Asymptotic for Orthogonal Polynomials with Respect to a Rational Modification of a Measure Supported on the Semi-Axis

Abstract

Given a sequence of orthogonal polynomials {Ln}n=0∞, orthogonal with respect to a positive Borel ν measure supported on R+, let {Qn}n=0∞ be the the sequence of orthogonal polynomials with respect to the modified measure r(x)dν(x), where r is certain rational function. This work is devoted to the proof of the relative asymptotic formula Qn(d)(z)Ln(d)(z)⇉n∏k=1N1ak+iz+akAk∏j=1N2z+bjbj+iBj, on compact subsets of C∖R+, where ak and bj are the zeros and poles of r, and the Ak, Bj are their respective multiplicities.