The analytic continuation of the Gaussian hypergeometric function 2F1(a,b;c;z) for arbitrary parameters

π and the hypergeometric functions of complex argument

“…The analytic continuation 2 F 1 is a subject of recent research like [2], or [13], where a complicated formula is found involving three determinations of 2 F 1 . Therefore we deem our finding on the value of 2 F 1 for z = 2 is of some interest not only to the analytic number theory, but also to fixing a benchmark for any computational efforts.…”

Tempered stable distributions and processes in finance: numerical analysis

“…Equation (2.12) implies the evaluation of the hypergeometric 2 F 1 (a, b; c; z) function only on the straight line represented by the subset I = {iy | y ∈ R} of the complex plane C. We do not need a general algorithm to evaluate the function on the entire complex plane C, but just on a subset of it. This can be done by means of the analytic continuation, without having recourse neither to numerical integration nor to numerical solution of a differential equation [17] (for a complete table of the analytic continuation formulas for arbitrary values of z ∈ C and of the parameters a, b, c, see [3] or [9]). The hypergeometric function belongs to the special function class and often occurs in many practical computational problems.…”

On the closed form solutions for non-extensive Value at Risk

“…where 2 F 1 is the Gauss hypergeometric function (Ghf) [30][31][32]. The series converges for |z| < 1 and can be analytically continued onto the entire complex plane cut along [1, ∞],…”