While flagitating some kids with my usual sutra of always replacing Bernoulli numbers with Bernoulli polynomials in some new variable, I derived (c47) intosum(sum(2*(-1)^i*x^(2*i-1)*bernpoly(y,2*i)/(2*i)!,i,0,inf) = sin(x*y)+cot(x/2)*cos(x*y)) inf ==== i 2 i - 1 \ 2 (- 1) x bernpoly(y, 2 i) (d47) > ---------------------------------- = / (2 i)! ==== i = 0 x sin(x y) + cot(-) cos(x y) 2 and (c48) intosum(sum(2*(-1)^i*x^(2*i)*bernpoly(y,2*i+1)/(2*i+1)!,i,0,inf) = cot(x/2)*sin(x*y)-cos(x*y)) inf ==== i 2 i \ 2 (- 1) x bernpoly(y, 2 i + 1) (d48) > ---------------------------------- = / (2 i + 1)! ==== i = 0 x cot(-) sin(x y) - cos(x y) 2 generalizing the expansions of cot(x/2) and -+1(!) respectively. Putting y=0 and y=1 in the first one must be equal, giving the cot halfangle formula. --rwg COURT PAINTERS PERSCRUTATION