Reduction of the multivariable Lauricella-D hypergeometric function to the Gauss 2F1 via finite-part integration

Abstract

This study presents a formal derivation of new reduction identities for the multivariable Lauricella-D hypergeometric function F_D using finite-part integration. By mapping Euler-type integral representations to complex contour integrals, we demonstrated a technique for decoupling coupled multivariable dependencies into linear superpositions of independent Gauss hypergeometric functions. We specifically established closed-form identities for the two-variable case over the interval [0, ∞) and the three-variable case over the finite domain [0,1]. The restriction of the latter to finite bounds was necessitated by the lack of existing mathematics on the asymptotic expansion of F_D⁽³⁾ as its arguments approach infinity, identifying a key area for further analytical development. The resulting identities provide computationally stable alternatives to high-dimensional summations and extend the utility of Lauricella functions in solving complex  systems where standard symbolic computation fails.