Whittaker Differential Equation

(1)

Let , where  denotes a Whittaker function. Then (1) becomes

(2)

Rearranging,

(3)

(4)

so

(5)

where  (Abramowitz and Stegun 1972, p. 505; Zwillinger 1997, p. 128). The solutions are known as Whittaker functions. Replacing  by , the solutions can also be written in the form

(6)

where  is a confluent hypergeometric function of the second kind and  is a generalized Laguerre polynomial.

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 505, 1972.

Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 128, 1997.