Expectation of beta function with binomial distribution

Question

Is there any way to express $E[\log(B(a+S_n,b+n-S_n))]$ where $B$ stands for beta function and $S_n \sim B(n,p)$ has a binomial distribution, in a nice way (without using multiple sums by direct expansion) or find a recursive formula for $n$ increasing ? I tried utilizing beta-binomial distribution and expanding the formula directly, but couldn't come up with a nice closed form expression.

Answer 1

Letting $f_n(a,b):=E\ln B(a+S_n,b+n-S_n)$, we get $f_0(a,b)=\ln B(a,b)$. Conditioning on $S_n$, we get the recursion $$f_{n+1}(a,b)=pf_n(a+1,b)+(1-p)f_n(a,b+1)$$ for $n=0,1,\dots$.