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.
Generalizations of Jacobsthal sums and hypergeometric series over finite fields