Question: Need help with a function of complex variables

Hi everybody,

Here's what I have:

After many calculations, I arrive to this expression
> S := Sum(exp(I*Pi*(2*p+1)*(2*m+1)/(2*n)), p = 0 .. n-1);

                    n - 1                                
                    -----    /1                         \
                     \       |- I Pi (2 p + 1) (2 m + 1)|
                      )      |2                         |
                     /    exp|--------------------------|
                    -----    \            n             /
                    p = 0                                
with {n,m}::posint and n > m.  Then I rewrite it this way
> exp(I*Pi*beta)*(Sum(exp((I*Pi*2)*p*beta), p = 0 .. n-1));

                                 /n - 1                   \
                                 |-----                   |
                                 | \                      |
                                 |  )                     |
                  exp(I Pi beta) | /    exp(2 I Pi p beta)|
                                 |-----                   |
                                 \p = 0                   /
> beta = (2*m+1)/(2*n);

                                      2 m + 1
                               beta = -------
                                        2 n  
if I am not wrong, this sum give
> St := exp(I*Pi*beta)*(1-exp((2*I)*Pi*n*beta))/(1-exp((2*I)*Pi*beta));

                   exp(I Pi beta) (1 - exp(2 I Pi n beta))
                   ---------------------------------------
                            1 - exp(2 I Pi beta)          
I have try a couples of hours to simplify it so I can get,
 using at the end the Euler's identity, the expression
> St1 := -I/sin(Pi*beta);

                                      I      
                               - ------------
                                 sin(Pi beta)
If you could show me the way, I have exhausted all I could think of
 so far.  In paper or with Maple would be nice.

Thanks

Mario Lemelin