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 $a>1$, $b\in (-1/3,1/3)\setminus \left\{0\right\}$. In this case we present the following positive continuous weights $p\left(\theta \right)=p(\theta ,a,b)$ on the unit circle for $y_n(x,a,b)$: $2\pi p(\theta ,a,b)=-1+2(a-1){\int }_0^1{e}^{-bucos\theta }cos(busin\theta ){(1-u)}^{a-2}du,$ where $\theta \in [0,2\pi ]$. Namely, we have ${\int }_0^{2\pi }{y}_n(e^{i\theta },a,b)y_m(e^{i\theta },a,b)p(\theta ,a,b)d\theta =C_n{\delta }_{n,m},C_n\ne 0,n,m=0,1,2,\dots .$ Notice that this orthogonality differs from the usual one for orthogonal polynomials on the unit circle. Some applications of the above orthogonality are given.