How to solve this differential equation?

December 16, 2009

How would you solve the differential equation

B'(x) = e^x B(x)

with the initial condition B(0) = 1? I know what the answer is supposed to be, but I don’t know how to directly solve it.

In case you’re wondering, B(x) is the exponential generating function for the Bell numbers, which count set partitions. The differential equation in question arises from noting that

\displaystyle B_{n+1} = \sum_{k=0}^n \binom n k B_{n-k}

(to make a partition of \{1, \dots, n+1\}, you can put anywhere from k = 0 to n elements in the same set with n+1; there are \binom n k ways to choose the k elements to include, and B_{n-k} ways to partition the rest).

15 Comments | combinatorics, math | Tagged: , , , , | Permalink
Posted by Brent

The Poisson distribution and Stirling numbers

September 16, 2008

While working on an assignment for my machine learning class, I rediscovered the fact that if X is a random variable from a Poisson distribution with parameter \lambda, then

\displaystyle E[X^n] = \sum_{k=1}^n S(n,k) \lambda^k,

where S(n,k) denotes a Stirling number of the second kind. (I actually prefer Knuth’s curly bracket notation, but I can’t seem to get it to work on this blog.) In particular, if \lambda = 1, then E[X^n] is the nth Bell number B_n, the number of ways of partitioning a set of size n into subsets!

As it turned out, this didn’t help me at all with my assignment, I just thought it was nifty.

2 Comments | combinatorics, grad school, learning, math | Tagged: , , , | Permalink
Posted by Brent

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