I know the following recurrence relation
$$a_n=\frac{a+na_{n-1}}{a-n}$$
with $a_0=1$ can be represented alternatively as an integral
$$a_n=a\int_0^1{x^{a-n-1}(2-x)^ndx}$$
Verifying this is easy, but is there any general technique to do this kind of transformations?