Weak quasiclassical asymptotics of polynomial solutions of three-term recurrence relations of high order

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Abstract

For polynomials $Q_{n}(z):=z^n + \cdots$ defined by three-term recurrence relations
$Q_{n+1}=zQ_n-a_{n-p+1}Q_{n-p}$,
$p\ge {1}$, of order $p+1$ with the coefficient $a_{n}\equiv a_{n,N}$ (the variable recurrence coefficient) depending on the parameter $N$,
the weak asymptotics of $Q_n (z)$ are investigated in the quasi-classical regime as $n \to \infty$,
$n/N \to t$, and $a_{n,N} \to a(t)$.
The case $p=1$ (orthogonal polynomials) was studied earlier. The results obtained (for $p=2$) are applied to the problem of eigenvalues distributions of ensembles of normal random matrices.

About the authors

Alexander Ivanovich Aptekarev

Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow

Email: aptekaa@gmail.com
Scopus Author ID: 6603809965
Doctor of physico-mathematical sciences, Professor

Victor Yur'evich Novokshenov

Institute of Mathematics with Computing Centre, Ufa Federal Research Centre, Russian Academy of Sciences, Ufa

Email: novik53@mail.ru
Doctor of physico-mathematical sciences, Professor

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