Do you remember anything at all about the gamma function?
The integral on the right hand side may be recognized as the gamma function.
Up to elementary factors, it is a special case of the double gamma function.
A good solution to this is the gamma function.
It is somewhat problematic that a large number of definitions have been given for the gamma function.
The gamma function is defined for all complex numbers.
However, the gamma function does not appear to satisfy any simple differential equation.
A thorough and systematic book devoted entirely to the subject of the gamma function.
See Particular values of the gamma function for calculated values.
A definite and generally applicable characterization of the gamma function was not given until 1922.