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Convolution structures for Laguerre polynomials. (English) Zbl 0347.33006


MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
Full Text: DOI

References:

[1] R. Askey,Orthogonal polynomials and positivity, inStudies in Applied Mathematics, Wave Propagation and Special Functions, SIAM, 1970, pp. 64-85.
[2] Askey, R.; Gasper, G., Linearization of the product of Jacobi polynomials. III, Canad. J. Math., 23, 332-338 (1971) · Zbl 0212.40904
[3] Askey, R.; Gasper, G., Jacobi polynomial expansions of Jacobi polynomials with nonnegative coefficients, Proc. Cambridge Philos. Soc., 70, 243-255 (1971) · Zbl 0217.11402 · doi:10.1017/S0305004100049847
[4] Askey, R.; Gasper, G., Certain rational functions whose power series have positive coefficients, Amer. Math. Monthly, 79, 327-341 (1972) · Zbl 0242.33023 · doi:10.2307/2978081
[5] Askey, R.; Wainger, S.; Haimo, D., A dual convolution structure for Jacobi polynomials, Orthogonal Expansions and their Continuous Analogues, 25-36 (1968), Carbondale, Illinois: Southern Illinois Univ. Press, Carbondale, Illinois · Zbl 0174.36305
[6] Askey, R.; Wainger, S., A convolution structure for Jacobi series, Amer. J. Math., 91, 463-485 (1969) · Zbl 0186.12303 · doi:10.2307/2373520
[7] Bailey, W. W., Contiguous hypergeometric functions of the type_3F_2(1), Proc. Glasgow Math. Assoc., 2, 62-65 (1954) · Zbl 0056.06702
[8] Bailey, W. N., Generalized Hypergeometric Series (1964), New York: Cambridge Univ. Press, New York · JFM 61.0406.01
[9] S. Bochner,Sturm-Liouville and heat equations whose eigenfunctions are ultraspherical polynomials or associated Bessel functions, Proc. Conf. on Diff. Eq., Univ. of Maryland, 1955, pp. 23-48. · Zbl 0075.28002
[10] Eagleson, G. K., A characterization theorem for positive definite sequences on the Krawtchouk polynomials, Austral. J. Statist., 11, 29-38 (1969) · Zbl 0184.22601
[11] Erdélyi, A., Higher Transcendental Functions (1953), New York: McGraw-Hill, New York · Zbl 0051.30303
[12] Gasper, G., Linearization of the product of Jacobi polynomials. II, Canad. J. Math., 22, 582-593 (1970) · Zbl 0198.39201
[13] Gasper, G., Positivity and the convolution structure for Jacobi series, Ann. of Math., 93, 112-118 (1971) · Zbl 0208.08101 · doi:10.2307/1970755
[14] Gasper, G., Banach algebras for Jacobi series and positivity of a kernel, Ann. of Math., 95, 261-280 (1972) · Zbl 0236.33013 · doi:10.2307/1970800
[15] Gillis, J.; Shimshoni, M., Triple product integrals of Laguerre functions, Math. Comp., 16, 50-62 (1962) · Zbl 0171.37704 · doi:10.2307/2003810
[16] Hille, E., On Laguerre’s series. First note, Proc. Nat. Acad. Sci. U.S.A., 12, 261-265 (1926) · JFM 52.0281.01 · doi:10.1073/pnas.12.4.261
[17] Karlin, S.; McGregor, J., The differential equations of birth-and-death processes, and the Stieltjes moment problem, Trans. Amer. Math. Soc., 85, 489-546 (1957) · Zbl 0091.13801 · doi:10.2307/1992942
[18] Karlin, S.; McGregor, J., The Hahn polynomials, formulas and applications, Scripta Math., 26, 33-46 (1961) · Zbl 0104.29103
[19] Kennedy, M., A stochastic process associated with the ultraspherical polynomials, Proc. Roy. Irish Acad. Sect. A, 61, 89-100 (1961) · Zbl 0104.11202
[20] Kingman, J. F. C., Random walks with spherical symmetry, Acta Math., 109, 11-53 (1963) · Zbl 0121.12803 · doi:10.1007/BF02391808
[21] O’Neil, R., Convolution operators and L(p,q) spaces, Duke Math. J., 30, 129-142 (1963) · Zbl 0178.47701 · doi:10.1215/S0012-7094-63-03015-1
[22] Szegö, G., Über gewisse Potenzreihen mit lauter positiven Koeffizienten, Math. Z., 37, 674-688 (1933) · Zbl 0007.34401 · doi:10.1007/BF01474608
[23] Szegö, G., Orthogonal Polynomials (1967), Providence, R. I.: Amer. Math. Soc., Providence, R. I. · JFM 61.0386.03
[24] Titchmarsh, E. C., Some integrals involving Hermite polynomials, J. London Math. Soc., 23, 15-16 (1948) · Zbl 0032.27503 · doi:10.1112/jlms/s1-23.1.15
[25] Vere-Jones, D., Finite bivariate distributions and semigroups of non-negative matrices, Quart. J. Math. Oxford, 22, 2, 247-270 (1971) · Zbl 0215.40201 · doi:10.1093/qmath/22.2.247
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