An integral representation for the Bessel form
The sequence of monic Bessel polynomials \(\{P_n (x)\}_{n \geq 0}\) is generated by the recurrence formula \[ P_{n+1} (x)= (x- \beta_n) P_n (x)- \gamma_n P_{n-1} (x), \qquad n\geq 0, \] with \(P_{-1} (x)= 0\), \(P_0 (x)=1\) and \(\beta_n\) and \(\gamma_n\) depending on a (generally speaking, complex) parameter \(\alpha\neq -n/2\). The purpose of the author is to establish an explicit formula for an orthogonalizing weight, that is, for an absolutely continuous on \(\mathbb{R}\) function \(U\) with rapid decay such that \[ \int_{-\infty}^{+\infty} P_n (x) P_m (x) U(x) dx=0 \qquad \text{for} \qquad m\neq n. \] Using the semi- classical character of the Bessel form, suitable formulations are obtained, although they are not proved for all values of the parameter \(\alpha\). This generalizes the work of \textit{K. H. Kwon}, \textit{S. S. Kim} and \textit{S. S. Han} [Bull. Lond. Math. Soc. 24, No. 4, 361-367 (1992; Zbl 0768.33007)], where the case of \(\alpha=1\) was studied.
scientific article; zbMATH DE number 770350