Stirling polynomial

Stirling’s polynomials $S_{k}(x)$ are defined by the generating function

\\left({t\\over{1-e^{-t}}}\\right)^{x+1}=\\sum_{k=0}{S_{k}(x)\\over k!}t^{k}.

The sequence $S_{k}(x-1)$ is of binomial type, since $S_{k}(x+y-1)=\\sum_{i=0}^{k}{k\\choose i}S_{i}(x-1)S_{k-i}(y-1)$ . Moreover, this basic recursion holds: $S_{k}(x)=(x-k){S_{k}(x-1)\\over x}+kS_{k-1}(x+1)$ .

These are the first polynomials:

1.
$S_{0}(x)=1$ ;
2.
$S_{1}(x)={1\\over 2}(x+1)$ ;
3.
$S_{2}(x)={1\\over 12}(3x^{2}+5x+2)$ ;
4.
$S_{3}(x)={1\\over 8}(x^{3}+2x^{2}+x)$ ;
5.
$S_{4}(x)={1\\over 240}(15x^{4}+30x^{3}+5x^{2}-18x-8)$ .

In addition we have these special values:

1.
$S_{k}(-m)={(-1)^{k}\\over{k+m-1\\choose k}}S_{k+m-1,m-1}$ , where $S_{m,n}$ denotes Stirling numbers of the second kind. Conversely, $S_{n,m}=(-1)^{n-m}{n\\choose m}S_{n-m}(-m-1)$ ;
2.
$S_{k}(-1)=\\delta_{k,0}$ ;
3.
$S_{k}(0)=(-1)^{k}B_{k}$ , where $B_{k}$ are Bernoulli’s numbers;
4.
$S_{k}(1)=(-1)^{k+1}((k-1)B_{k}+kB_{k-1})$ ;
5.
$S_{k}(2)={(-1)^{k}\\over 2}((k-1)(k-2)B_{k}+3k(k-2)B_{k-1}+2k(k-1)B_{k-2})$ ;
6.
$S_{k}(k)=k!$ ;
7.
$S_{k}(m)={(-1)^{k}\\over{m\\choose k}}s_{m+1,m+1-k}$ , where $s_{m,n}$ are Stirling numbers of the first kind. They may be recovered by $s_{n,m}=(-1)^{n-m}{n-1\\choose n-m}S_{n-m}(n-1)$ .

Explicit representations involving Stirling numbers can be deduced with Lagrange’s interpolation formula:

S_{k}(x)=\\sum_{n=0}^{k}(-1)^{k-n}S_{k+n,n}{{x+n\\choose n}{x+k+1\\choose k-n}%\\over{k+n\\choose n}}=\\sum_{n=0}^{k}(-1)^{n}s_{k+n+1,n+1}{{x-k\\choose n}{x-k-n-%1\\choose k-n}\\over{k+n\\choose k}}.

These following formulae hold as well:

{k+m\\choose k}S_{k}(x-m)=\\sum_{i=0}^{k}(-1)^{k-i}{k+m\\choose i}S_{k-i+m,m}S_{i%}(x),

{k-m\\choose k}S_{k}(x+m)=\\sum_{i=0}^{k}{k-m\\choose i}s_{m,m-k+i}S_{i}(x).