Some extension of the Bessel-type orthogonal polynomials

Abstract

We consider the perturbation of the classical Bessel moment functional by the addition of the linear functional $M_0\delta(x)+M_1\delta'(x)$, where $M_0$ and $M_1\in\bf R$. We give necessary and sufficient conditions in order for this functional to be a quasi-definite functional. In such a situation we analyze the corresponding sequence of monic orthogonal polynomials $B \alpha,M_0,M_1}_n(x)$. In particular, a hypergeometric representation $(_4F_2)$ for them is obtained.

Furthermore, we deduce a relation between the corresponding Jacobi matrices, as well as the asymptotic behavior of the ratio $B \alpha,M_0,M_1}_n(x)/B alpha_n(x)$, outside the closed contour $\Gamma$ containing the origin, and the difference between the new polynomials and the classical ones, inside $\Gamma$.