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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.

MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type 42C05 General theory of orthogonal functions and polynomials
Keywords:
Bessel polynomials
Full Text:
References:
 [1] A.J. Duran, Functions with given moments and weight functions for orthogonal polynomials, Rocky Mountain J. Math., to appear. · Zbl 0777.44003 [2] Dwight, H. B.: Tables of integrals and other mathematical data. (1961) · Zbl 0154.18410 [3] Evans, W. D.; Everitt, W. N.; Krall, A. M.; Kwon, K. H.; Littlejohn, L. L.: A solution to the general Bessel moment problem. World sci. Ser. appl. Anal. 1, 205-220 (1992) · Zbl 0858.33004 [4] Grosswald, E.: Bessel polynomials. Lecture notes in math. 698 (1978) · Zbl 0416.33008 [5] Kim, S. S.; Kwon, K. H.: Generalized weights for orthogonal polynomials. Diff. int. Eq. 4, No. 3, 601-608 (1991) · Zbl 0733.33006 [6] Kim, S. S.; Kwon, K. H.; Han, S. S.: Orthogonalizing weights of chebychev sets of polynomials. Bull. London math. Soc. 24, 361-367 (1992) · Zbl 0768.33007 [7] Krall, A. M.: The Bessel polynomial moment problem. Acta math. Acad. sci. Hungar. 38, 105-107 (1981) · Zbl 0431.33004 [8] Krall, H. L.; Frink, O.: A new class of orthogonal polynomials: the Bessel polynomials. Trans. amer. Math. soc. 65, 100-115 (1949) · Zbl 0031.29701 [9] Maroni, P.: Une théorie algébrique des polynômes orthogonaux. Application aux polynômes orthogonaux semiclassique. IMACS ann. Comput. appl. Math. 9, 95-130 (1991) · Zbl 0944.33500 [10] Maroni, P.: Variations around classical orthogonal polynomials. Connected problems. J. comput. Appl. math. 48, No. 1--2, 133-155 (1993) · Zbl 0790.33006 [11] Maroni, P.: Modified classical orthogonal polynomials associated with oscillating functions--open problems. Appl. numer. Math. 15, 259-283 (1994) · Zbl 0826.42020 [12] P. Maroni and E. Santi, Fibonacci polynomials, Pell, Lucas and Pell-Lucas polynomials as orthogonal polynomials, in preparation. [13] Stieltjes, T. J.: Ann. fac. Sci. Toulouse. 9, A1-A47 (1895)