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 $${\int }_0^{2\pi }{y}_n(e^{i\theta },a,b)y_m(e^{i\theta },a,b)p(\theta ,a,b)d\theta =C_n{\delta }_{n,m},C_n\ne 0,n,m\in {\mathbb{Z}}_{+}.$$ Notice that this orthogonality differs from the usual orthogonality of OPUC. Some applications of the above orthogonality are given.

16 pages