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 
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     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://archive.numdam.org/articles/10.1051/ps/2019029/}
}
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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://archive.numdam.org/articles/10.1051/ps/2019029/

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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.