rsm>Does anyone have any other examples of hypergeometric transformations or summation formulas with nonlinearly constrained parameters? On p122 of "A calculus of series rearrangements" in Algorithms and Complexity--New Directions and Recent Results, J. Traub & H. T. Kung, eds, Academic Press, 1976, I gave block([fancy_display:false], print(hyper_f[4,3]([1,a,b,c],[d+1,e+1,-e-d+c+b+a+1]) =(d*(-e-d+c+a)*(-e-d+c+b+a)*e*(1-((Gamma(d)*Gamma(-e-d+c+b+a)*Gamma(e)) /(Gamma(a)*Gamma(b)*Gamma(c)))))/((a-d)*(c-d)*(a-e)*(c-e))), print((-e-d+c+b+a)^2=-e^2-d^2+c^2+b^2+a^2),0)$ hyper_f ([1, a, b, c], [d + 1, e + 1, - e - d + c + b + a + 1]) = 4, 3 d (- e - d + c + a) (- e - d + c + b + a) e gamma(d) gamma(- e - d + c + b + a) gamma(e) (1 - --------------------------------------------) gamma(a) gamma(b) gamma(c) /((a - d) (c - d) (a - e) (c - e)) provided 2 2 2 2 2 2 (- e - d + c + b + a) = - e - d + c + b + a . Constraints on symmetric functions of parameters are typically equivalent to simpler constraints on the coefficients of the term ratio polynomials, which equally well characterize the series. For identities imposing nonlinear constraints on the coefficients themselves, see "Decision procedure for indefinite hypergeometric summation," Proc. Natl. Acad. Sci., Jan 1978, p 42. --Bill Gosper