Relative asymptotic of multiple orthogonal polynomials for Nikishin systems Original Research Article

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)N(σ1,…,σm) is such that for each kk, σkσk has a constant sign on its compact support View the MathML sourcesupp(σk)⊂R consisting of an interval View the MathML sourceΔ˜k, on which View the MathML source|σk′|>0 almost everywhere, and a discrete set without accumulation points in View the MathML sourceR∖Δ˜k. If View the MathML sourceCo(supp(σk))=Δk denotes the smallest interval containing View the MathML sourcesupp(σk), we assume that Δk∩Δk+1=0̸Δk∩Δk+1=0̸, k=1,…,m−1k=1,…,m−1. The second Nikishin system N(r1σ1,…,rmσm)N(r1σ1,…,rmσm) is a perturbation of the first by means of rational functions rkrk, k=1,…,mk=1,…,m, whose zeros and poles lie in View the MathML sourceC∖∪k=1mΔk.