Today, I found an interesting identity. First, let the number of ways of partitioning a set of n elements into m nonempty sets be S(n,m). Also, let
This can be seen either as the Taylor Series of f(x) or the generating function of B_n
The identity would be:
A special case of this identity would be:
Where e is euler’s constant or e=2.71828182846…
It looks really complicated with all the notation but in essence, the identity can be easily understood. I will explain the summation notation in a future post regarding Analytic Number Theory.