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An integral representation for the Bessel form. (English) Zbl 0827.33006
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 $\alpha\ne -n/2$. The purpose of the author is to establish an explicit formula for an orthogonalizing weight, that is, for an absolutely continuous on $\bbfR$ function $U$ with rapid decay such that $$\int_{-\infty}^{+\infty} P_n (x) P_m (x) U(x) dx=0 \qquad \text{for} \qquad m\ne n.$$ 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 {\it K. H. Kwon}, {\it S. S. Kim} and {\it S. S. Han} [Bull. Lond. Math. Soc. 24, No. 4, 361-367 (1992; Zbl 0768.33007)], where the case of $\alpha=1$ was studied.

33C45Orthogonal polynomials and functions of hypergeometric type
42C05General theory of orthogonal functions and polynomials
Full Text: DOI
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