On 5/27/08, Fred lunnon <fred.lunnon@gmail.com> wrote:
Neil --- re-revised --- apologies --- Fred [Read my lips: no more circle product theorems!]
Well, maybe just one more ... Writing x' for [(x+1)/phi^2] etc, we have already x o y = 3 x y - x y' - x' y; x o y o z = 8 x y z - 3 (x' y z + x y' z + x y z') + (x y' z' + x' y z' + x' y' z); and having nothing better to do, we might idly compute x o y o z o w = 21 (x y z w) - 8 (x y z w' + ...) + 3 (x y z' w' + ...) - 1 (x y' z' w' + ...) + 0 (x' y' z' w'); x o y o z o w o v = 55 (x y z w v) - 21 (x y z w v' + ...) + 8 (x y z w' v' + ...) - 3 (x y z' w' v' + ...) + 1 (x y' z' w' v' + ...) - 0 (x' y' z' w' v'); At which point dawns --- utterly pointless, but rather engaging --- a formula for the circle product of n arguments: THEOREM: x^1 o x^2 o ... o x^n = \sum_{0 <= m <= n} (-1)^(n-m) F_{2m} \sum_{n_C_m} x^1 x^2 ... x^m x^{m+1}' ... x^n' where the inner sum ranges over all monomials with m undashed and n-m dashed variables; and the coefficients are alternate Fibonacci numbers depending (apart from sign) only on m.