A family of Nikishin systems with periodic recurrence coefficients

TL;DR

This paper constructs a new family of multiple orthogonal polynomials, called Chebyshev-Nikishin polynomials, with periodic recurrence coefficients derived from Nikishin systems, and analyzes their asymptotic behavior and associated Riemann surface.

Contribution

It generalizes previous results to arbitrary p, establishing the orthogonality and asymptotics of Chebyshev-Nikishin polynomials using block Toeplitz matrices and Riemann surface techniques.

Findings

Construction of Chebyshev-Nikishin polynomials with periodic recurrence coefficients

Proof that these polynomials form a multiple orthogonal system

Derivation of strong asymptotics and Widom-type formulas

Abstract

Suppose we have a Nikishin system of $p$ measures with the $k$th generating measure of the Nikishin system supported on an interval $\Delta_k\subset\er$ with $\Delta_k\cap\Delta_{k+1}=\emptyset$ for all $k$. It is well known that the corresponding staircase sequence of multiple orthogonal polynomials satisfies a $(p+2)$-term recurrence relation whose recurrence coefficients, under appropriate assumptions on the generating measures, have periodic limits of period $p$. (The limit values depend only on the positions of the intervals $\Delta_k$.) Taking these periodic limit values as the coefficients of a new $(p+2)$-term recurrence relation, we construct a canonical sequence of monic polynomials $\{P_{n}\}_{n=0}^{\infty}$, the so-called \emph{Chebyshev-Nikishin polynomials}. We show that the polynomials $P_{n}$ themselves form a sequence of multiple orthogonal polynomials with respect to some Nikishin system of measures, with the $k$th generating measure being absolutely continuous on $\Delta_{k}$. In this way we generalize a result of the third author and Rocha \cite{LopRoc} for the case $p=2$. The proof uses the connection with block Toeplitz matrices, and with a certain Riemann surface of genus zero. We also obtain strong asymptotics and an exact Widom-type formula for the second kind functions of the Nikishin system for $\{P_{n}\}_{n=0}^{\infty}$.