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On the families of polynomials forming a part of the Askey-Wilson scheme and their probabilistic applications. (English) Zbl 1509.33020

Summary: In this paper, we review the properties of six families of orthogonal polynomials that form the main bulk of the collection called the Askey-Wilson scheme of polynomials. We give connection coefficients between them as well as the so-called linearization formulae and other useful important finite and infinite expansions and identities. An important part of the paper is the presentation of probabilistic models where most of these families of polynomials appear. These results were scattered within the literature in recent 15 years. We put them together to enable an overall outlook on these families and understand their crucial role in the attempts to generalize Gaussian distributions and find their bounded support generalizations. This paper is based on 65 items in the predominantly recent literature.

MSC:

33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
05A30 \(q\)-calculus and related topics
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
60E05 Probability distributions: general theory
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[1] Al-Salam, W. A. and Ismail, M. E. H., q-beta integrals and the q-Hermite polynomials, Pacific J. Math.135 (1988) 209-221. · Zbl 0658.33002
[2] Andrews, G. E., Askey, R. and Roy, R., Special Functions, , No. 71 (Cambridge Univ. Press, 1999), xvi+664 pp., ISBN: 0-521-62321-9; 0-521-78988-5. · Zbl 0920.33001
[3] Askey, R. A., Rahman, M. and Suslov, S. K., On a general q-Fourier transformation with nonsymmetric kernels, J. Comput. Appl. Math.68 (1996) 25-55. · Zbl 0871.33008
[4] Askey, R. and Wilson, J., Some Basic Hypergeometric Orthogonal Polynomials That Generalize Jacobi Polynomials, , Vol. 54, No. 319 (Amer. Math. Soc., 1985), iv+55 pp. · Zbl 0572.33012
[5] Atakishiyeva, M. and Atakishiyev, N., A non-standard generating function for continuous dual q-Hahn polynomials, Rev. Mat. Teor. Apl.18 (2011) 111-120. · Zbl 1307.33009
[6] Bożejko, M., Kümmerer, B and Speicher, R., q-Gaussian processes: Non-commutative and classical aspects, Comm. Math. Phys.185 (1997) 129-154. · Zbl 0873.60087
[7] Bressoud, D. M., A simple proof of Mehler’s formula for q-Hermite polynomials, Indiana Univ. Math. J.29 (1980) 577-580. · Zbl 0411.33010
[8] Bryc, W., Stationary random fields with linear regressions, Ann. Probab.29 (2001) 504-519. · Zbl 1014.60013
[9] Bryc, W., Matysiak, W. and Szabłowski, P. J., Probabilistic aspects of Al-Salam-Chihara polynomials. Proc. Amer. Math. Soc.133 (2005) 1127-1134(electronic). · Zbl 1074.33015
[10] Bryc, W., Matysiak, W. and Wesołowski, J., The bi-Poisson process: A quadratic harness, Ann. Probab.36 (2008) 623-646. · Zbl 1137.60036
[11] Bryc, W. and Wesołowski, J., Askey-Wilson polynomials, quadratic harnesses and martingales, Ann. Probab.38 (2010) 1221-1262. · Zbl 1201.60077
[12] Carlitz, L., Some polynomials related to theta functions, Ann. Mat. Pura Appl. (4)41 (1956) 359-373. · Zbl 0071.06202
[13] Carlitz, L., Some polynomials related to theta functions, Duke Math. J.24 (1957) 521-527. · Zbl 0079.09503
[14] Carlitz, L., Generating functions for certain q-orthogonal polynomials, Collect. Math.23 (1972) 91-104. · Zbl 0273.33012
[15] Chen, W. Y. C., Saad, H. L. and Sun, L. H., The bivariate Rogers-Szegö polynomials, J. Phys. A40 (2007) 6071-6084. · Zbl 1119.05011
[16] Corteel, S. and Williams, L. K., Staircase tableaux, the asymmetric exclusion process, and Askey-Wilson polynomials, Proc. Natl. Acad. Sci. USA107 (2010) 6726-6730. · Zbl 1205.05243
[17] Garrett, K., Ismail, M. E. H. and Stanton, D., Variants of the Rogers-Ramanujan identities, Adv. Appl. Math.23 (1999) 274-299. · Zbl 0944.33015
[18] Gasper, G. and Rahman, M., Positivity of the Poisson kernel for the continuous q-ultraspherical polynomials, SIAM J. Math. Anal.14 (1983) 409-420. · Zbl 0516.33009
[19] Ismail, M. E. H., Classical and Quantum Orthogonal Polynomials in One Variable.With two chapters by Walter Van Assche. With a foreword by Richard A. Askey, , Vol. 98 (Cambridge Univ. Press, 2005), xviii+706 pp., ISBN: 978-0-521-78201-2; 0-521-78201-5. · Zbl 1082.42016
[20] Ismail, M. E. H. and Masson, D. R., q-Hermite polynomials, biorthogonal rational functions, and q-beta integrals, Trans. Amer. Math. Soc.346 (1994) 63-116. · Zbl 0812.33012
[21] Ismail, M. E. H., Rahman, M. and Stanton, D., Quadratic q-exponentials and connection coefficient problems, Proc. Amer. Math. Soc.127 (1999) 2931-2941. · Zbl 0934.33025
[22] Ismail, M. E. H. and Stanton, D., Addition theorems for the q-exponential functions, Contemp. Math.254 (2000) 235-245. · Zbl 0958.33010
[23] Ismail, M. E. H. and Stanton, D., Tribasic integrals and identities of Rogers-Ramanujan type, Trans. Amer. Math. Soc.355 (2003) 4061-4091. · Zbl 1033.33012
[24] Ismail, M. E. H., Stanton, D. and Viennot, G., The combinatorics of q-Hermite polynomials and the Askey-Wilson integral, Eur. J. Combin.8 (1987) 379-392. · Zbl 0642.33006
[25] Kibble, W. F., An extension of a theorem of Mehler’s on Hermite polynomials, Proc. Cambridge Philos. Soc.41 (1945) 12-15. · Zbl 0060.19602
[26] Kim, D., Stanton, D. and Zeng, J., The combinatorics of the Al-Salam-Chihara q-Charlier polynomials, Sém. Lothar. Combin.54 (2005/2007) Art. B54i, 15 pp. (electronic). · Zbl 1203.05016
[27] Koekoek, R., Lesky, P. A. and Swarttouw, R. F., Hypergeometric Orthogonal Polynomials and Their q-analogues.With a foreword by Tom H. Koornwinder, (Springer-Verlag, 2010), xx+578 pp., ISBN: 978-3-642-05013-8. · Zbl 1200.33012
[28] R. Koekoek and R. F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, https://arxiv.org/pdf/math/9602214.pdf.
[29] Lancaster, H. O., The structure of bivariate distributions, Ann. Math. Statist.29 (1958) 719-736. · Zbl 0086.35102
[30] Lancaster, H. O., Correlation and complete dependence of random variables, Ann. Math. Statist.34 (1963) 1315-1321. · Zbl 0121.35905
[31] Lancaster, H. O., Correlations and canonical forms of bivariate distributions, Ann. Math. Statist.34 (1963) 532-538. · Zbl 0115.14104
[32] Lancaster, H. O., Joint probability distributions in the Meixner classes, J. Roy. Statist. Soc. Ser. B37 (1975) 434-443. · Zbl 0315.62009
[33] Matysiak, W. and Szabłowski, P. J., A few remarks on Bryc’s paper on random fields with linear regressions, Ann. Probab.30 (2002) 1486-1491. · Zbl 1022.60048
[34] W. Matysiak and P. J. Szabłowski, Bryc’s random fields: The existence and distributions analysis, preprint (2005), arXiv:math.PR/math/0507296.
[35] Mercer, J., Functions of positive and negative type and their connection with the theory of integral equations, Philos. Trans. Roy. Soc. A209 (1909) 415-446. · JFM 40.0408.02
[36] Nesterenko, M., Patera, J., Szajewska, M. and Tereszkiewicz, A., Orthogonal polynomials of compact simple Lie groups: Branching rules for polynomials, J. Phys. A43 (2010) 495207, 27 pp. · Zbl 1225.22013
[37] Rahman, M. and Tariq, Q. M., Poisson kernel for the associated continuous q-ultraspherical polynomials, Methods Appl. Anal.4 (1997) 77-90. · Zbl 0890.33009
[38] Rogers, L. J., On the expansion of certain infinite products, Proc. London Math. Soc.24 (1893) 337-352. · JFM 25.0432.01
[39] Rogers, L. J., Second memoir on the expansion of certain infinite products, Proc. London Math. Soc.25 (1894) 318-343.
[40] Rogers, L. J., Third memoir on the expansion of certain infinite products, Proc. London Math. Soc.26 (1895) 15-32. · JFM 26.0289.01
[41] Simon, B., The classical moment problem as a self-adjoint finite difference operator, Adv. Math.137 (1998) 82-203. · Zbl 0910.44004
[42] Slepian, D., On the symmetrized Kronecker power of a matrix and extensions of Mehler’s formula for Hermite polynomials, SIAM J. Math. Anal.3 (1972) 606-616. · Zbl 0222.33016
[43] Szabłowski, P. J., Probabilistic implications of symmetries of q-Hermite and Al-Salam-Chihara polynomials, Infin. Dimens. Anal. Quantum Probab. Relat. Top.11 (2008) 513-522. · Zbl 1175.60050
[44] Szabłowski, P. J., q-Gaussian distributions: Simplifications and simulations, J. Probab. Stat.2009 (2009) Art. ID 752430, 18 pp. · Zbl 1205.60035
[45] Szabłowski, P. J., Multidimensional q-normal and related distributions — Markov case, Electron. J. Probab.15 (2010) 1296-1318. · Zbl 1222.62061
[46] Szabłowski, P. J., On the structure and probabilistic interpretation of Askey-Wilson densities and polynomials with complex parameters, J. Funct. Anal.262 (2011) 635-659, http://arxiv.org/abs/1011.1541. · Zbl 1222.33016
[47] Szabłowski, P. J., q-Wiener and \((\alpha,q)\)-Ornstein-Uhlenbeck processes. A generalization of known processes, Theory Probab. Appl.56(4) (2011) 742-772, http://arxiv.org/abs/math/0507303. · Zbl 1276.60045
[48] Szabłowski, P. J., Expansions of one density via polynomials orthogonal with respect to the other, J. Math. Anal. Appl.383 (2011) 35-54, http://arxiv.org/abs/1011.1492. · Zbl 1232.33019
[49] Szabłowski, P. J., On summable form of Poisson-Mehler kernel for big q-Hermite and Al-Salam-Chihara polynomials, Infin. Dimens. Anal. Quantum Probab. Relat. Top.15 (2012) 1250014, http://arxiv.org/abs/1011.1848. · Zbl 1266.33018
[50] Szabłowski, P. J., Towards a q-analogue of the Kibble-Slepian formula in \(3\) dimensions, J. Funct. Anal.262 (2012) 210-233, http://arxiv.org/abs/1011.4929. · Zbl 1236.33035
[51] Szabłowski, P. J., On affinity relating two positive measures and the connection coefficients between polynomials orthogonalized by these measures, Appl. Math. Comput.219 (2013) 6768-6776. · Zbl 1290.33012
[52] Szabłowski, P. J., On the q-Hermite polynomials and their relationship with some other families of orthogonal polynomials, Demonstratio Math.66 (2013) 679-708, http://arxiv.org/abs/1101.2875. · Zbl 1290.33024
[53] Szabłowski, P. J., Askey-Wilson integral and its generalizations, Adv. Difference Equ.2014 (2014) 316, http://arxiv.org/abs/1112.4830. · Zbl 1347.33038
[54] Szabłowski, P. J., Befriending Askey-Wilson polynomials, Infin. Dimens. Anal. Quantum Probab. Relat. Top.17 (2014) 1450015, 25 pp., http://arxiv.org/abs/1111.0601. · Zbl 1301.33025
[55] Szabłowski, P. J., On Markov processes with polynomial conditional moments, Trans. Amer. Math. Soc.367 (2015) 8487-8519, http://arxiv.org/abs/1210.6055. · Zbl 1334.60154
[56] Szabłowski, P. J., Around Poisson-Mehler summation formula, Hacet. J. Math. Stat.45 (2016) 1729-1742, http://arxiv.org/abs/1108.3024. · Zbl 1370.41015
[57] Szabłowski, P. J., On stationary Markov processes with polynomial conditional moments, Stochastic Anal. Appl.35 (2017) 852-872, http://arxiv.org/abs/1312.4887. · Zbl 1377.60050
[58] Szabłowski, P. J., Markov processes, polynomial martingales and orthogonal polynomials, Stochastics90 (2018) 61-77. · Zbl 1498.60313
[59] Szabłowski, P. J., On three dimensional multivariate version of q-normal distribution and probabilistic interpretations of Askey-Wilson, Al-Salam-Chihara and q-ultraspherical polynomials, J. Math. Anal. Appl.474 (2019) 1021-1035. · Zbl 1478.60061
[60] P. J. Szabłowski, Multivariate generating functions involving Chebyshev polynomials and some of its generalizations involving q-Hermite ones, in print Colloq. Math., doi:10.4064, arXiv:1706.00316.
[61] Vilenkin, N. J. and Klimyk, A. U., Representation of Lie Groups and Special Functions. Recent Advances.Translated from the Russian manuscript by V. A. Groza and A. A. Groza. , Vol. 316 (Kluwer Academic Publishers Group, 1995), xvi+497 pp., ISBN: 0-7923-3210-5.
[62] Voiculescu, D., Lectures on free probability theory, in Lectures on Probability Theory and Statistics (Saint-Flour, 1998), , Vol. 1738 (Springer, 2000), pp. 279-349. · Zbl 1015.46037
[63] Voiculescu, D. V., Dykema, K. J. and Nica, A., Free Random Variables: A Noncommutative Probability Approach to Free Products with Applications to Random Matrices, Operator Algebras and Harmonic Analysis on Free Groups, , Vol. 1 (Amer. Math. Soc., 1992), vi+70 pp., ISBN: 0-8218-6999-X. · Zbl 0795.46049
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