Generalized Zernike or disc polynomials

Journal of Computational and Applied Mathematics

Volumn 174, Issue 1, 2005, Pages 135-163

  Wünsche, Alfred ^a  

^a HUMBOLDT UNIVERSITY   (Germany)

Author keywords

Jacobi polynomials; Laguerre 2D polynomials; Lie group SU(1,1,) and Lie algebra su(1,1); Lowering and raising operators; Recurrence relations; Rodrigues and operational definitions of dise polynomials

Indexed keywords

DIFFERENTIAL EQUATIONS; FUNCTIONS; MATHEMATICAL OPERATORS; MATHEMATICAL TRANSFORMATIONS; STANDARDIZATION; TWO DIMENSIONAL;

COMPLEX COORDINATES; DISC FUNCTIONS; JACOBI POLYNOMIALS; TWO DIMENSIONAL (2D) POLYNOMIALS;

POLYNOMIALS;

MATHEMATICAL METHOD;

EID: 8444225727     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2004.04.004     Document Type: Article

References (70)

1. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, National Bureau of Standards (series 55), New York, 1965.
2. Andrews G.E. Askey R. Roy R. Special Functions 1999 Cambridge University Press Cambridge
3. Appell P. Kampé de Fériet J. Fonctions hypergéométrique et hypersphérique, Polynomes d'Hermite 1926 Gauthier-Villars Paris
4. Barut A.O. Girardello L. Commun. Math. Phys. 21 1971 41 (republished in [60], p. 255)
5. Bezdidko S.N. Sov. J. Opt. Technol. 41 1974 425
6. Bhatia A.B. Wolf E. Proc. Camb. Phil. Soc. 50 1954 40
7. M. Born, E. Wolf, Principles of Optics, 7th Edition, Cambridge University Press, Cambridge, 1999 (first edition 1959).
8. Buchdahl H.A. An Introduction to Hamiltonian Optics 1970 Cambridge University Press Cambridge
9. R. Courant, D. Hilbert, Methoden der mathematischen Physik, (4. Aufl.), Springer, Berlin, 1993 (2nd Edition in two volumes 1930 and 1937, English Translation, Wiley-Interscience 1953 and 1962)
10. Dattoli G. Lorenzutta S. Maino G. Torre A. J. Math. Phys. 35 1994 4451
11. Dattoli G. Lorenzutta S. Maino G. Torre A. Ann. Numer. Math. 2 1995 211
12. Dattoli G. Torre A. Carpanese M. J. Math. Anal. Appl. 227 1998 98
13. Dattoli G. Torre A. Lorenzutta S. J. Math. Anal. Appl. 236 1999 399
14. Dodonov V.V. J. Phys. A 27 1994 6191
15. Dodonov V.V. Man'ko V.I. J. Math. Phys. 35 1994 4277
16. Dodonov V.V. Man'ko O.V. Man'ko V.I. Phys. Rev. A 49 1994 2993
17. Dodonov V.V. Man'ko O.V. Man'ko V.I. Phys. Rev. A 50 1994 813
18. Dunkl Ch.F. Xu Yuan Orthogonal Polynomials of Several Variables 2001 Cambridge University Press Cambridge
19. A. Erdélyi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1953.
20. Fernández F.M. Phys. Lett. A 237 1998 189
21. Fan Hong-yi Zou Hui Fan Yue Internat. J. Modern Phys. A 16 2001 369
22. Fan Hony-yi Ye Xiong Phys. Lett. A 175 1993 387
23. Fan Hong-yi Fan Yue Phys. Lett. A 246 1998 242
24. Fan Hong-yi Liu Zu-wei Ruan Tu-nan Commun. Theoret. Phys. (Beijing) 3 1984 175
25. N.Kh. Ibragimov, Transformation Groups in Mathematical Physics, Nauka, Moscow, 1983 (in Russian).
26. E. Inönü (Inonu), E.P. Wigner, Proc. Natl. Acad. Sci. (N.Y.) 39 (1953) 510
27. C.-J. Kim, R.R. Shannon, in: R. Shannon, J. Wyant (Eds.), Applied Optics and Optical Engineering, Vol. X, Academic Press, New York, 1987, 193-221.
28. J.R. Klauder, Bo-St. Skagerstam (Eds.), Coherent States (with a collection of reprinted papers), World Scientific, Singapore, 1985.
29. Koornwinder T.H. Two-variable analogues of the classical orthogonal polynomials In: Askey R.A. (Ed.) Theory and Application of Special Functions 1975 435-495 Academic Press New York
30. V.N. Mahajan, in: R.R. Shannon (Ed.), Engineering and Laboratory Notes, Opt. Soc. America, Vol. 17, 1994, p. S-21
31. V.N. Mahajan, Appl. Opt. 33 1994 8121
32. Man'ko V. Wünsche A. Quant. Semiclass. Opt. 9 1997 381
33. Miller W. Jr. Lie Theory of Special Functions 1968 Academic Press New York
34. Miller W. Jr. Symmetry, separation of variables, and special functions In: Askey R.A. (Ed.) Theory and Application of Special Functions 1975 305-351 Academic Press New York
35. Miller W. Jr. Symmetry and Separation of Variables 1977 Addison-Wesley Reading, MA
36. Myrick D.R. J. SIAM Appl. Math. 14 1966 476
37. Nieto M.M. Truax D.R. Phys. Rev. Lett. A 208 1995 8
38. B.R.A. Njiboer, Thesis, University of Groningen, 1942 (cited according to [5]).
39. Njiboer R.A. Physica 13 1947 605
40. Olver P.R. Applications of Lie Groups to Differential Equations 1986 Springer New York
41. L.V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982 (Russian origin, Nauka, Moscow, 1978)
42. Perelomov A.M. Commun. Math. Phys. 26 1972 222 (republished in [60], p. 281)
43. Perelomov A.M. Uspekhi Fiz. Nauk 123 1977 23 (in English: Sov. Phys. Usp. 20 (1977) 703)
44. Perelomov A. Generalized Coherent States 1986 Springer Berlin
45. Prata A. Rusch W.V.T. Appl. Opt. 28 1989 749
46. Rainville E.D. Special Functions 1960 Chelsea Publishing Company New York
47. Srivastava H.M. Manocha H.L. A Treatise On Generating Functions 1984 Ellis Horwood, Wiley New York
48. P.K. Suetin, Orthogonal Polynomials in Two Variables, Gordon and Breach, Amsterdam, 1999 (Russian original: Nauka, Moscow, 1988).
49. G. Szegö, Orthogonal Polynomials, 2nd Edition, American Mathematical Society, New York, 1959 (first Edition, 1939).
50. Torre A. Lorenzutta S. Maino G. Il Nuovo Cimento 111 B 1996 811
51. Vilenkin N.J. Special Functions and the Theory of Group Representation 1968 American Mathematical Society Providence, RI
52. N.Ja. Vilenkin, A.U. Klimyk, Representations of Lie Groups and Special Functions, Vol. 1: Simplest Lie Groups, Special Functions and Integral Transforms, Kluwer Academic, Dordrecht, 1991, Vol. 2: Class I Representations, Special Functions, and Integral Transforms, Kluwer Academic, Dordrecht, 1993, Vol. 3: Classical and Quantum Groups and Special Functions, Kluwer Academic, Dordrecht, 1992, Additional Vol.: Recent Advances, Kluwer Academic, Dordrecht, 1995.
53. M.A. Vorontzov, V.I. Shmal'gausen, Principles of Adaptive Optics, Nauka, Moscow, 1985 (in Russian).
54. Vourdas A. Wünsche A. J. Phys. A 31 1998 9431
55. Wang J.Y. Silva D.E. Appl. Opt. 19 1980 1510
56. E.W. Weisstein, CRC Concise Encyclopedia of Mathematics, 2nd Edition, Chapman & Hall/CRC, London, 2002 (see also: Eric Weisstein's World of Mathematics, A Wolfram Web Research).
57. Wünsche A. Acta Phys. Slovaca 46 1996 505
58. Wünsche A. J. Phys. A 31 1998 8267
59. Wünsche A. J. Phys. A 32 1999 3179
60. Wünsche A. J. Phys. A 33 2000 1603-1629
61. Wünsche A. Corrigendum J. Phys. A 33 2000 3531
62. Wünsche A. In: Dattoli G. Srivastava H.M. Cesarano C. (Eds.) Advanced Special Functions and Integration Methods, Proceedings of the Workshop Melfi, Italy, 2000 2001 157-198 Aracne Editrice Roma
63. Wünsche A. J. Comput. Appl. Math. 133 2001 665 (Special Issue: Proceedings of Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications, Patras, Greece, 1999)
64. Wünsche A. J. Opt. B: Quantum Semiclass. Opt. 4 2002 359
65. Wünsche A. J. Opt. B: Quantum Semiclass. Opt. 5 2003 S429
66. J.C. Wyant, K. Creath, in: R. Shannon, J. Wyant (Eds.), Applied Optics and Optical Engineering, Vol. XI, Academic Press, New York, 1992, 28-37.
67. Wybourne B. Classical Groups for Physicists 1974 Wiley-Interscience New York
68. Zernike F. Physica 1 1934 689
69. Zernike F. Brinkman H.C. Verh. Akad. Wet. Amst. (Proc. Sec. Sci.) 38 1935 11
70. S. Zhang, R.R. Shannon, in: R. Shannon, J. Wyant (Eds.), Applied Optics and Optical Engineering, Vol. XI, Academic Press, New York, 1992, 201 (Chapter 4).