Modified Spherical Bessel Function of the First Kind

A modified spherical Bessel function of the first kind (Abramowitz and Stegun 1972), also called a "spherical modified Bessel function of the first kind" (Arfken 1985), is the first solution to the modified spherical Bessel differential equation, given by

(1)

where  is a modified Bessel function of the first kind (Arfken 1985, p. 633).

For positive , the first few values for small nonnegative integer indices are

			(2)
			(3)
			(4)
			(5)
			(6)

(OEIS A094674 and A094675).

Writing

(7)

the  are given by the recurrence equation

(8)

together with

			(9)
			(10)

(Abramowitz and Stegun 1972, p. 443).

The parity of  is  (Arfken 1985, p. 633).

 is related to the spherical Bessel function of the first kind  by

(11)

for  and integer  (Arfken 1985, p. 633).

They also satisfy the differential identities

			(12)
			(13)

and the recurrence relations

			(14)
			(15)

(Arfken 1985, p. 634).

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). "Modified Spherical Bessel Functions." §10.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 443-445, 1972.

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 633-634, 1985.

Sloane, N. J. A. Sequences A094674 and A094675 in "The On-Line Encyclopedia of Integer Sequences."