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 { L n } n = 0 ∞ , orthogonal with respect to a positive Borel ν measure supported on R + , let { Q n } n = 0 ∞ be the the sequence of orthogonal polynomials with respect to the modified measure r ( x ) d ν ( x ) , where r is certain rational function. This work is devoted to the proof of the relative asymptotic formula Q n ( d ) ( z ) L n ( d ) ( z ) ⇉ n ∏ k = 1 N 1 a k + i z + a k A k ∏ j = 1 N 2 z + b j b j + i B j , on compact subsets of C ∖ R + , where a k and b j are the zeros and poles of r, and the A k , B j are their respective multiplicities