On the integration of the differential equations of five-parametric double-hypergeometric functions of second order

Abstract

Solutions of the differential equations associated with Appell’s hypergeometric function F₂ (a,b₁,b₂,c₁,c₂; x₁,x₂) are obtained by considering them as a Laplace transform of a product of two confluent hypergeometric functions. Since these differential equations may be transformed into the equations associated with Appell’s function F₃(a,b₁,b₂,c₁,c₂; x₁,x₂) and Horn’s function H₂(a,b,c,d,e; x₁,x₂), these equations are solved simultaneously. A set of 36 distinct solutions in terms of six types of series is given. This set is the smallest set which accounts for the general behavior of any of the three double‐hypergeometric second order functions F₂, F₃, and H₂. Various representations of the solutions in terms of series and integrals are given as well as connections between the solutions which continue the functions analytically.