Chebyshev spectral approximation for parabolic partial differential equations J. Egypt. Math. Soc. 5 83-101
25. Fox L and Parker I B 1972 Chebyshev Polynomials in Numerical Analysis (London: Oxford University Press)
26. Gasper G 1974 Projection formulas for orthogonal polynomials of a discrete variable J. Math. Anal. Appl. 45 176-98
27. Godoy E, Ronveaux A, Zarzo A and Area I 1997 Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: continuous case J. Comput. Appl. Math. 84 257-75
28. Gottlieb D and Orszag S A 1977 Numerical analysis of spectral methods: theory and applications CBMS - NSF Regional Conf. Series in Applied Mathematics (Philadelphia, PA: SIAM)
29. Haidvogel D B and Zang T 1979 The accurate solution of Poisson's equation by Chebyshev polynomials J. Comput. Phys. 30 167-80
30. Jirari A 1995 Second-order Sturm-Liouville difference equations and orthogonal polynomials Mem. Am. Math. Soc. 113
31. Karageorghis A 1988a A note on the Chebyshev coefficients of the general-order derivative of an infinitely differentiable function J. Comput. Appl. Math. 21 129-32
32. Karageorghis A 1988b A note on the Chebyshev coefficients of the moments of the general-order derivative of infinitely differentiable function J. Comput. Appl. Math. 21 383-6
33. Karageorghis A and Phillips T N 1989 On the coefficients of differentiated expansions of ultraspherical polynomials ICASE Report No 89-65 (Hampton, VA: NASA Langley Research Center)
34. Karageorghis A and Phillips T N 1992 Appl. Numer. Math. 9 133-41
35. Koepf W and Schmersau D 1998 Representations of orthogonal polynomials J. Comput. Appl. Math. 90 57-94
36. Lewanowicz S 1986 Recurrence relations for the coefficients in Jacobi series solutions of linear differential equations SIAM J. Math. Anal. 17 1037-52
37. Lewanowicz S 1991 A new approach to the problem of constructing recurrence relations for the Jacobi coefficients Appl. Math. 21 303-26
38. Lewanowicz S 1992 Quick construction of recurrence relations for the Jacobi coefficients J. Comput. Appl. Math. 43 355-72
39. Lewanowicz S 2002 Recurrences of the coefficients of series expansions with respect to classical orthogonal polynomials Appl. Math. 29 97-116
40. Lewanowicz S, Godoy E, Area I, Ronveaux A and Zarzo A 2000 Recurrence relations for the coefficients of the Fourier series expansions with respect to q-classical orthogonal polynomials Numer. Algorithms 23 31-50
41. Lewanowicz S and Woźny P 2001 Algorithms for construction of recurrence relations for the coefficients of expansions in series of classical orthogonal polynomials Preprint Institute of Computer Science, University of Wroclaw, online at http://www.ii.uni.wroc.pl/sle/publ.html
42. Luke Y 1969 The Special Functions and Their Applications vol 1 (New York: Academic)
43. Luke Y 1975 Mathematical Functions and Their Approximations (New York: Academic)
44. Oliver J 1969 The numerical solution of linear recurrence relations Numer. Math. 11 349-60
45. Peyret R 2002 Spectral Methods for Incompressible Viscous Flow (Berlin: Springer)
46. Phillips T N 1988 On the Legendre coefficients of a general-order derivative of an infinitely differentials function IMA. J. Numer. Anal 8 455-9
47. Phillips T N and Karageorghis A 1990 On the coefficients of integrated expansions of ultraspherical polynomials SIAM J. Numer. Anal. 8 823-30
48. Rainville E D 1960 Special Functions (New York: Macmillan)
49. Ronveaux A, Belmehdi S, Godoy E and Zarzo A 1996 Recurrence relation approach for connection coefficients. Applications to classical discrete orthogonal polynomials CRM Proc. Lecture Notes 9 319-35
50. Ronveaux A, Zarzo A and Godoy E 1995 Recurrence relations for connection coefficients between two families of orthogonal polynomials J. Comput. Appl. Math. 62 67-73
51. Sánchez-Ruiz J and Dehesa J S 1998 Expansions in series of orthogonal polynomials J. Comput. Appl. Math. 89 155-70
52. Scraton R E 1972 A modification of Miller's recurrence algorithms BIT 12 242-51
53. Siyyam H I and Syam M I 1997 An accurate solution of the Poisson equation by the Chebyshev-Tau method J. Comput. Appl. Math. 85 1-10
54. Szegö G 1985 Orthogonal polynomials Am. Math. Soc. Coll. Publ. 23 58-90
55. Voigt R G, Gottlieb D and Hussaini M Y 1984 Spectral Methods for Partial Differential Equations (Philadelphia, PA: SIAM)
56. Weixlbaumer C 2001 Solutions of Difference Equations with Polynomial Coefficients (Australia: RISC Linz Johannes Kepler Universität)
57. Wimp J 1984 Computations With Recurrence Relations (London: Pitman)
58. Wolfram Research Inc. 1999 Mathematica Version 4.0 Wolfram Research, Champaign, IL