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User:Paul August/Subpage 12 - Wikipedia, the free encyclopedia

User:Paul August/Subpage 12

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< User:Paul August

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Stirling polynomials $S k (x)$ 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 $S k (x - 1)$ 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:

$S_k(x)= (x-k) {S_k(x-1) \over x} + k S_{k-1}(x-1).$

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

S m, n

denotes Stirling numbers of the second kind. Conversely,

$S_{n,m}=(-1)^{n-m} {n \choose m} S_{n-m}(-m-1).$

$S_k(-1)= \delta_{k,0}.\,$
$S_k(0)= (-1)^k B_k,\,$

where

B k

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

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:

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

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

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