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Stirling polynomials Sk(x) are a polynomial sequence defined by the generating function:

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

The sequence Sk(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} + k S_{k-1}(x-1).

These are the first polynomials:

S_0(x)=1\,\!;
S_1(x)= {1 \over 2}(x+1);
S_2(x)= {1 \over 12} (3 x^2 + 5x+ 2);
S_3(x)= {1 \over 8} (x^3+2 x^2+x);
S_4(x)= {1 \over 240} (15 x^4+ 30 x^3+5 x^2- 18x-8).

In addition we have these special values:

  • S_k(-m)= {(-1)^k \over {k+m-1 \choose k}} S_{k+m-1,m-1},
where m is a positive integer and Sm,n denotes Stirling numbers of the second kind. Conversely,
S_{n,m}=(-1)^{n-m} {n \choose m} S_{n-m}(-m-1).
  • S_k(-1)= \delta_{k,0}.\,
  • S_k(0)= (-1)^k B_k,\,
where Bk are Bernoulli's numbers.
  • S_k(1)= (-1)^{k+1} ((k-1) B_k+ k B_{k-1}).\,
  • S_k(2)= {(-1)^{k}\over 2} ((k-1)(k-2) B_k+ 3 k(k-2) B_{k-1}+ 2 k(k-1) B_{k-2}).\,
  • S_k(k)= k!\,
  • S_k(m)= {(-1)^k \over {m \choose k}} s_{m+1, m+1-k},
where sm,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).