Asymptotic behavior of semiclassical jointly-orthogonal polynomials of Bessel-Laguerre type

Abstract

which gives a picture of the distribution of zeros. As follows from our results, n zeros of the polynomial Q 2n tend to zero as n → ∞, and the other n zeros fill an interval bounded away from zero; for the polynomials Q2n, n zeros “go” to infinity and the other zeros are distributed along a curve of Szegő type, exactly as is the case for the classical Bessel polynomials. The formulas of strong asymptotics (Theorem 1 and Theorem 2) can be proved by using the saddlepoint method [4]. The application of potential theory [5] makes it possible to find measures for the distribution of zeros (equilibriummeasures) for the polynomials (3a) and (3b) (Corollaries 1 and 2). To write out the asymptotic formulas, we need the functions