Harmonic–Multifactorial Identity via Gamma and Pochhammer Structures

We establish a closed-form identity linking ratios of multifactorials to finite weighted sums involving harmonic number differences. This identity unifies factorial and double-factorial cases under a Gamma–Pochhammer framework. The proof relies on Gamma function reductions and a finite hypergeometric summation identity, with digamma reformulations and asymptotic consequences. Keywords: Multifactorials, Gamma Function, Harmonic Numbers, Hypergeometric Series, Special Functions, Number Theory.