Relative asymptotic of multiple orthogonal polynomials for Nikishin systems

We prove the relative asymptotic behavior for the ratio of two sequences of multiple orthogonal polynomials with respect to the Nikishin systems of measures. The first Nikishin system N(σ1,…,σm) is such that for each k, σk has a constant sign on its compact support supp(σk)⊂R consisting of an interval Δ˜k, on which |σk′|>0 almost everywhere, and a discrete set without accumulation points in R∖Δ˜k. If Co(supp(σk))=Δk denotes the smallest interval containing supp(σk), we assume that Δk∩Δk+1=0̸, k=1,…,m−1. The second Nikishin system N(r1σ1,…,rmσm) is a perturbation of the first by means of rational functions rk, k=1,…,m, whose zeros and poles lie in C∖∪k=1mΔk.