Bessel Polynomial

Krall and Fink (1949) defined the Bessel polynomials as the function

			(1)
			(2)

where  is a modified Bessel function of the second kind. They are very similar to the modified spherical bessel function of the second kind . The first few are

			(3)
			(4)
			(5)
			(6)
			(7)

(OEIS A001497). These functions satisfy the differential equation

(8)

Carlitz (1957) subsequently considered the related polynomials

(9)

This polynomial forms an associated Sheffer sequence with

(10)

This gives the generating function

(11)

The explicit formula is

			(12)
			(13)

where  is a double factorial and  is a confluent hypergeometric function of the first kind. The first few polynomials are

			(14)
			(15)
			(16)
			(17)

(OEIS A104548).

The polynomials satisfy the recurrence formula

(18)

REFERENCES:

Carlitz, L. "A Note on the Bessel Polynomials." Duke Math. J. 24, 151-162, 1957.

Grosswald, E. Bessel Polynomials. New York: Springer-Verlag, 1978.

Krall, H. L. and Fink, O. "A New Class of Orthogonal Polynomials: The Bessel Polynomials." Trans. Amer. Math. Soc. 65, 100-115, 1949.

Roman, S. "The Bessel Polynomials." §4.1.7 in The Umbral Calculus. New York: Academic Press, pp. 78-82, 1984.

Sloane, N. J. A. Sequences A001497, A001498, and A104548 in "The On-Line Encyclopedia of Integer Sequences."