A couple of years ago on this list, Bill Gosper asked about hypergeometric identities in which the parameters are constrained nonlinearly, instead of linearly. The example he gave was the summation formula

c + b + a + 4 hyper_f ([c - a + 1, c + b, c - b + 1, c + a, -------------], 5, 4 3

3 c + b + a + 1 1 [b + a, c, c + -, -------------], -) 2 3 4

(b + a - 1)! (2 c + 1)! = -------------------------------------------, c (c + b + a + 1) (c + a - 1)! (c + b - 1)!

which holds if c = a*b. I've just written a paper on how Euler's transformation can be extended from the Gauss hypergeometric function to hypergeometric functions of higher order. It's at http://www.arxiv.org/math.CA/0302084 . The natural analogues of Euler's transformation turn out to involve, lo and behold, nonlinear parametric constraints. As a corollary, I've derived a curious nonlinearly parametrized summation formula for 2F1(-1), namely 2F1 (6t^2 - 11t + 6, 4; -6t^2 + 5t + 3; -1) = -t (6t+1) (6t^2 - 5t - 2). This can alternatively be derived from classical contiguous function relations, by uniformizing a genus-0 algebraic curve. But it certainly looks mysterious. In the course of writing the paper, I discovered an even more intriguing identity discovered by Niblett (a student of Chaundy's, I think) in the 1950s. The citation is J. D. Niblett, "Some Hypergeometric Identities", Pacific J. Math. 2 (1953), 219-224. Niblett's identity is a quadratic function transformation, which expresses 5F4(x) in terms of 3F2(-4x/(1-x)^2) whenever the hypergeometric parameters satisfy a certain nonlinear condition. There are three free parameters. Does anyone have any other examples of hypergeometric transformations or summation formulas with nonlinearly constrained parameters? How to verify any such identity algorithmically is an interesting problem. --Robert