Reduction identities for the generalized hypergeometric function 2F2 arising from the solution to the diffusion equation through finite-part integration

Abstract

We obtain new reduction identities for the generalized hypergeometric function involving linear combinations of ₂F₂(z) over the field of polynomials in z with integer coefficients. These identities are derived from a linear superposition of a separable solution to the diffusion equation that is expressed as a parametrized integral which diverges outside its strip of analyticity in the complex parameter space. The relevant solution to the diffusion equation from which the identities are derived is the regularized limit at the poles of the analytic continuation of the integral. The regularized limit coincides with the finite part of the integral when extended outside of its strip of analyticity. The known contour integral representation of the finite-part integral at pole singularities then establishes that the finite part is a legitimate solution to the diffusion equation. The enforcement of this fact leads to the identities reported in this paper.