Positive orthogonalizing weights on the unit circle for the generalized Bessel polynomials

Abstract

In this paper we study the generalized Bessel polynomials $y_n(x,a,b)$ (in the notation of Krall and Frink). Let $a>1$, $b\in(-1/3,1/3)\backslash\{ 0\}$. In this case we present the following positive continuous weights $p(\theta) = p(\theta,a,b)$ on the unit circle for $y_n(x,a,b)$: $$ 2\pi p(\theta,a,b) = -1 + 2(a-1) \int_0^1 e^{-bu\cos\theta} \cos(bu\sin\theta) (1-u)^{a-2} du, $$ where $\theta\in[0,2\pi]$. Namely, we have $$ \int_0^{2\pi} y_n(e^{i\theta},a,b) y_m(e^{i\theta},a,b) p(\theta,a,b) d\theta = C_n \delta_{n,m},\qquad C_n\not=0,\ n,m\in\mathbb{Z}_+. $$ Notice that this orthogonality differs from the usual orthogonality of OPUC. Some applications of the above orthogonality are given.