Analytic Continuation of Appell's Hypergeometric Series F2 to the Vicinity of the Singular Point x = 1, y = 1

Abstract

The infinite series, absolutely convergent if |x| + |y| < 1, for Appell's F2 (α, β, β′, γ, γ′; x, y) is analytically continued into a linear combination of four infinite series in powers of (x - 1) and (y - 1); each of the latter four series is absolutely convergent if |x − 1| + |y − 1| < 1. The analytic continuation is carried out by manipulation of the Mellin-Barnes integral representations for the hypergeometric functions appearing in the course of the calculation.

The Fractional Orthogonal Derivative

“…Remark 5.1. It is well-known [7, 5.9 (10)] that the F 2 (a 0 ; b 1 , b 2 ; c 1 , c 2 ; x, y) function is a solution of the following system of partial differential equations…”

“…The functions F 2 , F 3 , H 2 and F P are all five-parametric double-hypergeometric functions which are solutions of a system of two partial differential equations of second order [14]. This system is associated with the Appell function F 2 (a, b 1 , b 2 , c 1 , c 2 ; x, y) [7, 5.9 (10)]. Note that the orthogonal polynomials (2.6) and (2.8) used for the fractional orthogonal derivative are also expressed in terms of F 2 functions.…”

Integrals containing confluent hypergeometric functions with applications to perturbed singular potentials

“…Unfortunately, the direct use of the series representation (2.1) for the Appell's hypergeometric series F 2 does not converge for the variables needed in the computation of Landau and Lifshitz's integral (1.1), namely h < k + k ′ . However, there are several analytic continuations of Appell's hypergeometric series F 2 available in the literature [4], [12]- [14]. In the next section we shall give several new analytic continuation of F 2 that can be used directly in the computation of the integral (1.1).…”

“…However, there are several analytic continuations of Appell's hypergeometric series F 2 available in the literature [4], [12]- [14]. In the next section we shall give several new analytic continuation of F 2 that can be used directly in the computation of the integral (1.1).…”