Bill Gosper: Then I bashed on it so hard that it got unnervingly ungrotesque: (*) EllipticK[1/4 (2 + (-2 + t)/Sqrt[1 - t])] == (1 - t)^(1/4) EllipticK[t] How could anything this simple be new? WDS: Let s=Sqrt[1-t]. Then (*) is rewritten as EllipticK[ 1/2 - (1+s^2)/(4s) ] == EllipticK[ -(1-s)^2/(4s) ] == s^(1/2) EllipticK[1-s^2] Now let s=1-m. Then this is EllipticK[ -m^2/(4(1-m)) ] == (1-m)^(1/2) EllipticK[(2-m)m]. Gosper: WDS formula affliected by "branch bug" Empirically, only works for m west of 1: gosper.org/braunch.pdf WDS: Ok. That demonstrates empirically that my formulas are correct. They work for Re(m)<1, and I claim they also work for Re(m)>1 PROVIDED you choose the correct branch of the EllipticK function [the arbitrary choice made by your software is the "wrong" branch] -- proof: analytic continuation. Gosper: t->1-z is safe and really nice: WDS: Gosper was trying to say this: (**) (1-t)^(-1/4) EllipticK[- ( t^(1/4) - t^(-1/4))^2/4] == EllipticK[t] Now. I think these formulas should be provable, thus demonstrating the correctness of everything, as follows. You note that y=K(x) and y=K(sqrt(1-x^2)) are the basis solutions of the linear differential equation (EQ 1.3.8 in Borwein book "pi & the AGM") (x^3 - x) * y'' + (3*x^3-1) * y' + x*y = 0. Using (**), compute the LHS of the diffl equation using the LHS of (**), and verify it yields 0 if the LHS of the diffl equation yields 0 using the RHS of (**). If so, then we've proven anything obeying the diffl equation must obey the Gosper identity. This proof (if it works) will also prove a "twin" of your identity, for the other solution of the diffl eqn (or, it won't work for the other solution, in which case the initial conditions will be crucial to the proof). By the way, other K() identities in the Borwein book are on pages 10 & 16, probably also many more in there somewhere. It might be yours is equivalent to one Borweins knew about, I have not attempted to decide that.