On 4/3/08, Bill Gosper <rwmgosper@yahoo.com> wrote:
rwg>It's pretty clear that any product or sum over a period of a rational function of trigs comes out in closed form. You can interconvert such products and sums via trig, calculus, and generating function hacks. [...] So we get sums/prods over periods of some algebraic functions of trigs as well as rational.
I stand amazed. Is there some automatic algorithm lurking under there?
(Reminder: T_n(x) = cos(n acos x) = cosh(n acosh x) = 2 x T_n-1 - T_n-2, T_0(x)=1, T_1(x)=x. T_n(T_m(x)) = T_nm(x), so e.g. T_(n/2)(x) := T_n(sqrt((x+1)/2)) = sqrt((T_n(x)+1)/2) .)
This last line does seem to work --- e.g. T_2(x) = 2x^2 - 1, T_4(x) = 8x^4 - 8x^2 + 1, 2x^2 - 1 = 2((x+1)^2) - 4(x+1) + 1 = sqrt(4x^4 - 4x^2 + 1) --- but I still don't quite understand why. Something to do with underlying exponential functions? WFL