I'm working with a Kummer equation of the form:
$z\frac{\partial^2 w}{\partial z^2}+(b-z)\frac{\partial w}{\partial z} -aw=0,$
which is solved by Kummer's Confluent Hypergeometric function:
${}_{1}F_{1}(a,b,z)=M(a,b,z).$
There is an identity, called Kummer's transformation, which states that:
$M(a,b,z)=e^{z}M(b-a,b,-z).$
I can currently prove this using the integral form of Kummer's equation:
$M(a,b,z)=\frac{\Gamma(b)}{\Gamma(b-a)\Gamma(a)}\int\limits_{0}^{1}e^{zt}t^{a-1}(1-t)^{b-a-1}\mathrm{d}t \text{, for }b>a>0.$
But this is only valid under the conditions $b>a>0$. I have not found any source that quotes these same conditions for the transformation so I assume it can be proved by some other means. Any ideas?
Cheers,
Hamzaan