Whittaker Function

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

$$M_{k,m}(z)\equiv z^{1/2+m}e^{-z/2}\left[{1+{{1\over 2}+m-k\o... ...+m-k)({3\over 2}+m-k)\over 2!(2m+1)(2m+2)} z^2+\ldots}\right],$$ (1)

and $M_{k,-m}(z)$, where $M_{k,m}(z)$ is a Confluent Hypergeometric Function. In terms of Confluent Hypergeometric Functions, the Whittaker functions are

$$M_{k,m}(z)=e^{-z/2}z^{m+1/2} {}_1F_1({\textstyle{1\over 2}}+ m-k,1+ 2m;z)$$ (2)

$$W_{k,m}(z)=e^{-z/2}z^{m+1/2} U({\textstyle{1\over 2}}+ m-k,1+ 2m;z)$$ (3)

(see Whittaker and Watson 1990, pp. 339-351). However, the Confluent Hypergeometric Function disappears when $2m$ is an Integer, so Whittaker functions are often defined instead. The Whittaker functions are related to the Parabolic Cylinder Functions. When $\vert{\rm arg\ } z\vert< 3\pi/2$ and $2m$ is not an Integer,

$$W_{k,m}(z)={\Gamma(-2m)\over \Gamma({\textstyle{1\over 2}}-m... ...amma(2m)\over \Gamma({\textstyle{1\over 2}}+m-k)} M_{k,-m}(z).$$ (4)

When $\vert\arg (-z)\vert< 3\pi/2$ and $2m$ is not an Integer,

$$W_{-k,m}(-z)={\Gamma(-2m)\over \Gamma({\textstyle{1\over 2}}... ...ma(2m)\over \Gamma({\textstyle{1\over 2}}+m+k)} M_{-k,-m}(-z).$$ (5)

Whittaker functions satisfy the Recurrence Relations

$$W_{k,m}(z)=z^{1/2}W_{k-1/2,m-1/2}(z)+({\textstyle{1\over 2}}-k+m)W_{k-1,m}(z)$$ (6)

$$W_{k,m}(z)=z^{1/2}W_{k-1/2,m+1/2}(z)+({\textstyle{1\over 2}}-k-m)W_{k-1,m}(z)$$ (7)

$$zW_{k,m}'(z)=(k-{\textstyle{1\over 2}}z)W_{k,m}(z)-[m^2-(k-{\textstyle{1\over 2}})^2]W_{k-1,m}(z).$$ (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.