Whittaker Function

The Whittaker functions arise as solutions to the Whittaker differential equation. The linearly independent solutions to this equation are

			(1)
			(2)

and , where is a confluent hypergeometric function of the second kind and  is a Pochhammer symbol. In terms of confluent hypergeometric functions of the first and second kinds, these solutions are

			(3)
			(4)

(Abramowitz and Stegun 1972, p. 505; Whittaker and Watson 1990, pp. 339-351).

These functions are implemented in the Wolfram Language as WhittakerM[k, m, z] and WhittakerW[k, m, z], respectively.

Whittaker and Watson (1990, p. 340) define

(5)

whenever  and  is not an integer.

A particular case is given by

(6)

for  (Whittaker and Watson 1990, p. 341, adjusting the normalization of  to conform to the modern convention).

The Whittaker functions are related to the parabolic cylinder functions through

(7)

When  and  is not an integer,

(8)

When  and  is not an integer,

(9)

Whittaker functions satisfy the recurrence relations

			(10)
			(11)
			(12)

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). "Confluent Hypergeometric Functions." Ch. 13 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 503-515, 1972.

Becker, P. A. "On the Integration of Products of Whittaker Functions with Respect to the Second Index." J. Math. Phys. 45, 761-773, 2004.

Iyanaga, S. and Kawada, Y. (Eds.). "Whittaker Functions." Appendix A, Table 19.II in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1469-1471, 1980.

Meijer, C. S. "Über die Integraldarstellungen der Whittakerschen Funktion  und der Hankelschen und Besselschen Funktionen." Nieuw Arch. Wisk. 18, 35-57, 1936.

Whittaker, E. T. "An Expression of Certain Known Functions as Generalised Hypergeometric Functions." Bull. Amer. Math. Soc. 10, 125-134, 1904.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.