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Help with this indefinite sum

Posted on 2008-10-14 10:29 By hweengee (6)
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Given that
3*(sum(cot(m*Pi/(2*M+1))^2, m = 1 .. M))

is a polynomial of degree r in M, find r and the coefficient of Mr. (The coefficient is an integer)

Any ideas how to go about solving this problem? When i input this particular indefinite sum into Maple it does not generate a polynomial.

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hexagonal numbers

Comment on 2008-10-14 11:14 from Joe Riel (Maple Rating 4 1248)

I don't know how to show this analytically in Maple, but generating the sequence of integers (via rounding) is easy enough.  The sequence corresponds to the hexagonal numbers, m*(2*m-1).

Axel Vogt's picture

no need for floats: write as cos

Comment on 2008-10-14 13:36 from Axel Vogt (Maple Rating 4 948)

for example

-3*Sum(cos(m*Pi/(2*M+1))^2/(-1+cos(m*Pi/(2*M+1))^2),m = 1 .. M);
map(convert,%,cos); eval(%,M=7): value(%); combine(%,trig); simplify(%);

gives 91

smells like using Chebyshev polynomials or similar

without rounding

Comment on 2008-10-14 14:23 from jakubi (Maple Rating 4 911)

The integers of the sequence arise simply:

S:=3*(Sum(cot(m*Pi/(2*M+1))^2, m = 1 .. M));
l:=[seq(simplify(value(subs(M=i,S))),i=1..5)];
                       l := [1, 6, 15, 28, 45]

rec:=gfun:-listtorec(l,u(N));#N starts at 0

  rec := [{(N + 2) u(N + 2) + (-4 + 2 N) u(N + 1) + (-6 - 3 N) u(N),

        u(0) = 1, u(1) = 6}, ogf]

SN:=rsolve(op(1,rec),u(N));

                                            2
                         SN := 1 + 3 N + 2 N

SM:=collect(subs(N=M-1,SN),M);

                                     2
                            SM := 2 M  - M

perfect. thanks.

Comment on 2008-10-15 05:57 from hweengee (6)

perfect. thanks.

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