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.



How to prove the following Whittaker formula
If you start from the expression for $W\_{k,m}(z)$ given [here](https://en.wikipedia.org/wiki/Whittaker_function),


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


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](https://en.wikipedia.org/wiki/Confluent_hypergeometric_function#Kummer's_equation)
$$U(a,b,z)=\frac{\Gamma(1-b)}{\Gamma(a+1-b)}M(a,b,z)+\frac{\Gamma(b-1)}{\Gamma(a)}z^{1-b}M(a+1-b,2-b,z),$$
with $a:=m-k+\frac{1}{2}$ and $b:=1+2m$, and also use the formula for $M\_{k,m}$ in the [first link](https://en.wikipedia.org/wiki/Whittaker_function),
$$M\_{k,m}\left(z\right) = e^{-\tfrac{z}{2}}z^{m+\tfrac{1}{2}}M\left(m-k+\frac{1}{2}, 1+2m, z\right),$$
you end up with $W\_{k,m}$ as the linear combination of $M\_{k,m}$ and $W\_{k,-m}$ as you wrote.


