Factorials and Stirling numbers in the algebra of formal Laurent series (Q1175979)

The recursion equation \(a(n,i+1)=a(n,i)+ia(n-1,i)\) for all integers \(n\geq 1\), \(i\in\mathbb{Z}\), with the initial values \(a(0,i)=1\) for all \(i\in\mathbb{Z}\), \(a(n,0)=0\) for all \(n\geq 1\), is solved by \[ a(n,i)=\begin{cases} S(n-i,-i) & \text{for all \(i<0,\)}\\ | s(i,i-n)| & \text{for \(0\leq n\leq i\),}\end{cases} \] where \(s,S\) are the Stirling numbers of the first and second kind, respectively, defined by \(x^ n=\sum^ n_{i=0}S(n,i)x^{(i)}\) and \(x^{(n)}=\sum^ n_{i=0}s(n,i)x^ i\) where \(x^{(n)}=x(x-1)\ldots(x-n+1)\) stands for the falling factorial powers. The author wants to fill the ``wedge'' for \(n\geq i\) by generalizing \(x^{(n)}\) to \(\{x^{(n)}\}\), a lower Laurent series, for all integers \(n\). This comprises the Stirling numbers of both kinds, and the new numbers are closely related to Bernoulli numbers of general order. \(\{x^{(n)}\}\) can also be used to construct a binomial series \({x\brace n}\), allowing for Vandermonde convolution.