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

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.