Background
Suppose that the national mint mints coins with bias $p_i \sim \rm{Beta}(A,B)$ for some unknown constants $A$, $B$.
Given $n$ coins, you flip each coin a certain number of times. Coin $i$ comes up heads $a_i$ times and tails $b_i$ times. (The flips are clearly exchangeable, so we don't need the actual sequence of flips.)
If I did the math right, I figured out that the likelihood of $A, B$ given the statistics $(a_i, b_i)$ is:
$L(A,B \mid a_i, b_i) = \frac{\prod_i\rm{Beta}(A + a_i, B + b_i)}{\rm{Beta}(A, B)^n}.$
(Edit in 2012: This is the product of $n$ Pólya–Eggenberger urn schemes.)
I would like to summarize the statistics $(a_i, b_i)$ even further, into a finite set of numbers if possible.
Question
Can $\prod_i\Gamma(A + a_i)$ be written using sufficient statistics calculated from the $a_i$?
Example
For example, $\prod_i\exp(A + a_i)$ can be written $\exp(nA + S)$ where $n, S$ are sufficient statistics calculated from the $a_i$.