Stirling Number - Inversion Relationships

Inversion Relationships

The Stirling numbers of the first and second kinds can be considered to be inverses of one another:

and

where is the Kronecker delta. These two relationships may be understood to be matrix inverse relationships. That is, let s be the lower triangular matrix of Stirling numbers of first kind, so that it has matrix elements

Then, the inverse of this matrix is S, the lower triangular matrix of Stirling numbers of second kind. Symbolically, one writes

where the matrix elements of S are

Note that although s and S are infinite, so calculating a product entry involves an infinite sum, the matrix multiplications work because these matrices are lower triangular, so only a finite number of terms in the sum are nonzero.

A generalization of the inversion relationship gives the link with the Lah numbers

with the conventions and if .