For a given $\alpha,\beta,a,b>0$, I'm trying to find the value of $p$ that satisfies the following two equations

$$\frac{1}{2}\bigg[a\bigg(b-\frac{\alpha}{\alpha+\beta+1}\bigg)+1\bigg] = I_p(\alpha,\beta+1)$$

$$\frac{1}{2}\bigg[a\bigg(b-\frac{\alpha+1}{\alpha+\beta+1}\bigg)+1\bigg] = I_p(\alpha+1,\beta)$$

where $I_p(\alpha,\beta)$ is the regularized incomplete beta function.

My attempt involved application of the consecutive neighbors identities: $$I_p(\alpha+1,\beta) = I_p(\alpha,\beta) - \frac{p^\alpha(1-p)^{\beta}}{\alpha B(\alpha,\beta)}$$

$$I_p(\alpha,\beta+1) = I_p(\alpha,\beta) + \frac{p^\alpha(1-p)^{\beta}}{\beta B(\alpha,\beta)}$$

but that just seems to complicate things. Any ideas or hints?


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