Mathematics – Classical Analysis and ODEs
Scientific paper
2011-10-13
Mathematics
Classical Analysis and ODEs
29 pages
Scientific paper
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 $p=2$. We also obtain strong asymptotics and an exact Widom-type formula for the second kind functions of the Nikishin system for $\{P_{n}\}_{n=0}^{\infty}$.
Delvaux Steven
García Abey López
Lagomasino Guillermo López
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