Several authors [1, 5, 9] have investigated the algebraic and
transcendental values of the Gaussian hypergeometric series
for rational parameters a, b, c and algebraic and rational values of
z ∈ (0, 1). This led to several new identities such as
where Γ(x) denotes the gamma function. It was pointed out by the present authors
 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  to produce further examples of the type
(1·2) and (1·3). Thus, we find the following results:
The result (1·6) is of particular interest because the argument and value of the F
function are both rational.