Whittaker functions

The functions $M _ {\lambda , \mu } ( z)$ and $W _ {\lambda , \mu } ( z)$ which are solutions of the Whittaker equation

$$\tag{* } w ^ {\prime\prime} + \left ( \frac{ {1 / 4 } - \mu ^ {2} }{z ^ {2} } + { \frac \lambda {z} } - { \frac{1}{4} } \right ) w = 0.$$

The function $W _ {\lambda , \mu }$ satisfies the equation

$$W _ {\lambda , \mu } ( z) = \ \frac{\Gamma (- 2 \mu ) }{\Gamma \left ( { \frac{1}{2} } - \lambda - \mu \right ) } M _ {\lambda , \mu } ( z) + \frac{\Gamma ( 2 \mu ) }{\Gamma \left ( { \frac{1}{2} } - \lambda + \mu \right ) } M _ {\lambda , - \mu } ( z).$$

The pairs of functions $M _ {\lambda , \mu } ( z) , M _ {\lambda , - \mu } ( z)$ and $W _ {\lambda , \mu } ( z) , W _ {- \lambda , \mu } ( z)$ are linearly independent solutions of the equation (*). The point $z = 0$ is a branching point for $M _ {\lambda , \mu } ( z)$, and $z = \infty$ is an essential singularity.

Relation with other functions:

with the degenerate hypergeometric function:

$$M _ {\lambda , \mu } ( z) = \ z ^ {\mu + 1/2 } e ^ {-z/2} \Phi \left ( \mu - \lambda + \frac{1}{2} ; \ 2 \mu + 1; z \right ) ,$$

with the modified Bessel functions and the Macdonald function:

$$M _ {0, \mu } ( z) = \ 2 ^ {2 \mu } \Gamma ( \mu + 1) \sqrt z I _ \mu \left ( { \frac{z}{2} } \right ) ,$$

$$W _ {0, \mu } ( z) = \sqrt { \frac{z} \pi } K _ \mu \left ( { \frac{z}{2} } \right ) ;$$

with the probability integral:

$$W _ {- {1 / 4 } , {1 / 4 } } ( z) = \ 2 z ^ {1/4} e ^ {z/2} \mathop{\rm Erfc} ( \sqrt z );$$

with the Laguerre polynomials:

$$W _ {n + \mu + 1/2, \mu } ( z) = \ n! (- 1) ^ {n} z ^ {\mu + 1/2 } e ^ {-z/2} L _ {n} ^ {2 \mu } ( z).$$

References

[1]	H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953)
[2]	E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952)

Comments

The Whittaker function $W _ {\lambda , \mu }$ can be expressed in terms of the $\Psi$-function introduced in confluent hypergeometric function:

$$W _ {\lambda , \mu } ( z) = e ^ {- z/2 } z ^ {\mu + 1/2 } \Psi ( \mu - \lambda + 1/2; 2 \mu + 1; z).$$

Thus, the special cases discussed in confluent hypergeometric function can be rewritten as special cases for the Whittaker functions. See also the references given there.