Multivariate Jacobi and Laguerre polynomials, infinite-dimensional extensions, and their probabilistic connections with multivariate Hahn and Meixner polynomials

Bernoulli

Volumn 17, Issue 3, 2011, Pages 1095-1125

  Griffiths, Robert C ^a   Spanò, Dario ^b  

^a Department of Statistics ^*  (United Kingdom)

^b UNIVERSITY OF WARWICK   (United Kingdom)

Author keywords

Beta stacy; Dirichlet distribution; Hahn polynomials; Jacobi polynomials; Laguerre polynomials; Meixner polynomials; Multivariate orthogonal polynomials; Size biased random discrete distributions

Indexed keywords

EID: 79960237408     PISSN: 13507265     EISSN: None     Source Type: Journal    
DOI: 10.3150/10-BEJ305     Document Type: Article

References (23)

1. Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series 55. Washington, DC: US Government Printing Office. MR0167642
2. Andrews, G. E., Askey, R. and Roy, R. (1999). Special Functions. Encyclopedia of Mathematics and its Applications 71. Cambridge: Cambridge Univ. Press. MR1688958
3. Diaconis, P., Khare, K. and Saloff-Coste, L. (2008). Gibbs sampling, exponential families and orthogonal polynomials. Statist. Sci. 23 151-178. MR2446500
4. Doksum, K. (1974). Tailfree and neutral random probabilities and their posterior distributions. Ann. Probab. 2 183-201. MR0373081
5. Dunkl, C. F. and Xu, Y. (2001). Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and Its Applications 81. Cambridge: Cambridge Univ. Press. MR1827871
6. Erdélyi, A. (1938). On some expansions in Laguerre polynomials. J. London Math. Soc. s1-13 154-156.
7. Exton, H. (1976). Multiple Hypergeometric Functions and Applications. Chichester: Ellis Horwood. MR0422713
8. Griffiths, R. C. and Spanò, D. (2010). Diffusion processes and coalescent trees. In Probability and Mathematical Genetics: Papers in Honour of Sir John Kingman (N. H. Bingham and C. M. Goldie, eds.), Chapter 15, 358-380. London Mathematical Society Lecture Notes Series. Cambridge: Cambridge Univ. Press.
9. Griffiths, R. and Spanò, D. (2010). n-kernel orthogonal polynomials on the Dirichlet, Dirichletmultinomial, Poisson-Dirichlet and Ewens' Sampling Distributions, and positive-definite sequences. Available at http://arxiv.org/abs/1003.5131v1.
10. Griffiths, R. C. (1979). On the distribution of allele frequencies in a diffusion model. Theoret. Population Biol. 15 140-158. MR0528914
11. Karlin, S. and McGregor, J. (1975). Linear growth models with many types and multidimensional Hahn polynomials. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) 261-288. New York: Academic Press. MR0406574
12. Karlin, S. and McGregor, J. L. (1961). The Hahn polynomials, formulas and an application. Scripta Math. 26 33-46. MR0138806
13. Khare, K. and Zhou, H. (2009). Rates of convergence of some multivariate Markov chains with polynomial eigenfunctions. Ann. Appl. Probab. 19 737-777. MR2521887
14. Kingman, J. F. C., Taylor, S. J., Hawkes, A. G., Walker, A. M., Cox, D. R., Smith, A. F. M., Hill, B. M., Burville, P. J. and Leonard, T. (1975). Random discrete distributions. J. Roy. Statist. Soc. Ser. B 37 1-22. MR0368264
15. Koornwinder, T. (1975). Two-variable analogues of the classical orthogonal polynomials. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) 435-495. New York: Academic Press. MR0402146
16. Koornwinder, T. H. and Schwartz, A. L. (1997). Product formulas and associated hypergroups for orthogonal polynomials on the simplex and on a parabolic biangle. Constr. Approx. 13 537-567. MR1466065 (Pubitemid 127450283)
17. Lauricella, G. (1893). Sulle funzioni ipergeometriche a piu' variabili. Rend. Circ. Mat. Palermo 7 111-158.
18. Pitman, J. (1995). Exchangeable and partially exchangeable random partitions. Probab. Theory Related Fields 102 145-158. MR1337249
19. Pitman, J. (1996). Random discrete distributions invariant under size-biased permutation. Adv. in Appl. Probab. 28 525-539. MR1387889 (Pubitemid 126504690)
20. Pitman, J. (2006). Combinatorial Stochastic Processes. Lecture Notes in Mathematics 1875. Berlin: Springer. MR2245368
21. Sauer, T. (2007). Jacobi polynomials in Bernstein form. J. Comput. Appl. Math. 199 149-158. MR2267539 (Pubitemid 44442685)
22. Waldron, S. (2006). On the Bernstein-Bézier form of Jacobi polynomials on a simplex. J. Approx. Theory 140 86-99. MR2226679 (Pubitemid 43671014)
23. Walker, S. and Muliere, P. (1997). Beta-Stacy processes and a generalization of the Pólya-urn scheme. Ann. Statist. 25 1762-1780. MR1463547