I've been trying to find analytical solutions for the zeros of the following expression Zeros in terms of variables n, m and f (here I represented two forms of the expression R, I've worked with both but I couldn't find a general solution, the code is intended to be used on Mathematica):
R[x_, n_, m_] :=
x^(m - 1)*(Sum[
Binomial[n - 1, k]*x^(k - m + 1)*(1 - x)^(n - 1 - k), {k, m,
n - 1}] +
n*(Sum[Binomial[n - 1, k]*x^(k - m + 1)*(1 - x)^(n - 1 - k), {k,
0, m - 1}]) + m*Binomial[n - 1, m - 1]*(1 - x)^(n - m))=
1 - (1 - N)* (Sum[
Binomial[n - 1, k]*x^(k)*(1 - x)^(n - 1 - k), {k, 0,
m - 1}]) + m*Binomial[n - 1, m - 1]*x^(m - 1)*(1 - x)^(n - m);
Zeros[x_, n_, m_, f_] := R[x, n, m] - n/f;
I also tried solving the equation by using Reduce[R[x, n, m] - n/F == 0, x] but he couldn't evaluate it and get to a conclusion. I'd really appreciate some suggestions.
Thanks for your time.