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An integral representation for the Bessel form. (English) Zbl 0827.33006

The sequence of monic Bessel polynomials ${\left\{{P}_{n}\left(x\right)\right\}}_{n\ge 0}$ is generated by the recurrence formula

${P}_{n+1}\left(x\right)=\left(x-{\beta }_{n}\right){P}_{n}\left(x\right)-{\gamma }_{n}{P}_{n-1}\left(x\right),\phantom{\rule{2.em}{0ex}}n\ge 0,$

with ${P}_{-1}\left(x\right)=0$, ${P}_{0}\left(x\right)=1$ 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 $ℝ$ function $U$ with rapid decay such that

${\int }_{-\infty }^{+\infty }{P}_{n}\left(x\right){P}_{m}\left(x\right)U\left(x\right)dx=0\phantom{\rule{2.em}{0ex}}\text{for}\phantom{\rule{2.em}{0ex}}m\ne n·$

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 K. H. Kwon, S. S. Kim and S. S. Han [Bull. Lond. Math. Soc. 24, No. 4, 361-367 (1992; Zbl 0768.33007)], where the case of $\alpha =1$ was studied.

##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type 42C05 General theory of orthogonal functions and polynomials
##### Keywords:
Bessel polynomials