Complex zeros of Bessel function derivatives and associated orthogonal polynomials

Abstract

We introduce a sequence of orthogonal polynomials whose associated moments are the Rayleigh-type sums, involving the zeros of the Bessel derivative $J_{\hat{I}}{½}^{\prime }$ of order $\hat{I}½$. We also discuss the fundamental properties of those polynomials such as recurrence, orthogonality, etc. Consequently, we obtain a formula for the Hankel determinant, elements of which are chosen as the aforementioned Rayleigh-type sums. As an application, we complete the Hurwitz-type theorem for $J_{\hat{I}}{½}^{\prime }$, which deals with the number of complex zeros of $J_{\hat{I}}{½}^{\prime }$ depending on the range of $\hat{I}½$.

33 pages, 4 figures