Legendre’s relation (19.7.1) can be written
The complete cases of and have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). Upper signs apply if , and lower signs if :
is symmetric only in and , but either (nonzero) or (nonzero) can be moved to the third position by using
or the corresponding equation with and interchanged.
where both summations extend over the three cyclic permutations of .
Connection formulas for are given in Carlson (1977b, pp. 99, 101, and 123–124).
Let be real and nonnegative, with at most one of them 0. Change-of-parameter relations can be used to shift the parameter of from either circular region to the other, or from either hyperbolic region to the other (§19.20(iii)). The latter case allows evaluation of Cauchy principal values (see (19.20.14)).
and may be permuted. Also,
For each value of , permutation of produces three values of , one of which lies in the same region as and two lie in the other region of the same type. In (19.21.12), if is the largest (smallest) of , and , then and lie in the same region if it is circular (hyperbolic); otherwise and lie in different regions, both circular or both hyperbolic. If , then and ; hence