A family of Nikishin systems with periodic recurrence coefficients

Abstract

Suppose we have a Nikishin system of $p$ measures with the $k$th generating measure of the Nikishin system supported on an interval $Δ_k\subset\er$ with ${\Delta }_k\cap {\Delta }_{k+1}=\varnothing$ 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 $\{P_n{\}}_{n=0}^{\infty }$.

30 pages, minor changes

References in corpus (2)

• Nikishin systems are perfect
• High order three-term recursions, Riemann-Hilbert minors and Nikishin systems on star-like sets