Asymptotic for orthogonal polynomials with respect to a rational modification of a measure supported on the semi-axis
Given a sequence of orthogonal polynomials $\{L_n\}_{n=0}^\infty$, orthogonal with respect to a positive Borel $ν$ measure supported on $\mathbb{R}_+$, let $\{Q_n\}_{n=0}^\infty$ be the sequence of orthogonal polynomials with respect to the modified measure $r(x)dν(x)$, where $r$ is certain rational function, and {$L_n(-1) = Q_n(-1)= (-1)^n$}. This work is devoted to the proof of the relative asymptotic $$ \frac{Q_n^{(d)}(z)}{L_n^{(d)}(z)} \unifn \prod_{k=1}^{N_1}\left(\frac{\sqrt{a_k}+i}{\sqrt{z}+\sqrt{a_k}}\right)^{A_k}\prod_{j=1}^{N_2} \left(\frac{\sqrt{z}+\sqrt{b_j}}{\sqrt{b_j}+i}\right)^{B_j},$$ on compact subsets of $\mathbb{C}\setminus\mathbb{R}_+$, where $a_k$ and $b_j$ are the zeros and poles of $r$, and the $A_k$, $B_j$ are their respective multiplicities.