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.