In this paper we introduce a sequence of varying orthogonal polynomials related to a Laguerre weight where this absolutely continuous measure is perturbed by a sequence of nonnegative masses located at the origin. The main objective is to obtain asymptotic relations between the zeros of these polynomials and the zeros of the Bessel functions of the first kind (or linear combinations of them). This is done through Mehler-Heine type formulas. With these relations we can easily compute asymptotically the zeros of these polynomials. We show some numerical experiments.
We consider the Sobolev inner productwhere dψ (α,β) (x) = (1 − x) α (1 + x) β dx with α, β > −1, and ψ is a measure involving a rational modification of a Jacobi weight and with a mass point outside the interval (−1, 1). We study the asymptotic behaviour of the polynomials which are orthogonal with respect to this inner product on different regions of the complex plane. In fact, we obtain the outer and inner strong asymptotics for these polynomials as well as the Mehler-Heine asymptotics which allow us to obtain the asymptotics of the largest zeros of these polynomials. We also show that in a certain sense the above inner product is also equilibrated.
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