## Whittaker Function

Solutions to the Whittaker Differential Equation. The linearly independent solutions are

 (1)

and , where is a Confluent Hypergeometric Function. In terms of Confluent Hypergeometric Functions, the Whittaker functions are

 (2)

 (3)

(see Whittaker and Watson 1990, pp. 339-351). However, the Confluent Hypergeometric Function disappears when is an Integer, so Whittaker functions are often defined instead. The Whittaker functions are related to the Parabolic Cylinder Functions. When and is not an Integer,

 (4)

When and is not an Integer,

 (5)

Whittaker functions satisfy the Recurrence Relations
 (6)

 (7)

 (8)

See also Confluent Hypergeometric Function, Kummer's Formulas, Pearson-Cunningham Function, Schlömilch's Function, Sonine Polynomial

References

Abramowitz, M. and Stegun, C. 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.

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.

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