NEW PROPERTIES OF THE P. E. APPELL HYPERGEOMETRIC SERIES F2(α;β, β′;γ, γ′;x, y) TO THE VICINITY OF THE SINGULAR POINT (1, 1) AND NEAR THE BOUNDARY OF ITS DOMAIN OF CONVERGENCE D2:|x|+|y|<1

Abstract

Exact analytical representations for Appell's series F2(x, y) to the vicinity of the singular point (1, 1) and near the boundary of its domain of convergence are given. It is shown that Appell's functions F2(1, 1) and F3(1, 1) have the property of mirror-like symmetry with respect to the center j0 ↦ -(1/2) under the change j ↦ -j-1, z∈ℤ, and they correlate between each other.