Consider the expression $$I_{p}(\alpha,\beta+1) - I_{p}(\alpha+1,\beta) = c$$ where $I_p(a,b)$ is the regularized incomplete beta function.
Question: Given $\alpha,\beta,$ and $c>0$, what is $p$?
Attempt: I need to invert the CDF's, but the problem is the arguments of the CDFs are not the same. I'm not sure how to proceed.
Using the inductive property
$$I_p(\alpha,\beta+1)=I_p(\alpha,\beta)+\frac{p^{\alpha}(1-p)^{\beta}}{\beta\,B(\alpha,\beta)}$$
$$I_p(\alpha+1,\beta)=I_p(\alpha,\beta)-\frac{p^{\alpha}(1-p)^{\beta}}{\alpha\,B(\alpha,\beta)}$$
It is obtained:
$$p^{\alpha}\,(1-p)^{\beta}=c\,(\frac{1}{\alpha}+\frac{1}{\beta})\,B(\alpha,\beta)\:$$
From here there is no general solution in a closed form $\forall\,\alpha$,$\beta$ but last equation is very easy to solve numerically