In Sewall Wright's Evolution of Mendalian Population, the equation for the nonrecurrent mutation is $$\frac{\phi(x)}n = \binom n {nx}\int_0^1 q^{nx}(1-q)^{n(1-x)}\phi(q)\,dq,\quad \forall x\in[0,1],$$ where $n>0$ and $\binom a b$ is the binomial coefficient. We are to solve for a nonzero function $\phi\ne0$. It is an eigenvalue problem of a compact operator. But what is a general method for solving this integral equation? dirichlet Is there a transform, say Mellin transform, that does the trick?
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Maybe try searching for Fredholm integral equations of the second kind. – J. Heller Jun 28 '20 at 18:49
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@J.Heller: Yes, it is an homogeneous Fredholm integral equation of the second kind. You can just view it as an eigenfunction problem with eigenvalue of $1$ of a compact operator. However, is there a way to analytically find the solution? – Hans Jun 28 '20 at 21:33