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

The sequence of monic Bessel polynomials {P n (x)} n0 is generated by the recurrence formula

P n+1 (x)=(x-β n )P n (x)-γ n P n-1 (x),n0,

with P -1 (x)=0, P 0 (x)=1 and β n and γ n depending on a (generally speaking, complex) parameter α-n/2. The purpose of the author is to establish an explicit formula for an orthogonalizing weight, that is, for an absolutely continuous on function U with rapid decay such that

- + P n (x)P m (x)U(x)dx=0formn·

Using the semi- classical character of the Bessel form, suitable formulations are obtained, although they are not proved for all values of the parameter α. 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 α=1 was studied.

33C45Orthogonal polynomials and functions of hypergeometric type
42C05General theory of orthogonal functions and polynomials