The Bernoulli polynomials of the second kind [ 1] [ 2] ψn (x ) , also known as the Fontana-Bessel polynomials [ 3] , are the polynomials defined by the following generating function:
z
(
1
+
z
)
x
ln
(
1
+
z
)
=
∑
n
=
0
∞
z
n
ψ
n
(
x
)
,
|
z
|
<
1.
{\displaystyle {\frac {z(1+z)^{x}}{\ln(1+z)}}=\sum _{n=0}^{\infty }z^{n}\psi _{n}(x),\qquad |z|<1.}
The first five polynomials are:
ψ
0
(
x
)
=
1
ψ
1
(
x
)
=
x
+
1
2
ψ
2
(
x
)
=
1
2
x
2
−
1
12
ψ
3
(
x
)
=
1
6
x
3
−
1
4
x
2
+
1
24
ψ
4
(
x
)
=
1
24
x
4
−
1
6
x
3
+
1
6
x
2
−
19
720
{\displaystyle {\begin{array}{l}\displaystyle \psi _{0}(x)=1\\[2mm]\displaystyle \psi _{1}(x)=x+{\frac {1}{2}}\\[2mm]\displaystyle \psi _{2}(x)={\frac {1}{2}}x^{2}-{\frac {1}{12}}\\[2mm]\displaystyle \psi _{3}(x)={\frac {1}{6}}x^{3}-{\frac {1}{4}}x^{2}+{\frac {1}{24}}\\[2mm]\displaystyle \psi _{4}(x)={\frac {1}{24}}x^{4}-{\frac {1}{6}}x^{3}+{\frac {1}{6}}x^{2}-{\frac {19}{720}}\end{array}}}
Some authors define these polynomials slightly differently[ 4] [ 5]
z
(
1
+
z
)
x
ln
(
1
+
z
)
=
∑
n
=
0
∞
z
n
n
!
ψ
n
∗
(
x
)
,
|
z
|
<
1
,
{\displaystyle {\frac {z(1+z)^{x}}{\ln(1+z)}}=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}\psi _{n}^{*}(x),\qquad |z|<1,}
so that
ψ
n
∗
(
x
)
=
ψ
n
(
x
)
n
!
{\displaystyle \psi _{n}^{*}(x)=\psi _{n}(x)\,n!}
and may also use a different notation for them (the most used alternative notation is bn (x ) ).
The Bernoulli polynomials of the second kind were largely studied by the Hungarian mathematician Charles Jordan,[ 1] [ 2] but their history may also be traced back to the much earlier works[ 3] .
Integral representations
The Bernoulli polynomials of the second kind may be represented via these integrals[ 1] [ 2]
ψ
n
(
x
)
=
∫
x
x
+
1
(
u
n
)
d
u
=
∫
0
1
(
x
+
u
n
)
d
u
{\displaystyle \psi _{n}(x)=\int \limits _{x}^{x+1}\!{\binom {u}{n}}\,du=\int \limits _{0}^{1}{\binom {x+u}{n}}\,du}
as well as[ 3]
ψ
n
(
x
)
=
(
−
1
)
n
+
1
π
∫
0
∞
π
cos
π
x
−
sin
π
x
ln
z
(
1
+
z
)
n
⋅
z
x
d
z
ln
2
z
+
π
2
,
−
1
≤
x
≤
n
−
1
ψ
n
(
x
)
=
(
−
1
)
n
+
1
π
∫
−
∞
+
∞
π
cos
π
x
−
v
sin
π
x
(
1
+
e
v
)
n
⋅
e
v
(
x
+
1
)
v
2
+
π
2
d
v
,
−
1
≤
x
≤
n
−
1
{\displaystyle {\begin{array}{l}\displaystyle \psi _{n}(x)={\frac {(-1)^{n+1}}{\pi }}\int \limits _{0}^{\infty }{\frac {\pi \cos \pi x-\sin \pi x\ln z}{(1+z)^{n}}}\cdot {\frac {z^{x}dz}{\ln ^{2}z+\pi ^{2}}},\qquad -1\leq x\leq n-1\,\\[3mm]\displaystyle \psi _{n}(x)={\frac {(-1)^{n+1}}{\pi }}\int \limits _{-\infty }^{+\infty }{\frac {\pi \cos \pi x-v\sin \pi x}{\,(1+e^{v})^{n}}}\cdot {\frac {e^{v(x+1)}}{v^{2}+\pi ^{2}}}\,dv,\qquad -1\leq x\leq n-1\,\end{array}}}
These polynomials are, therefore, up to a constant, the antiderivative of the binomial coefficient and also that of the falling factorial .[ 1] [ 2] [ 3]
For an arbitrary n , these polynomials may be computed explicitly via the following summation formula[ 1] [ 2] [ 3]
ψ
n
(
x
)
=
1
(
n
−
1
)
!
∑
l
=
0
n
−
1
s
(
n
−
1
,
l
)
l
+
1
x
l
+
1
+
G
n
,
n
=
1
,
2
,
3
,
…
{\displaystyle \psi _{n}(x)={\frac {1}{(n-1)!}}\sum _{l=0}^{n-1}{\frac {s(n-1,l)}{l+1}}x^{l+1}+G_{n},\qquad n=1,2,3,\ldots }
where where s (n ,l ) are the signed Stirling numbers of the first kind and G n are the Gregory coefficients .
The Bernoulli polynomials of the second kind satisfy the recurrence relation[ 1] [ 2]
ψ
n
(
x
+
1
)
−
ψ
n
(
x
)
=
ψ
n
−
1
(
x
)
{\displaystyle \psi _{n}(x+1)-\psi _{n}(x)=\psi _{n-1}(x)}
or equivalently
Δ
ψ
n
(
x
)
=
ψ
n
−
1
(
x
)
{\displaystyle \Delta \psi _{n}(x)=\psi _{n-1}(x)}
The repeated difference produces[ 1] [ 2]
Δ
m
ψ
n
(
x
)
=
ψ
n
−
m
(
x
)
{\displaystyle \Delta ^{m}\psi _{n}(x)=\psi _{n-m}(x)}
Symmetry property
The main property of the symmetry reads[ 2] [ 4]
ψ
n
(
1
2
n
−
1
+
x
)
=
(
−
1
)
n
ψ
n
(
1
2
n
−
1
−
x
)
{\displaystyle \psi _{n}({\tfrac {1}{2}}n-1+x)=(-1)^{n}\psi _{n}({\tfrac {1}{2}}n-1-x)}
Some further properties and particular values
Some properties and particular values of these polynomials include
ψ
n
(
0
)
=
G
n
ψ
n
(
1
)
=
G
n
−
1
+
G
n
ψ
n
(
−
1
)
=
(
−
1
)
n
+
1
∑
m
=
0
n
|
G
m
|
=
(
−
1
)
n
C
n
ψ
n
(
n
−
2
)
=
−
|
G
n
|
ψ
n
(
n
−
1
)
=
(
−
1
)
n
ψ
n
(
−
1
)
=
1
−
∑
m
=
1
n
|
G
m
|
ψ
2
n
(
n
−
1
)
=
M
2
n
ψ
2
n
(
n
−
1
+
y
)
=
ψ
2
n
(
n
−
1
−
y
)
ψ
2
n
+
1
(
n
−
1
2
+
y
)
=
−
ψ
2
n
+
1
(
n
−
1
2
−
y
)
ψ
2
n
+
1
(
n
−
1
2
)
=
0
{\displaystyle {\begin{array}{l}\displaystyle \psi _{n}(0)=G_{n}\\[2mm]\displaystyle \psi _{n}(1)=G_{n-1}+G_{n}\\[2mm]\displaystyle \psi _{n}(-1)=(-1)^{n+1}\sum _{m=0}^{n}|G_{m}|=(-1)^{n}C_{n}\\[2mm]\displaystyle \psi _{n}(n-2)=-|G_{n}|\\[2mm]\displaystyle \psi _{n}(n-1)=(-1)^{n}\psi _{n}(-1)=1-\sum _{m=1}^{n}|G_{m}|\\[2mm]\displaystyle \psi _{2n}(n-1)=M_{2n}\\[2mm]\displaystyle \psi _{2n}(n-1+y)=\psi _{2n}(n-1-y)\\[2mm]\displaystyle \psi _{2n+1}(n-{\tfrac {1}{2}}+y)=-\psi _{2n+1}(n-{\tfrac {1}{2}}-y)\\[2mm]\displaystyle \psi _{2n+1}(n-{\tfrac {1}{2}})=0\end{array}}}
where C n are the Cauchy numbers of the second kind and M n are the central difference coefficients .[ 1] [ 2] [ 3]
Expansion into a Newton series
The expansion of the Bernoulli polynomials of the second kind into a Newton series reads[ 1] [ 2]
ψ
n
(
x
)
=
G
0
(
x
n
)
+
G
1
(
x
n
−
1
)
+
G
2
(
x
n
−
2
)
+
…
+
G
n
{\displaystyle \psi _{n}(x)=G_{0}{\binom {x}{n}}+G_{1}{\binom {x}{n-1}}+G_{2}{\binom {x}{n-2}}+\ldots +G_{n}}
Some series involving the Bernoulli polynomials of the second kind
The digamma function Ψ(x ) may be expanded into a series with the Bernoulli polynomials of the second kind
in the following way[ 3]
Ψ
(
v
)
=
ln
(
v
+
a
)
+
∑
n
=
1
∞
(
−
1
)
n
ψ
n
(
a
)
(
n
−
1
)
!
(
v
)
n
,
ℜ
(
v
)
>
−
a
,
{\displaystyle \Psi (v)=\ln(v+a)+\sum _{n=1}^{\infty }{\frac {(-1)^{n}\psi _{n}(a)\,(n-1)!}{(v)_{n}}},\qquad \Re (v)>-a,}
and hence[ 3]
γ
=
−
ln
(
a
+
1
)
−
∑
n
=
1
∞
(
−
1
)
n
ψ
n
(
a
)
n
,
ℜ
(
a
)
>
−
1
{\displaystyle \gamma =-\ln(a+1)-\sum _{n=1}^{\infty }{\frac {(-1)^{n}\psi _{n}(a)}{n}},\qquad \Re (a)>-1}
and
γ
=
∑
n
=
1
∞
(
−
1
)
n
+
1
2
n
{
ψ
n
(
a
)
+
ψ
n
(
−
a
1
+
a
)
}
,
a
>
−
1
{\displaystyle \gamma =\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{2n}}{\Big \{}\psi _{n}(a)+\psi _{n}{\Big (}-{\frac {a}{1+a}}{\Big )}{\Big \}},\quad a>-1}
where γ is Euler's constant . Furthermore, we also have[ 3]
Ψ
(
v
)
=
1
v
+
a
−
1
2
{
ln
Γ
(
v
+
a
)
+
v
−
1
2
ln
2
π
−
1
2
+
∑
n
=
1
∞
(
−
1
)
n
ψ
n
+
1
(
a
)
(
v
)
n
(
n
−
1
)
!
}
,
ℜ
(
v
)
>
−
a
,
{\displaystyle \Psi (v)={\frac {1}{v+a-{\tfrac {1}{2}}}}\left\{\ln \Gamma (v+a)+v-{\frac {1}{2}}\ln 2\pi -{\frac {1}{2}}+\sum _{n=1}^{\infty }{\frac {(-1)^{n}\psi _{n+1}(a)}{(v)_{n}}}(n-1)!\right\},\qquad \Re (v)>-a,}
where Γ(x ) is the gamma function . The Hurwitz and Riemann zeta functions may be expanded into these
polynomials as follows[ 3]
ζ
(
s
,
v
)
=
(
v
+
a
)
1
−
s
s
−
1
+
∑
n
=
0
∞
(
−
1
)
n
ψ
n
+
1
(
a
)
∑
k
=
0
n
(
−
1
)
k
(
n
k
)
(
k
+
v
)
−
s
{\displaystyle \zeta (s,v)={\frac {(v+a)^{1-s}}{s-1}}+\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+1}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(k+v)^{-s}}
and
ζ
(
s
)
=
(
a
+
1
)
1
−
s
s
−
1
+
∑
n
=
0
∞
(
−
1
)
n
ψ
n
+
1
(
a
)
∑
k
=
0
n
(
−
1
)
k
(
n
k
)
(
k
+
1
)
−
s
{\displaystyle \zeta (s)={\frac {(a+1)^{1-s}}{s-1}}+\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+1}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(k+1)^{-s}}
and also
ζ
(
s
)
=
1
+
(
a
+
2
)
1
−
s
s
−
1
+
∑
n
=
0
∞
(
−
1
)
n
ψ
n
+
1
(
a
)
∑
k
=
0
n
(
−
1
)
k
(
n
k
)
(
k
+
2
)
−
s
{\displaystyle \zeta (s)=1+{\frac {(a+2)^{1-s}}{s-1}}+\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+1}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(k+2)^{-s}}
The Bernoulli polynomials of the second kind are also involved in the following relationship[ 3]
(
v
+
a
−
1
2
)
ζ
(
s
,
v
)
=
−
ζ
(
s
−
1
,
v
+
a
)
s
−
1
+
ζ
(
s
−
1
,
v
)
+
∑
n
=
0
∞
(
−
1
)
n
ψ
n
+
2
(
a
)
∑
k
=
0
n
(
−
1
)
k
(
n
k
)
(
k
+
v
)
−
s
{\displaystyle {\big (}v+a-{\tfrac {1}{2}}{\big )}\zeta (s,v)=-{\frac {\zeta (s-1,v+a)}{s-1}}+\zeta (s-1,v)+\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+2}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}(k+v)^{-s}}
between the zeta functions, as well as in various formulas for the Stieltjes constants , e.g.[ 3]
γ
m
(
v
)
=
−
ln
m
+
1
(
v
+
a
)
m
+
1
+
∑
n
=
0
∞
(
−
1
)
n
ψ
n
+
1
(
a
)
∑
k
=
0
n
(
−
1
)
k
(
n
k
)
ln
m
(
k
+
v
)
k
+
v
{\displaystyle \gamma _{m}(v)=-{\frac {\ln ^{m+1}(v+a)}{m+1}}+\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+1}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}{\frac {\ln ^{m}(k+v)}{k+v}}}
and
γ
m
(
v
)
=
1
1
2
−
v
−
a
{
(
−
1
)
m
m
+
1
ζ
(
m
+
1
)
(
0
,
v
+
a
)
−
(
−
1
)
m
ζ
(
m
)
(
0
,
v
)
−
∑
n
=
0
∞
(
−
1
)
n
ψ
n
+
2
(
a
)
∑
k
=
0
n
(
−
1
)
k
(
n
k
)
ln
m
(
k
+
v
)
k
+
v
}
{\displaystyle \gamma _{m}(v)={\frac {1}{{\tfrac {1}{2}}-v-a}}\left\{{\frac {(-1)^{m}}{m+1}}\,\zeta ^{(m+1)}(0,v+a)-(-1)^{m}\zeta ^{(m)}(0,v)-\sum _{n=0}^{\infty }(-1)^{n}\psi _{n+2}(a)\sum _{k=0}^{n}(-1)^{k}{\binom {n}{k}}{\frac {\ln ^{m}(k+v)}{k+v}}\right\}}
which are both valid for
ℜ
(
a
)
>
−
1
{\displaystyle \Re (a)>-1}
and
v
∈
C
∖
{
0
,
−
1
,
−
2
,
…
}
{\displaystyle v\in \mathbb {C} \setminus \!\{0,-1,-2,\ldots \}}
.
See also
References
^ a b c d e f g h i Jordan, Charles (1928), "Sur des polynomes analogues aux polynomes de Bernoulli, et sur des formules de sommation analogues à celle de Maclaurin-Euler", Acta Sci. Math. (Szeged) , 4 : 130–150
^ a b c d e f g h i j Jordan, Charles (1965). The Calculus of Finite Differences (3rd Edition) . Chelsea Publishing Company.
^ a b c d e f g h i j k l Blagouchine, Iaroslav V. (2018), "Three notes on Ser's and Hasse's representations for the zeta-functions" (PDF) , Integers (Electronic Journal of Combinatorial Number Theory) , 18A (#A3): 1–45 arXiv
^ a b Roman, S. (1984). The Umbral Calculus . New York: Academic Press.
^ Weisstein, Eric W. Bernoulli Polynomial of the Second Kind . From MathWorld--A Wolfram Web Resource.
Mathematics