What's the power series for ursine? -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Bill Gosper Sent: Sunday, February 21, 2010 2:34 PM To: math-fun@mailman.xmission.com Subject: [math-fun] Showing off to some kids my oft-quoted but never proved claim that "cyclotomic" sums and products of arbitrary trigs over a full period always have closed forms: sum(sin(x+%pi*j/n)*sin(y+%pi*j/n)/(sin(w+%pi*j/n)*sin(z+%pi*j/n)),j,1,n) = n*(cos((n-2)*z+y+x)/sin(n*z)+cos(y-x)*(cot(n*w)-cot(n*z))-cos(y+x+(n-2)*w)/sin(n*w))/(2*sin(z-w)) n %pi j %pi j ==== sin(x + -----) sin(y + -----) \ n n
----------------------------- = / %pi j %pi j ==== sin(w + -----) sin(z + -----) j = 1 n n
cos((n - 2) z + y + x) n (---------------------- + cos(y - x) (cot(n w) - cot(n z)) sin(n z) cos(y + x + (n - 2) w) - ----------------------)/(2 sin(z - w)), sin(n w) which is sort of interesting for the "n-2"s, and its w=z limit, sum(sin(x+%pi*j/n)*sin(y+%pi*j/n)/sin(z+%pi*j/n)^2,j,1,n) = n*sin((n-2)*z+y+x)/sin(n*z)+n^2*sin(z-x)*sin(z-y)/sin(n*z)^2 n %pi j %pi j ==== sin(x + -----) sin(y + -----) \ n n
----------------------------- = / 2 %pi j ==== sin (z + -----) j = 1 n
2 n sin((n - 2) z + y + x) n sin(z - x) sin(z - y) ------------------------ + ------------------------, sin(n z) 2 sin (n z) which is sort of interesting for the n^2. I wish to thank Mma 7 for being no help at all simplifying the RHSs (which were initially mucho messy), thus making me feel useful. (A package to do these seems feasible, but decent simplifying would be ursine.) --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun