How to prove the following Whittaker formula

Question

I am a theoretical physicist and I need help in proving the alternate Whittaker formula

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

in case where $| \arg z | < \frac { 3 } { 2 } \pi$ and 2$m$ is not an integer. Taking into account the appropriate definitions

$M _ { k . u } ( z ) = z ^ { \mu + 1 / 2 } e ^ { - z / 2 } \Phi \left( \frac { 1 } { 2 } - k + \mu , 2 \mu + 1 ; z \right) , \quad | \arg z | < \pi$ $W _ { k , \mu } ( z ) = z ^ { \mu + 1 / 2 } e ^ { - z / 2 }\Psi \left( \frac { 1 } { 2 } - k + \mu , 2 \mu + 1 ; z \right) , \quad | \arg z | < \pi$

Interestingly, I found this function as I was working my thesis related to the article "Creation of scalar and Dirac particles in the presence of a time varying electric field in an anisotropic Bianchi type I universe Victor M. Villalba* and Walter Greiner" where they used this formula and I tried to find a way to referred references and I found the author himself who discovered these functions in his book of "A course in modern analysis by Whittaker and Watson" p.346, chap.16.

Any help is appreciated.

Answer 1

If you start from the expression for $W_{k,m}(z)$ given here,

$$W_{k,m}\left(z\right) = e^{-\tfrac{z}{2}}z^{m+\tfrac{1}{2}}U\left(m-k+\frac{1}{2}, 1+2m, z\right)$$

and express the Tricomi function $U\left(m-k+\frac{1}{2}, 1+2m, z\right)$ in terms of the Kummer functions $M$ as here $$U(a,b,z)=\frac{\Gamma(1-b)}{\Gamma(a+1-b)}M(a,b,z)+\frac{\Gamma(b-1)}{\Gamma(a)}z^{1-b}M(a+1-b,2-b,z),$$ with $a:=m-k+\frac{1}{2}$ and $b:=1+2m$, and also use the formula for $M_{k,m}$ in the first link, $$M_{k,m}\left(z\right) = e^{-\tfrac{z}{2}}z^{m+\tfrac{1}{2}}M\left(m-k+\frac{1}{2}, 1+2m, z\right),$$ you end up with $W_{k,m}$ as the linear combination of $M_{k,m}$ and $W_{k,-m}$ as you wrote.