On the generalized Kesten–McKay distributions
ESAIM: Probability and Statistics, Tome 24 (2020), pp. 56-68.

We examine the properties of distributions with the density of the form: $$ where c, a1, …, a$$ are some parameters and A$$ a suitable constant. We find general forms of A$$, of k-th moment and of k-th polynomial orthogonal with respect to such measures. We also calculate Cauchy transforms of these measures. We indicate connections of such distributions with distributions and polynomials forming the so called Askey–Wilson scheme. On the way we prove several identities concerning rational symmetric functions. Finally, we consider the case of parameters a1, …, a$$ forming conjugate pairs and give some multivariate interpretations based on the obtained distributions at least for the cases n = 2, 4, 6.

DOI : 10.1051/ps/2019029
Classification : 60E05, 05E05, 62H05, 60J10
Mots-clés : Kesten–McKay, Bernstein-Szegö distributions, Chebyshev polynomials, orthogonal polynomials, Askey–Wilson polynomials, moments, symmetric rational functions, multivariate distributions, Cauchy (Hilbert) transform 
@article{PS_2020__24_1_56_0,
     author = {Szab{\l}owski, Pawe{\l} J.},
     title = {On the generalized {Kesten{\textendash}McKay} distributions},
     journal = {ESAIM: Probability and Statistics},
     pages = {56--68},
     publisher = {EDP-Sciences},
     volume = {24},
     year = {2020},
     doi = {10.1051/ps/2019029},
     mrnumber = {4069296},
     zbl = {1447.60043},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps/2019029/}
}
TY  - JOUR
AU  - Szabłowski, Paweł J.
TI  - On the generalized Kesten–McKay distributions
JO  - ESAIM: Probability and Statistics
PY  - 2020
SP  - 56
EP  - 68
VL  - 24
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps/2019029/
DO  - 10.1051/ps/2019029
LA  - en
ID  - PS_2020__24_1_56_0
ER  - 
%0 Journal Article
%A Szabłowski, Paweł J.
%T On the generalized Kesten–McKay distributions
%J ESAIM: Probability and Statistics
%D 2020
%P 56-68
%V 24
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ps/2019029/
%R 10.1051/ps/2019029
%G en
%F PS_2020__24_1_56_0
Szabłowski, Paweł J. On the generalized Kesten–McKay distributions. ESAIM: Probability and Statistics, Tome 24 (2020), pp. 56-68. doi : 10.1051/ps/2019029. http://www.numdam.org/articles/10.1051/ps/2019029/

[1] M. Bożejko, B. Kümmerer and R. Speicher, q-Gaussian processes: non-commutative and classical aspects. Comm. Math. Phys. 185 (1997) 129–154 | DOI | MR | Zbl

[2] W. Bryc, W. Matysiak and P.J. Szabłowski, Probabilistic aspects of Al-Salam-Chihara polynomials. Proc. Amer. Math. Soc. 133 (2005) 1127–1134 | DOI | MR | Zbl

[3] J.A. Cima, A.L. Matheson and W.T. Ross, The Cauchy transform. Math. Surv. Monogr. 125. American Mathematical Society, Providence, RI (2006) | MR | Zbl

[4] A.M. Delgado, J.S. Geronimo, P. Iliev and Y. Xu, On a two-variable class of Bernstein-Szegő measures. Constr. Approx. 30 (2009) 71–91 | DOI | MR | Zbl

[5] M.E.H. Ismail, Classical and quantum orthogonal polynomials in one variable. With two chapters by Walter Van Assche. With a foreword by Richard A. Askey. Encyclopedia of Mathematics and its Applications, 98. Cambridge University Press, Cambridge (2005) | MR | Zbl

[6] H. Kesten, Symmetric random walks on groups. Trans. Amer. Math. Soc. 92 (1959) 336–354 | DOI | MR | Zbl

[7] R. Koekoek, P.A. Lesky and R.F. Swarttouw, Hypergeometric orthogonal polynomials and their $q$-analogues. With a foreword by Tom H. Koornwinder. Springer Monographs in Mathematics. Springer-Verlag, Berlin (2010) | DOI | MR | Zbl

[8] B.D. Mckay, The expected eigenvalue distribution of a large regular graph. Linear Algebra Appl. 40 (1981) 203–216 | DOI | MR | Zbl

[9] I. Oren and U. Smilansky, Trace formulas and spectral statistics for discrete Laplacians on regular graphs (II). J. Phys. A 43 (2010) 225205 | DOI | MR | Zbl

[10] I. Oren, A. Godel and U. Smilansky, Trace formulae and spectral statistics for discrete Laplacians on regular graphs. I. J. Phys. A 42 (2009) 415101 | DOI | MR | Zbl

[11] S. Sodin, Random matrices, nonbacktracking walks, and orthogonal polynomials. J. Math. Phys.48 (2007) 123503 | DOI | MR | Zbl

[12] P.J. Szabłowski, Expansions of one density via polynomials orthogonal with respect to the other. J. Math. Anal. Appl. 383 (2011) 35–54 | DOI | MR | Zbl

[13] P.J. Szabłowski, q−Wiener and (α, q)− Ornstein–Uhlenbeck processes. A generalization of known processes, Theory Probab. Appl. 56 (2011) 742–772 | Zbl

[14] P.J. Szabłowski, On affinity relating two positive measures and the connection coefficients between polynomials orthogonalized by these measures. Appl. Math. Comput. 219 (2013) 6768–6776 | MR | Zbl

[15] P.J. Szabłowski, On the q−Hermite polynomials and their relationship with some other families of orthogonal polynomials. Dem. Math. 66 (2013) 679–708 | MR | Zbl

[16] P.J. Szabłowski, Askey-Wilson integral and its generalizations. Adv. Differ. Equ. 2014 (2014) 316 | DOI | MR | Zbl

[17] P.J. Szabłowski, Befriending Askey–Wilson polynomials. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 17 (2014) 1450015 | DOI | MR | Zbl

[18] P.J. Szabłowski, Moments of q−Normal and conditional q−Normal distributions. Stoch. Porob. Lett. 106 (2015) 65–72 | DOI | MR | Zbl

[19] G. Szegö, Orthogonal Polynomials. American Mathematical Society Colloquium Publications, v. 23. American Mathematical Society, New York (1939) | Zbl

Cité par Sources :

The author is deeply grateful to the unknown referee for careful checking of all formulae and moreover pointing out numerous misprint and misspellings found in the paper.