Special values of the hypergeometric series II

Abstract

Several authors [1, 5, 9] have investigated the algebraic and transcendental values of the Gaussian hypergeometric series

(formula here)

for rational parameters a, b, c and algebraic and rational values of z ∈ (0, 1). This led to several new identities such as

(formula here)

and

(formula here)

where Γ(x) denotes the gamma function. It was pointed out by the present authors [6] that these results, and others like it, could be derived simply by combining certain classical F transformation formulae with the singular values of the complete elliptic integral of the first kind K(k), where k denotes the modulus.

Here, we pursue the methods used in [6] to produce further examples of the type (1·2) and (1·3). Thus, we find the following results:

(formula here)

The result (1·6) is of particular interest because the argument and value of the F function are both rational.