Zeros of varying Laguerre-Krall orthogonal polynomials (Q2838961)

The authors consider the sequence \(\{ L_n^{(\alpha,M_n)} \}_n\) of varying Laguerre-Krall orthogonal polynomials, which is orthogonal with respect to the varying inner products NEWLINE\[NEWLINE(p,q)= \frac{1}{\Gamma(\alpha+1)} \int_0^\infty p(x)q(x) x^\alpha e^{-x}\, dx + M_n p(0)q(0),NEWLINE\]NEWLINE with \(\alpha > -1\) and \(\{M_n\}_n\) such that \( \lim_{n\rightarrow \infty } M_n n^\beta=M>0, \;\beta \in \mathbb{R}.\)NEWLINENEWLINENEWLINEThe aim of the paper is to characterize the local asymptotic behaviour around the origin of these polynomials through a Mehler-Heine type formula. The authors obtain three possible cases which are related to the size of the sequences of masses \(\{M_n\}_n\), that is, they depend on the relation between \(\beta\) and \(\alpha+1\). As a consequence, asymptotic relations between the zeros of these polynomials and the zeros of the Bessel functions of the first kind are found.NEWLINENEWLINENEWLINESimilar results for the varying generalized Hermite polynomials are deduced as a consequence of the Laguerre case. Finally, the results obtained for the varying Laguerre-Krall orthogonal polynomials are illustrated by some numerical experiments.