A family of Nikishin systems with periodic recurrence coefficients

Suppose we have a Nikishin system of p measures with the kth generating measure of the Nikishin system supported on an interval Delta(k) subset of R with Delta(k) boolean AND Delta(k+ 1) = empty set 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)(infinity), the so-called 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 kth generating measure being absolutely continuous on Delta(k). In this way we generalize a result of the third author and Rocha [22] 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 functions of the second kind of the Nikishin system for {P-n}(n=0)(infinity). Bibliography: 27 titles.

A family of Nikishin systems with periodic recurrence coefficients Articles