Properties of the polynomials associated with the Jacobi polynomials
Author:
S. Lewanowicz
Journal:
Math. Comp. 47 (1986), 669682
MSC:
Primary 33A65; Secondary 33A45
MathSciNet review:
856711
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: Power forms and Jacobi polynomial forms are found for the polynomials associated with Jacobi polynomials. Also, some differentialdifference equations and evaluations of certain integrals involving are given.
 [1]
Richard
Askey and George
Gasper, Jacobi polynomial expansions of Jacobi polynomials with
nonnegative coefficients, Proc. Cambridge Philos. Soc.
70 (1971), 243–255. MR 0296369
(45 #5430)
 [2]
Richard
Askey and Jet
Wimp, Associated Laguerre and Hermite polynomials, Proc. Roy.
Soc. Edinburgh Sect. A 96 (1984), no. 12,
15–37. MR
741641 (85f:33014), http://dx.doi.org/10.1017/S0308210500020412
 [3]
Pierre
Barrucand and David
Dickinson, On the associated Legendre polynomials, Orthogonal
Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill.,
1967) Southern Illinois Univ. Press, Carbondale, Ill., 1968,
pp. 43–50. MR 0232975
(38 #1298)
 [4]
Joaquin
Bustoz and Mourad
E. H. Ismail, The associated ultraspherical polynomials and their
𝑞analogues, Canad. J. Math. 34 (1982),
no. 3, 718–736. MR 663314
(84c:33013), http://dx.doi.org/10.4153/CJM19820496
 [5]
A. Erdélyi et al., Higher Transcendental Functions, Vols. 1 and 2, McGrawHill, New York, 1953.
 [6]
Walter
Gautschi, Minimal solutions of threeterm
recurrence relations and orthogonal polynomials, Math. Comp. 36 (1981), no. 154, 547–554. MR 606512
(82m:33006), http://dx.doi.org/10.1090/S00255718198106065126
 [7]
C.
C. Grosjean, The orthogonality property of the Lommel polynomials
and a twofold infinity of relations between Rayleigh’s
𝜎sums, J. Comput. Appl. Math. 10 (1984),
no. 3, 355–382. MR 755809
(85m:33011), http://dx.doi.org/10.1016/03770427(84)90044X
 [8]
C.
C. Grosjean, Theory of recursive generation of systems of
orthogonal polynomials: an illustrative example, Proceedings of the
international conference on computational and applied mathematics (Leuven,
1984), 1985, pp. 299–318. MR 793963
(86g:33015), http://dx.doi.org/10.1016/03770427(85)900263
 [9]
C. C. Grosjean, private communication, 1985.
 [10]
Shyam
L. Kalla, Salvador
Conde, and Yudell
L. Luke, Integrals of Jacobi
functions, Math. Comp. 38
(1982), no. 157, 207–214. MR 637298
(83a:33005), http://dx.doi.org/10.1090/S00255718198206372988
 [11]
Stanisław
Lewanowicz, Recurrence relations for
hypergeometric functions of unit argument, Math. Comp. 45 (1985), no. 172, 521–535. MR 804941
(86m:33004), http://dx.doi.org/10.1090/S00255718198508049412
 [12]
Stanisław
Lewanowicz, Recurrence relations for the coefficients in Jacobi
series solutions of linear differential equations, SIAM J. Math. Anal.
17 (1986), no. 5, 1037–1052. MR 853515
(87k:33011), http://dx.doi.org/10.1137/0517074
 [13]
Yudell
L. Luke, Mathematical functions and their approximations,
Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New
YorkLondon, 1975. MR 0501762
(58 #19039)
 [14]
Paul
Nevai, A new class of orthogonal
polynomials, Proc. Amer. Math. Soc.
91 (1984), no. 3,
409–415. MR
744640 (85f:42036), http://dx.doi.org/10.1090/S0002993919840744640X
 [15]
S. Paszkowski, Polynômes Associés aux Polynômes Orthogonaux Classiques, Publication ANO136, Univ. Sci. Tech. de Lille, U.E.R. d'I.E.E.A., 1984.
 [16]
Earl
D. Rainville, Special functions, The Macmillan Co., New York,
1960. MR
0107725 (21 #6447)
 [17]
Lucy
Joan Slater, Generalized hypergeometric functions, Cambridge
University Press, Cambridge, 1966. MR 0201688
(34 #1570)
 [18]
G. Szegö, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1939.
 [19]
Jet
Wimp, Computation with recurrence relations, Applicable
Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984.
MR 727118
(85f:65001)
 [1]
 R. Askey & G. Gasper, "Jacobi polynomial expansions of Jacobi polynomials with nonnegative coefficients," Proc. Cambridge Philos. Soc., v. 70, 1971, pp. 243255. MR 0296369 (45:5430)
 [2]
 R. Askey & J. Wimp, "Associated Laguerre and Hermite polynomials," Proc. Roy. Soc. Edinburgh Sect. A, v. 96, 1984, pp. 1537. MR 741641 (85f:33014)
 [3]
 P. Barrucand & D. Dickinson, "On the associated Legendre polynomials," in Orthogonal Expansions and Their Continuous Analogues (D. T. Haimo, ed.), Southern Illinois Univ. Press, Carbondale, Ill., 1967, pp. 4350. MR 0232975 (38:1298)
 [4]
 J. Bustoz & M. E. H. Ismail, "The associated ultraspherical polynomials and their qanalogues," Canad. J. Math., v. 34, 1982, pp. 718736. MR 663314 (84c:33013)
 [5]
 A. Erdélyi et al., Higher Transcendental Functions, Vols. 1 and 2, McGrawHill, New York, 1953.
 [6]
 W. Gautschi, "Minimal solutions of threeterm recurrence relations and orthogonal polynomials," Math. Comp., v. 36, 1981, pp. 547554. MR 606512 (82m:33006)
 [7]
 C. C. Grosjean, "The orthogonality property of the Lommel polynomials and a twofold infinity of relations between Rayleigh's sums," J. Comput. Appl. Math., v. 10, 1984, pp. 355382. MR 755809 (85m:33011)
 [8]
 C. C. Grosjean, "Theory of recursive generation of systems of orthogonal polynomials: An illustrative example," J. Comput. Appl. Math., v. 12/13, 1985, pp. 299318. MR 793963 (86g:33015)
 [9]
 C. C. Grosjean, private communication, 1985.
 [10]
 S. L. Kalla, S. Conde & Y. L. Luke, "Integrals of Jacobi functions," Math. Comp., v. 39, 1982, pp. 207214. MR 637298 (83a:33005)
 [11]
 S. Lewanowicz, "Recurrence relations for hypergeometric functions of unit argument," Math. Comp., v. 45, 1985, pp. 521535. MR 804941 (86m:33004)
 [12]
 S. Lewanowicz, "Recurrence relations for the coefficients in Jacobi series solutions of linear differential equations," SIAM J. Math. Anal., v. 17, 1986. (To appear.) MR 853515 (87k:33011)
 [13]
 Y. L. Luke, Mathematical Functions and Their Approximations, Academic Press, New York, 1975. MR 0501762 (58:19039)
 [14]
 P. Nevai, "A new class of orthogonal polynomials," Proc. Amer. Math. Soc., v. 91, 1984, pp. 409415. MR 744640 (85f:42036)
 [15]
 S. Paszkowski, Polynômes Associés aux Polynômes Orthogonaux Classiques, Publication ANO136, Univ. Sci. Tech. de Lille, U.E.R. d'I.E.E.A., 1984.
 [16]
 E. D. Rainville, Special Functions, Macmillan, New York, 1960. MR 0107725 (21:6447)
 [17]
 L. J. Slater, Generalized Hypergeometric Functions, Cambridge Univ. Press, Cambridge, 1966. MR 0201688 (34:1570)
 [18]
 G. Szegö, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1939.
 [19]
 J. Wimp, Computation with Recurrence Relations, Pitman, Boston, 1984. MR 727118 (85f:65001)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198608567118
PII:
S 00255718(1986)08567118
Article copyright:
© Copyright 1986
American Mathematical Society
