In mathematics , the Stirling polynomials are a family of polynomials that generalize important sequences of numbers appearing in combinatorics and analysis , which are closely related to the Stirling numbers , the Bernoulli numbers , and the generalized Bernoulli polynomials . There are multiple variants of the Stirling polynomial sequence considered below most notably including the Sheffer sequence form of the sequence,
S
k
(
x
)
{\displaystyle S_{k}(x)}
Stirling (convolution) polynomials ,
σ
n
(
x
)
{\displaystyle \sigma _{n}(x)}
ordinary generating function and that are of use in generalizing the Stirling numbers (of both kinds) to arbitrary complex -valued inputs. We consider the "convolution polynomial " variant of this sequence and its properties second in the last subsection of the article. Still other variants of the Stirling polynomials are studied in the supplementary links to the articles given in the references.
Definition and examples [ edit ] For nonnegative integers k , the Stirling polynomials, S k x ), are a Sheffer sequence for
(
g
(
t
)
,
f
¯
(
t
)
)
:=
(
e
−
t
,
log
(
t
1
−
e
−
t
)
)
{\displaystyle (g(t),{\bar {f}}(t)):=\left(e^{-t},\log \left({\frac {t}{1-e^{-t}}}\right)\right)}
[1]
(
t
1
−
e
−
t
)
x
+
1
=
∑
k
=
0
∞
S
k
(
x
)
t
k
k
!
.
{\displaystyle \left({t \over {1-e^{-t}}}\right)^{x+1}=\sum _{k=0}^{\infty }S_{k}(x){t^{k} \over k!}.}
The Stirling polynomials are a special case of the Nørlund polynomials (or generalized Bernoulli polynomials ) [2]
(
t
e
t
−
1
)
a
e
z
t
=
∑
k
=
0
∞
B
k
(
a
)
(
z
)
t
k
k
!
,
{\displaystyle \left({t \over {e^{t}-1}}\right)^{a}e^{zt}=\sum _{k=0}^{\infty }B_{k}^{(a)}(z){t^{k} \over k!},}
given by the relation
S
k
(
x
)
=
B
k
(
x
+
1
)
(
x
+
1
)
{\displaystyle S_{k}(x)=B_{k}^{(x+1)}(x+1)}
The first 10 Stirling polynomials are given in the following table:
k Sk (x )
0
1
{\displaystyle 1}
1
1
2
(
x
+
1
)
{\displaystyle {\scriptstyle {\frac {1}{2}}}(x+1)}
2
1
12
(
3
x
2
+
5
x
+
2
)
{\displaystyle {\scriptstyle {\frac {1}{12}}}(3x^{2}+5x+2)}
3
1
8
(
x
3
+
2
x
2
+
x
)
{\displaystyle {\scriptstyle {\frac {1}{8}}}(x^{3}+2x^{2}+x)}
4
1
240
(
15
x
4
+
30
x
3
+
5
x
2
−
18
x
−
8
)
{\displaystyle {\scriptstyle {\frac {1}{240}}}(15x^{4}+30x^{3}+5x^{2}-18x-8)}
5
1
96
(
3
x
5
+
5
x
4
−
5
x
3
−
13
x
2
−
6
x
)
{\displaystyle {\scriptstyle {\frac {1}{96}}}(3x^{5}+5x^{4}-5x^{3}-13x^{2}-6x)}
6
1
4032
(
63
x
6
+
63
x
5
−
315
x
4
−
539
x
3
−
84
x
2
+
236
x
+
96
)
{\displaystyle {\scriptstyle {\frac {1}{4032}}}(63x^{6}+63x^{5}-315x^{4}-539x^{3}-84x^{2}+236x+96)}
7
1
1152
(
9
x
7
−
84
x
5
−
98
x
4
+
91
x
3
+
194
x
2
+
80
x
)
{\displaystyle {\scriptstyle {\frac {1}{1152}}}(9x^{7}-84x^{5}-98x^{4}+91x^{3}+194x^{2}+80x)}
8
1
34560
(
135
x
8
−
180
x
7
−
1890
x
6
−
840
x
5
+
6055
x
4
+
8140
x
3
+
884
x
2
−
3088
x
−
1152
)
{\displaystyle {\scriptstyle {\frac {1}{34560}}}(135x^{8}-180x^{7}-1890x^{6}-840x^{5}+6055x^{4}+8140x^{3}+884x^{2}-3088x-1152)}
9
1
7680
(
15
x
9
−
45
x
8
−
270
x
7
+
182
x
6
+
1687
x
5
+
1395
x
4
−
1576
x
3
−
2684
x
2
−
1008
x
)
{\displaystyle {\scriptstyle {\frac {1}{7680}}}(15x^{9}-45x^{8}-270x^{7}+182x^{6}+1687x^{5}+1395x^{4}-1576x^{3}-2684x^{2}-1008x)}
Yet another variant of the Stirling polynomials is considered in [3] Stirling convolution polynomials below). In particular, the article by I. Gessel and R. P. Stanley defines the modified Stirling polynomial sequences,
f
k
(
n
)
:=
S
(
n
+
k
,
n
)
{\displaystyle f_{k}(n):=S(n+k,n)}
g
k
(
n
)
:=
c
(
n
,
n
−
k
)
{\displaystyle g_{k}(n):=c(n,n-k)}
c
(
n
,
k
)
:=
(
−
1
)
n
−
k
s
(
n
,
k
)
{\displaystyle c(n,k):=(-1)^{n-k}s(n,k)}
unsigned Stirling numbers of the first kind , in terms of the two Stirling number triangles for non-negative integers
n
≥
1
,
k
≥
0
{\displaystyle n\geq 1,\ k\geq 0}
k
≥
0
{\displaystyle k\geq 0}
f
k
(
n
)
{\displaystyle f_{k}(n)}
g
k
(
n
)
{\displaystyle g_{k}(n)}
n
∈
Z
+
{\displaystyle n\in \mathbb {Z} ^{+}}
2
k
{\displaystyle 2k}
double factorial term
(
1
⋅
3
⋅
5
⋯
(
2
k
−
1
)
)
/
(
2
k
)
!
{\displaystyle (1\cdot 3\cdot 5\cdots (2k-1))/(2k)!}
Below
B
k
(
x
)
{\displaystyle B_{k}(x)}
Bernoulli polynomials and
B
k
=
B
k
(
0
)
{\displaystyle B_{k}=B_{k}(0)}
Bernoulli numbers under the convention
B
1
=
B
1
(
0
)
=
−
1
2
;
{\displaystyle B_{1}=B_{1}(0)=-{\tfrac {1}{2}};}
s
m
,
n
{\displaystyle s_{m,n}}
Stirling number of the first kind ; and
S
m
,
n
{\displaystyle S_{m,n}}
Stirling numbers of the second kind .
Special values:
S
k
(
−
m
)
=
(
−
1
)
k
(
k
+
m
−
1
k
)
S
k
+
m
−
1
,
m
−
1
0
<
m
∈
Z
S
k
(
−
1
)
=
δ
k
,
0
S
k
(
0
)
=
(
−
1
)
k
B
k
S
k
(
1
)
=
(
−
1
)
k
+
1
(
(
k
−
1
)
B
k
+
k
B
k
−
1
)
S
k
(
2
)
=
(
−
1
)
k
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
!
{\displaystyle {\begin{aligned}S_{k}(-m)&={\frac {(-1)^{k}}{k+m-1 \choose k}}S_{k+m-1,m-1}&&0<m\in \mathbb {Z} \\[6pt]S_{k}(-1)&=\delta _{k,0}\\[6pt]S_{k}(0)&=(-1)^{k}B_{k}\\[6pt]S_{k}(1)&=(-1)^{k+1}((k-1)B_{k}+kB_{k-1})\\[6pt]S_{k}(2)&={\tfrac {(-1)^{k}}{2}}((k-1)(k-2)B_{k}+3k(k-2)B_{k-1}+2k(k-1)B_{k-2})\\[6pt]S_{k}(k)&=k!\\[6pt]\end{aligned}}}
If
m
∈
N
{\displaystyle m\in \mathbb {N} }
m
<
k
{\displaystyle m<k}
S
k
(
m
)
=
(
m
+
1
)
(
k
m
+
1
)
∑
j
=
0
m
(
−
1
)
m
−
j
s
m
+
1
,
m
+
1
−
j
B
k
−
j
k
−
j
.
{\displaystyle S_{k}(m)={(m+1)}{\binom {k}{m+1}}\sum _{j=0}^{m}(-1)^{m-j}s_{m+1,m+1-j}{\frac {B_{k-j}}{k-j}}.}
If
m
∈
N
{\displaystyle m\in \mathbb {N} }
m
≥
k
{\displaystyle m\geq k}
[4]
S
k
(
m
)
=
(
−
1
)
k
B
k
(
m
+
1
)
(
0
)
,
{\displaystyle S_{k}(m)=(-1)^{k}B_{k}^{(m+1)}(0),}
S
k
(
m
)
=
(
−
1
)
k
(
m
k
)
s
m
+
1
,
m
+
1
−
k
.
{\displaystyle S_{k}(m)={(-1)^{k} \over {m \choose k}}s_{m+1,m+1-k}.}
The sequence
S
k
(
x
−
1
)
{\displaystyle S_{k}(x-1)}
binomial type , since
S
k
(
x
+
y
−
1
)
=
∑
i
=
0
k
(
k
i
)
S
i
(
x
−
1
)
S
k
−
i
(
y
−
1
)
.
{\displaystyle S_{k}(x+y-1)=\sum _{i=0}^{k}{k \choose i}S_{i}(x-1)S_{k-i}(y-1).}
S
k
(
x
)
=
(
x
−
k
)
S
k
(
x
−
1
)
x
+
k
S
k
−
1
(
x
+
1
)
.
{\displaystyle S_{k}(x)=(x-k){S_{k}(x-1) \over x}+kS_{k-1}(x+1).}
Explicit representations involving Stirling numbers can be deduced with Lagrange's interpolation formula :
S
k
(
x
)
=
∑
n
=
0
k
(
−
1
)
k
−
n
S
k
+
n
,
n
(
x
+
n
n
)
(
x
+
k
+
1
k
−
n
)
(
k
+
n
n
)
=
∑
n
=
0
k
(
−
1
)
n
s
k
+
n
+
1
,
n
+
1
(
x
−
k
n
)
(
x
−
k
−
n
−
1
k
−
n
)
(
k
+
n
k
)
=
k
!
∑
j
=
0
k
(
−
1
)
k
−
j
∑
m
=
j
k
(
x
+
m
m
)
(
m
j
)
L
k
+
m
(
−
k
−
j
)
(
−
j
)
{\displaystyle {\begin{aligned}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}}\\[6pt]&=\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}}\\[6pt]&=k!\sum _{j=0}^{k}(-1)^{k-j}\sum _{m=j}^{k}{x+m \choose m}{m \choose j}L_{k+m}^{(-k-j)}(-j)\\[6pt]\end{aligned}}}
L
n
(
α
)
{\displaystyle L_{n}^{(\alpha )}}
Laguerre polynomials .
The following relations hold as well:
(
k
+
m
k
)
S
k
(
x
−
m
)
=
∑
i
=
0
k
(
−
1
)
k
−
i
(
k
+
m
i
)
S
k
−
i
+
m
,
m
⋅
S
i
(
x
)
,
{\displaystyle {k+m \choose k}S_{k}(x-m)=\sum _{i=0}^{k}(-1)^{k-i}{k+m \choose i}S_{k-i+m,m}\cdot S_{i}(x),}
(
k
−
m
k
)
S
k
(
x
+
m
)
=
∑
i
=
0
k
(
k
−
m
i
)
s
m
,
m
−
k
+
i
⋅
S
i
(
x
)
.
{\displaystyle {k-m \choose k}S_{k}(x+m)=\sum _{i=0}^{k}{k-m \choose i}s_{m,m-k+i}\cdot S_{i}(x).}
By differentiating the generating function it readily follows that
S
k
′
(
x
)
=
−
∑
j
=
0
k
−
1
(
k
j
)
S
j
(
x
)
B
k
−
j
k
−
j
.
{\displaystyle S_{k}^{\prime }(x)=-\sum _{j=0}^{k-1}{k \choose j}S_{j}(x){\frac {B_{k-j}}{k-j}}.}
Stirling convolution polynomials [ edit ] Definition and examples [ edit ] Another variant of the Stirling polynomial sequence corresponds to a special case of the convolution polynomials studied by Knuth's article [5] Concrete Mathematics reference. We first define these polynomials through the Stirling numbers of the first kind as
σ
n
(
x
)
=
[
x
x
−
n
]
⋅
1
x
(
x
−
1
)
⋯
(
x
−
n
)
.
{\displaystyle \sigma _{n}(x)=\left[{\begin{matrix}x\\x-n\end{matrix}}\right]\cdot {\frac {1}{x(x-1)\cdots (x-n)}}.}
It follows that these polynomials satisfy the next recurrence relation given by
(
x
+
1
)
σ
n
(
x
+
1
)
=
(
x
−
n
)
σ
n
(
x
)
+
x
σ
n
−
1
(
x
)
,
n
≥
1.
{\displaystyle (x+1)\sigma _{n}(x+1)=(x-n)\sigma _{n}(x)+x\sigma _{n-1}(x),\ n\geq 1.}
These Stirling "convolution " polynomials may be used to define the Stirling numbers,
[
x
x
−
n
]
{\displaystyle \scriptstyle {\left[{\begin{matrix}x\\x-n\end{matrix}}\right]}}
{
x
x
−
n
}
{\displaystyle \scriptstyle {\left\{{\begin{matrix}x\\x-n\end{matrix}}\right\}}}
n
≥
0
{\displaystyle n\geq 0}
arbitrary complex values of
x
{\displaystyle x}
n
≥
0
{\displaystyle n\geq 0}
n σn (x )
0
1
x
{\displaystyle {\frac {1}{x}}}
1
1
2
{\displaystyle {\frac {1}{2}}}
2
1
24
(
3
x
−
1
)
{\displaystyle {\frac {1}{24}}(3x-1)}
3
1
48
(
x
2
−
x
)
{\displaystyle {\frac {1}{48}}(x^{2}-x)}
4
1
5760
(
15
x
3
−
30
x
2
+
5
x
+
2
)
{\displaystyle {\frac {1}{5760}}(15x^{3}-30x^{2}+5x+2)}
5
1
11520
(
3
x
4
−
10
x
3
+
5
x
2
+
2
x
)
{\displaystyle {\frac {1}{11520}}(3x^{4}-10x^{3}+5x^{2}+2x)}
6
1
2903040
(
63
x
5
−
315
x
4
+
315
x
3
+
91
x
2
−
42
x
−
16
)
{\displaystyle {\frac {1}{2903040}}(63x^{5}-315x^{4}+315x^{3}+91x^{2}-42x-16)}
7
1
5806080
(
9
x
6
−
63
x
5
+
105
x
4
+
7
x
3
−
42
x
2
−
16
x
)
{\displaystyle {\frac {1}{5806080}}(9x^{6}-63x^{5}+105x^{4}+7x^{3}-42x^{2}-16x)}
8
1
1393459200
(
135
x
7
−
1260
x
6
+
3150
x
5
−
840
x
4
−
2345
x
3
−
540
x
2
+
404
x
+
144
)
{\displaystyle {\frac {1}{1393459200}}(135x^{7}-1260x^{6}+3150x^{5}-840x^{4}-2345x^{3}-540x^{2}+404x+144)}
9
1
2786918400
(
15
x
8
−
180
x
7
+
630
x
6
−
448
x
5
−
665
x
4
+
100
x
3
+
404
x
2
+
144
x
)
{\displaystyle {\frac {1}{2786918400}}(15x^{8}-180x^{7}+630x^{6}-448x^{5}-665x^{4}+100x^{3}+404x^{2}+144x)}
10
1
367873228800
(
99
x
9
−
1485
x
8
+
6930
x
7
−
8778
x
6
−
8085
x
5
+
8195
x
4
+
11792
x
3
+
2068
x
2
−
2288
x
−
768
)
{\displaystyle {\frac {1}{367873228800}}(99x^{9}-1485x^{8}+6930x^{7}-8778x^{6}-8085x^{5}+8195x^{4}+11792x^{3}+2068x^{2}-2288x-768)}
Generating functions [ edit ] This variant of the Stirling polynomial sequence has particularly nice ordinary generating functions of the following forms:
(
z
e
z
e
z
−
1
)
x
=
∑
n
≥
0
x
σ
n
(
x
)
z
n
(
1
z
ln
1
1
−
z
)
x
=
∑
n
≥
0
x
σ
n
(
x
+
n
)
z
n
.
{\displaystyle {\begin{aligned}\left({\frac {ze^{z}}{e^{z}-1}}\right)^{x}&=\sum _{n\geq 0}x\sigma _{n}(x)z^{n}\\\left({\frac {1}{z}}\ln {\frac {1}{1-z}}\right)^{x}&=\sum _{n\geq 0}x\sigma _{n}(x+n)z^{n}.\end{aligned}}}
More generally, if
S
t
(
z
)
{\displaystyle {\mathcal {S}}_{t}(z)}
power series that satisfies
ln
(
1
−
z
S
t
(
z
)
t
−
1
)
=
−
z
S
t
(
z
)
t
{\displaystyle \ln \left(1-z{\mathcal {S}}_{t}(z)^{t-1}\right)=-z{\mathcal {S}}_{t}(z)^{t}}
S
t
(
z
)
x
=
∑
n
≥
0
x
σ
n
(
x
+
t
n
)
z
n
.
{\displaystyle {\mathcal {S}}_{t}(z)^{x}=\sum _{n\geq 0}x\sigma _{n}(x+tn)z^{n}.}
We also have the related series identity [6]
∑
n
≥
0
(
−
1
)
n
−
1
σ
n
(
n
−
1
)
z
n
=
z
ln
(
1
+
z
)
=
1
+
z
2
−
z
2
12
+
⋯
,
{\displaystyle \sum _{n\geq 0}(-1)^{n-1}\sigma _{n}(n-1)z^{n}={\frac {z}{\ln(1+z)}}=1+{\frac {z}{2}}-{\frac {z^{2}}{12}}+\cdots ,}
and the Stirling (Sheffer) polynomial related generating functions given by
∑
n
≥
0
(
−
1
)
n
+
1
m
⋅
σ
n
(
n
−
m
)
z
n
=
(
z
ln
(
1
+
z
)
)
m
{\displaystyle \sum _{n\geq 0}(-1)^{n+1}m\cdot \sigma _{n}(n-m)z^{n}=\left({\frac {z}{\ln(1+z)}}\right)^{m}}
∑
n
≥
0
(
−
1
)
n
+
1
m
⋅
σ
n
(
m
)
z
n
=
(
z
1
−
e
−
z
)
m
.
{\displaystyle \sum _{n\geq 0}(-1)^{n+1}m\cdot \sigma _{n}(m)z^{n}=\left({\frac {z}{1-e^{-z}}}\right)^{m}.}
Properties and relations [ edit ] For integers
0
≤
k
≤
n
{\displaystyle 0\leq k\leq n}
r
,
s
∈
C
{\displaystyle r,s\in \mathbb {C} }
(
r
+
s
)
σ
n
(
r
+
s
+
t
n
)
=
r
s
∑
k
=
0
n
σ
k
(
r
+
t
k
)
σ
n
−
k
(
s
+
t
(
n
−
k
)
)
{\displaystyle (r+s)\sigma _{n}(r+s+tn)=rs\sum _{k=0}^{n}\sigma _{k}(r+tk)\sigma _{n-k}(s+t(n-k))}
and
n
σ
n
(
r
+
s
+
t
n
)
=
s
∑
k
=
0
n
k
σ
k
(
r
+
t
k
)
σ
n
−
k
(
s
+
t
(
n
−
k
)
)
.
{\displaystyle n\sigma _{n}(r+s+tn)=s\sum _{k=0}^{n}k\sigma _{k}(r+tk)\sigma _{n-k}(s+t(n-k)).}
When
n
,
m
∈
N
{\displaystyle n,m\in \mathbb {N} }
σ
n
(
m
)
{\displaystyle \sigma _{n}(m)}
Stirling numbers
{
n
m
}
=
(
−
1
)
n
−
m
+
1
n
!
(
m
−
1
)
!
σ
n
−
m
(
−
m
)
(
when
m
<
0
)
[
n
m
]
=
n
!
(
m
−
1
)
!
σ
n
−
m
(
n
)
(
when
m
>
n
)
,
{\displaystyle {\begin{aligned}\left\{{\begin{matrix}n\\m\end{matrix}}\right\}&=(-1)^{n-m+1}{\frac {n!}{(m-1)!}}\sigma _{n-m}(-m)\ ({\text{when }}m<0)\\\left[{\begin{matrix}n\\m\end{matrix}}\right]&={\frac {n!}{(m-1)!}}\sigma _{n-m}(n)\ ({\text{when }}m>n),\end{aligned}}}
and their relations to the Bernoulli numbers given by
σ
n
(
m
)
=
(
−
1
)
m
+
n
−
1
m
!
(
n
−
m
)
!
∑
0
≤
k
<
m
[
m
m
−
k
]
B
n
−
k
n
−
k
,
n
≥
m
>
0
σ
n
(
m
)
=
−
B
n
n
⋅
n
!
,
m
=
0.
{\displaystyle {\begin{aligned}\sigma _{n}(m)&={\frac {(-1)^{m+n-1}}{m!(n-m)!}}\sum _{0\leq k<m}\left[{\begin{matrix}m\\m-k\end{matrix}}\right]{\frac {B_{n-k}}{n-k}},\ n\geq m>0\\\sigma _{n}(m)&=-{\frac {B_{n}}{n\cdot n!}},\ m=0.\end{aligned}}}
^ See section 4.8.8 of The Umbral Calculus (1984) reference linked below.
^ See Norlund polynomials on MathWorld.
^ Gessel & Stanley (1978). "Stirling polynomials". J. Combin. Theory Ser. A . 53 : 24–33. doi :10.1016/0097-3165(78)90042-0 . ^ Section 4.4.8 of The Umbral Calculus .
^ Knuth, D. E. (1992). "Convolution Polynomials". Mathematica J . 2 : 67–78. arXiv :math/9207221 Bibcode :1992math......7221K . convolution polynomial families defined by special generating functions of the form
F
(
z
)
x
{\displaystyle F(z)^{x}}
F
(
0
)
=
1
{\displaystyle F(0)=1}
binomial power series ,
B
t
(
z
)
=
1
+
z
B
t
(
z
)
t
{\displaystyle {\mathcal {B}}_{t}(z)=1+z{\mathcal {B}}_{t}(z)^{t}}
tree polynomials , the Bell numbers ,
B
(
n
)
{\displaystyle B(n)}
Laguerre polynomials . For
F
n
(
x
)
:=
[
z
n
]
F
(
z
)
x
{\displaystyle F_{n}(x):=[z^{n}]F(z)^{x}}
n
!
⋅
F
n
(
x
)
{\displaystyle n!\cdot F_{n}(x)}
binomial type
z
F
n
(
x
+
t
n
)
(
x
+
t
n
)
=
[
z
n
]
F
t
(
z
)
x
{\displaystyle {\frac {zF_{n}(x+tn)}{(x+tn)}}=[z^{n}]{\mathcal {F}}_{t}(z)^{x}}
t
∈
C
{\displaystyle t\in \mathbb {C} }
F
t
(
z
)
{\displaystyle {\mathcal {F}}_{t}(z)}
functional equation of the form
F
t
(
z
)
=
F
(
x
F
t
(
z
)
t
)
{\displaystyle {\mathcal {F}}_{t}(z)=F\left(x{\mathcal {F}}_{t}(z)^{t}\right)}
^ Section 7.4 of Concrete Mathematics .
Erdeli, A.; Magnus, W.; Oberhettinger, F. & Tricomi, F. G. Higher Transcendental Functions. Volume III . New York. Graham; Knuth & Patashnik (1994). Concrete Mathematics: A Foundation for Computer Science . S. Roman (1984). The Umbral Calculus . 
![{\displaystyle {\begin{aligned}S_{k}(-m)&={\frac {(-1)^{k}}{k+m-1 \choose k}}S_{k+m-1,m-1}&&0<m\in \mathbb {Z} \\[6pt]S_{k}(-1)&=\delta _{k,0}\\[6pt]S_{k}(0)&=(-1)^{k}B_{k}\\[6pt]S_{k}(1)&=(-1)^{k+1}((k-1)B_{k}+kB_{k-1})\\[6pt]S_{k}(2)&={\tfrac {(-1)^{k}}{2}}((k-1)(k-2)B_{k}+3k(k-2)B_{k-1}+2k(k-1)B_{k-2})\\[6pt]S_{k}(k)&=k!\\[6pt]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab11d50b1266b6fea254e93291e75bd9d27d23b2)
![{\displaystyle {\begin{aligned}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}}\\[6pt]&=\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}}\\[6pt]&=k!\sum _{j=0}^{k}(-1)^{k-j}\sum _{m=j}^{k}{x+m \choose m}{m \choose j}L_{k+m}^{(-k-j)}(-j)\\[6pt]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b294559318300e34f8b63e528ecaa5d67e169760)
![{\displaystyle \sigma _{n}(x)=\left[{\begin{matrix}x\\x-n\end{matrix}}\right]\cdot {\frac {1}{x(x-1)\cdots (x-n)}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2d0037b8db31ce9bc95c552c3d5ecf3e013079)
![{\displaystyle \scriptstyle {\left[{\begin{matrix}x\\x-n\end{matrix}}\right]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c854ccf1db132e85dd532931b5eb0be57701b2c)
![{\displaystyle {\begin{aligned}\left\{{\begin{matrix}n\\m\end{matrix}}\right\}&=(-1)^{n-m+1}{\frac {n!}{(m-1)!}}\sigma _{n-m}(-m)\ ({\text{when }}m<0)\\\left[{\begin{matrix}n\\m\end{matrix}}\right]&={\frac {n!}{(m-1)!}}\sigma _{n-m}(n)\ ({\text{when }}m>n),\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef5f2ea4399342d521307f2157c2a858f5c7002)
![{\displaystyle {\begin{aligned}\sigma _{n}(m)&={\frac {(-1)^{m+n-1}}{m!(n-m)!}}\sum _{0\leq k<m}\left[{\begin{matrix}m\\m-k\end{matrix}}\right]{\frac {B_{n-k}}{n-k}},\ n\geq m>0\\\sigma _{n}(m)&=-{\frac {B_{n}}{n\cdot n!}},\ m=0.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ffb378e8d1a37613f664c9a7db301eca4c6dbdc)
![{\displaystyle F_{n}(x):=[z^{n}]F(z)^{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ef6b79d2221950286dadafc59714241f1c793b5)
![{\displaystyle {\frac {zF_{n}(x+tn)}{(x+tn)}}=[z^{n}]{\mathcal {F}}_{t}(z)^{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/884463c817f173365c4fb7fe5f364ed4d101392f)
