module sfsums; % Calculation of infinite sums of reciprocal
% Powers, see e.g. Abramowitz/Stegun ch 23.
%
% Author: Winfried Neun, Sep 1993
algebraic <<
let{
sum((-1)^~k /(2*(~k)-1)^~n,~k,1,infinity) =>
Pi^n*abs(Euler(n-1))/(factorial(n-1) * 2^(n+1))
when fixp n and n > 0 and not evenp n,
sum((-1)^~k /(2*(~k)-1)^2,~k,1,infinity) => - Catalan,
sum((-1)^~k /(2*(~k)+1)^2,~k,0,infinity) => Catalan,
sum(1/(2*(~k)-1)^~n,~k,1,infinity) => Zeta(n) *(1-2^(-n))
when fixp n and n > 0 and evenp n,
sum(1/~k^~s,~k,1,infinity) => Zeta(s),
sum((-1)^~k/~k^~n,~k,1,infinity) => Zeta(n)* (1-2^(1-n))
when fixp n and n > 0 and evenp n
} ;
% from Abigail Leeves Sep 15, 97
let {
prod((1+(1/~n**2)),~n,~r,infinity) =>
(((sinh(pi))/pi)/(prod((1+(1/~n**2)),~n,1,(~r-1))))
when (fixp r and r>=1 and r<15),
prod((1+(1/~n**3)),~n,~r,infinity) =>
(((cosh((sqrt(3)*pi)/2))/pi)/(prod((1+(1/~n**3)),~n,1,(~r-1))))
when (fixp r and r>=1 and r< 15),
prod((1+(1/~n**4)),~n,~r,infinity) =>
(((cosh(sqrt(2)*pi))-(cos(sqrt(2)*pi)))/(2*pi**2))/
(prod((1+(1/~n**4)),~n,1,(~r-1)))
when (fixp r and r>=1 and r<15),
prod((1+(1/~n**5)),~n,~r,infinity) =>
((((Gamma(exp((2*pi*i)/5)))*(Gamma(exp((6*pi*i)/5))))**-2)/
(prod((1+(1/~n**5)),~n,1,(~r-1))))
when (fixp r and r>=1 and r<15),
prod((1-(4/~n**2)),~n,~r,infinity) =>
(1/6)/(prod((1-(4/~n**2)),~n,3,(~r-1)))
when (fixp r and r<15 and r>=3),
prod((1-(8/~n**3)),~n,~r,infinity) =>
((sinh(sqrt(3)*pi))/(42*sqrt(3)*pi))/(prod((1-(8/~n**3)),~n,3,(~r-1)))
when (fixp r and r<15 and r>=3),
prod((1-(16/~n**4)),~n,~r,infinity) =>
((sinh(2*pi))/(120*pi))/(prod((1-(16/~n**4)),~n,3,(~r-1)))
when (fixp r and r<15 and r>=3),
prod((1-(32/~n**5)),~n,~r,infinity) =>
((1/1240)*((Gamma(2*exp((pi*i)/5)))*
(Gamma(2*exp((7*pi*i)/5)))**-2))/(prod((1-(32/~n**5)),~n,3,(~r-1)))
when (fixp r and r<15 and r>=3)
};
>>;
endmodule;
end;