Positive orthogonalizing weights on the unit circle for the generalized Bessel polynomials (Q6651977)

The author gives a thorough dicscussion of the generaized Bessel polynomials and questions about real and complex orthogonality, that started with the important paper by \textit{H. L. Krall} and \textit{O. Frink} [Trans. Am. Math. Soc. 65, 100--115 (1949; Zbl 0031.29701)], given in the form\N\[\Ny_n(x)=y_n(x,a,b)={}_2F_0(-n,n+a-1; - ;-\frac{x}{b}),\ (a\not\in\mathbb{N}^{+},\ b\not= 0).\N\]\NThey are orthogonal on the unit circle:\N\[\N\int_{0}^{2\pi}\, y_n(e^{i\theta},a,b)y_m(e^{i\theta},a,b)\rho (\theta,a,b)d\theta=-\frac{c_n}{b}\delta_{n,m}, \ c_n\not= 0,;n,m\in\mathbb{N}, \tag{\(\ast\)}\N\]\Nwith\N\[\N2\pi p(\theta,a,b)=-1+\frac{2(a-1)}{b}\cos{\theta}-\frac{2\pi i}{b} (e^{i\theta} \rho(e^{i\theta},a,b)+e^{-i\theta} \rho(e^{-i\theta},a,b))\N\]\N\Nfor \(\theta\in [0,2\pi ]\) and\N\[\N2\pi i\rho(z,a,b)= a-1-\frac{b}{z}\,{}_1F_1(1;a;-b/z) ,\ z\in\mathbb{C}\setminus\{0\}.\N\]\NThe main result is:\N\N{Theorem 1.} let \(x_0\) be the unique number from \([0,\pi/2]\), such that \(2\cos{x_0}=e^{x_0}\); let \(a>1,b\in (-x_0,x_0)\setminus \{0\} \supseteq (-1/3,1/3) \setminus \{0\}\) be fixed numbers.\N\NThen the generalized Bessel polynomials \(y_n(x,a,b)\) satisfy the orthogonality relations \((\ast)\).\N\NThe layout of the paper is as folllows:\N\N{\S1. Introduction} (\(3\frac{1}{2}\) pages)\N\NNotations and a historical overview of the different types of orthogonalty studied since 1949.\N\N{\S2. Positive weights for Bessel poynomials} (\(4\) pafes)\N\NThe main Theorem 1 and its proof.\N\N{\S3. applications of C-orthogonality relations} (\(3\) pages)\N\N{References} (\(20\) items)