Are you sure this step:
is correct? You skipped a lot of steps there. I'm not sure I followed them all.
Just double-check your work, is all.
The idea is very elegant. Nice work!
Ah, well. Those special functions, including the beta functions, are all related to the confluent hypergeometric functions. It's probably equivalent, through an amazing amount of difficult analysis. It even has similar arguments to the functions as what AlexisM has already computed.