On Quotients of Gamma and Beta Random Variables and Related Hypergeometric Identities

This study presents the development of a comprehensive framework wherein probabilistic and analytical techniques collaboratively produce identities pertinent to special functions. The fundamental premise is that ratios of random variables, particularly within the family of Gamma and Beta distributions, inherently generate integral representations and hypergeometric structures that can be harnessed to derive non-trivial summation formulas. A pivotal outcome of this investigation is that, in the context of independent and identically distributed random variables, symmetry plays a crucial role. A key property of ratios of i.i.d. random variables translates into straightforward probabilistic assertions that directly lead to precise analytic identities, thereby enabling the recovery of classical results such as Kummer's and Watson's summation theorems as consequences of distributional symmetry. When the assumption of identical distribution is relaxed, hypergeometric representations continue to exist; however, the absence of symmetry impedes closed-form evaluations of the same nature. This contrast underscores symmetry as the fundamental mechanism underlying exact summation formulas and elucidates why such identities are contingent upon specific parameter configurations. More generally, this methodology offers a probabilistic interpretation of hypergeometric identities and establishes a conceptual connection between probability theory and the theory of special functions. Furthermore, it suggests that more expansive constructions, based on non-identically distributed variables or iterated processes, could be fruitfully explored. In particular, examining ratios may lead to new identities or alternative derivations of classical results.