So I'm writing a paper involving Gamma and Beta functions, Dirichlet integrals, Generalized Hypergeometric functions, and a small grab-bag of other miscellaneous special functions. There's this grouping of results that I am most certianly going to generalize; in fact I feel rather like quoting Guass, whom said "My results, I've had them for some time. Problem is, I don't yet know how I am to arrive at them." Blah, blah, blah... what I've proved is in the attached PDF. Have at it. Any conjectures?