Differential equations and recursion relations for Laguerre functions on symmetric cones

Ding, Hongming

doi:10.1090/S0002-9947-07-04062-7

Differential equations and recursion relations for Laguerre functions on symmetric cones
HTML articles powered by AMS MathViewer

by Hongming Ding PDF

Trans. Amer. Math. Soc. 359 (2007), 3239-3250 Request permission

Abstract:

We obtain the differential equation and recurrence relations satisfied by the Laguerre functions $l_{\mathbf {m}}^{\nu }$ on an arbitrary symmetric cone $\Omega$.

References

George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
Mark Davidson and Gestur Ólafsson, Differential recursion relations for Laguerre functions on Hermitian matrices, Integral Transforms Spec. Funct. 14 (2003), no. 6, 469–484. MR 2017655, DOI 10.1080/10652460310001600582
Mark Davidson, Gestur Ólafsson, and Genkai Zhang, Laguerre polynomials, restriction principle, and holomorphic representations of $\textrm {SL}(2,\Bbb R)$, Acta Appl. Math. 71 (2002), no. 3, 261–277. MR 1903539, DOI 10.1023/A:1015283100541
Mark Davidson, Gestur Ólafsson, and Genkai Zhang, Laplace and Segal-Bargmann transforms on Hermitian symmetric spaces and orthogonal polynomials, J. Funct. Anal. 204 (2003), no. 1, 157–195. MR 2004748, DOI 10.1016/S0022-1236(03)00101-0
Hongming Ding and Kenneth I. Gross, Operator-valued Bessel functions on Jordan algebras, J. Reine Angew. Math. 435 (1993), 157–196. MR 1203914
Hongming Ding, Kenneth I. Gross, and Donald St. P. Richards, Ramanujan’s master theorem for symmetric cones, Pacific J. Math. 175 (1996), no. 2, 447–490. MR 1432840, DOI 10.2140/pjm.1996.175.447
Jacques Faraut and Adam Korányi, Analysis on symmetric cones, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications. MR 1446489
Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
Bertram Kostant, On Laguerre polynomials, Bessel functions, Hankel transform and a series in the unitary dual of the simply-connected covering group of $\textrm {Sl}(2,\textbf {R})$, Represent. Theory 4 (2000), 181–224. MR 1755901, DOI 10.1090/S1088-4165-00-00096-0
Michel Lassalle, Coefficients binomiaux généralisés et polynômes de Macdonald, J. Funct. Anal. 158 (1998), no. 2, 289–324 (French). MR 1648471, DOI 10.1006/jfan.1998.3281
Bent Ørsted and Gen Kai Zhang, Generalized principal series representations and tube domains, Duke Math. J. 78 (1995), no. 2, 335–357. MR 1333504, DOI 10.1215/S0012-7094-95-07815-6
Fulvio Ricci and Anita Tabacco Vignati, Bergman spaces on some tube-type domains and Laguerre operators on symmetric cones, J. Reine Angew. Math. 449 (1994), 81–101. MR 1268580
Zhi Min Yan, Generalized hypergeometric functions and Laguerre polynomials in two variables, Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991) Contemp. Math., vol. 138, Amer. Math. Soc., Providence, RI, 1992, pp. 239–259. MR 1199131, DOI 10.1090/conm/138/1199131
Gen Kai Zhang, Some recurrence formulas for spherical polynomials on tube domains, Trans. Amer. Math. Soc. 347 (1995), no. 5, 1725–1734. MR 1249896, DOI 10.1090/S0002-9947-1995-1249896-7