The polynomials are related by
M
n
(
x
)
=
2
⋅
n
!
g
n
(
x
)
{\displaystyle M_{n}(x)=2\cdot {n!}\,g_{n}(x)}
and we have
g
n
(
1
)
=
1
{\displaystyle g_{n}(1)=1}
for
n
⩾
1
{\displaystyle n\geqslant 1}
. Also
g
2
k
(
1
2
)
=
g
2
k
+
1
(
1
2
)
=
1
2
(
2
k
−
1
)
!
!
(
2
k
)
!
!
=
1
2
⋅
1
⋅
3
⋯
(
2
k
−
1
)
2
⋅
4
⋯
(
2
k
)
{\displaystyle g_{2k}({\frac {1}{2}})=g_{2k+1}({\frac {1}{2}})={\frac {1}{2}}{\frac {(2k-1)!!}{(2k)!!}}={\frac {1}{2}}\cdot {\frac {1\cdot 3\cdots (2k-1)}{2\cdot 4\cdots (2k)}}}
.
Explicit formulas are
g
n
(
x
)
=
∑
k
=
1
n
2
k
−
1
(
n
−
1
n
−
k
)
(
x
k
)
=
∑
k
=
0
n
−
1
2
k
(
n
−
1
k
)
(
x
k
+
1
)
{\displaystyle g_{n}(x)=\sum _{k=1}^{n}2^{k-1}{\binom {n-1}{n-k}}{\binom {x}{k}}=\sum _{k=0}^{n-1}2^{k}{\binom {n-1}{k}}{\binom {x}{k+1}}}
g
n
(
x
)
=
∑
k
=
0
n
−
1
(
n
−
1
k
)
(
k
+
x
n
)
{\displaystyle g_{n}(x)=\sum _{k=0}^{n-1}{\binom {n-1}{k}}{\binom {k+x}{n}}}
g
n
(
m
)
=
1
2
∑
k
=
0
m
(
m
k
)
(
n
−
1
+
m
−
k
m
−
1
)
=
1
2
∑
k
=
0
min
(
n
,
m
)
m
n
+
m
−
k
(
n
+
m
−
k
k
,
n
−
k
,
m
−
k
)
{\displaystyle g_{n}(m)={\frac {1}{2}}\sum _{k=0}^{m}{\binom {m}{k}}{\binom {n-1+m-k}{m-1}}={\frac {1}{2}}\sum _{k=0}^{\min(n,m)}{\frac {m}{n+m-k}}{\binom {n+m-k}{k,n-k,m-k}}}
(the last one immediately shows
n
g
n
(
m
)
=
m
g
m
(
n
)
{\displaystyle ng_{n}(m)=mg_{m}(n)}
, a kind of reflection formula), and
M
n
(
x
)
=
(
n
−
1
)
!
∑
k
=
1
n
k
2
k
(
n
k
)
(
x
k
)
{\displaystyle M_{n}(x)=(n-1)!\sum _{k=1}^{n}k2^{k}{\binom {n}{k}}{\binom {x}{k}}}
, which can be also written as
M
n
(
x
)
=
∑
k
=
1
n
2
k
(
n
k
)
(
n
−
1
)
n
−
k
(
x
)
k
{\displaystyle M_{n}(x)=\sum _{k=1}^{n}2^{k}{\binom {n}{k}}(n-1)_{n-k}(x)_{k}}
, where
(
x
)
n
=
n
!
(
x
n
)
=
x
(
x
−
1
)
⋯
(
x
−
n
+
1
)
{\displaystyle (x)_{n}=n!{\binom {x}{n}}=x(x-1)\cdots (x-n+1)}
denotes the falling factorial.
In terms of the Gaussian hypergeometric function , we have[ 4]
g
n
(
x
)
=
x
⋅
2
F
1
(
1
−
n
,
1
−
x
;
2
;
2
)
.
{\displaystyle g_{n}(x)=x\!\cdot {}_{2}\!F_{1}(1-n,1-x;2;2).}
As stated above, for
m
,
n
∈
N
{\displaystyle m,n\in \mathbb {N} }
, we have the reflection formula
n
g
n
(
m
)
=
m
g
m
(
n
)
{\displaystyle ng_{n}(m)=mg_{m}(n)}
.
The polynomials
M
n
(
x
)
{\displaystyle M_{n}(x)}
can be defined recursively by
M
n
(
x
)
=
2
x
M
n
−
1
(
x
)
+
(
n
−
1
)
(
n
−
2
)
M
n
−
2
(
x
)
{\displaystyle M_{n}(x)=2xM_{n-1}(x)+(n-1)(n-2)M_{n-2}(x)}
, starting with
M
−
1
(
x
)
=
0
{\displaystyle M_{-1}(x)=0}
and
M
0
(
x
)
=
1
{\displaystyle M_{0}(x)=1}
.
Another recursion formula, which produces an odd one from the preceding even ones and vice versa, is
M
n
+
1
(
x
)
=
2
x
∑
k
=
0
⌊
n
/
2
⌋
n
!
(
n
−
2
k
)
!
M
n
−
2
k
(
x
)
{\displaystyle M_{n+1}(x)=2x\sum _{k=0}^{\lfloor n/2\rfloor }{\frac {n!}{(n-2k)!}}M_{n-2k}(x)}
, again starting with
M
0
(
x
)
=
1
{\displaystyle M_{0}(x)=1}
.
As for the
g
n
(
x
)
{\displaystyle g_{n}(x)}
, we have several different recursion formulas:
(
1
)
g
n
(
x
+
1
)
−
g
n
−
1
(
x
+
1
)
=
g
n
(
x
)
+
g
n
−
1
(
x
)
{\displaystyle \displaystyle (1)\quad g_{n}(x+1)-g_{n-1}(x+1)=g_{n}(x)+g_{n-1}(x)}
(
2
)
(
n
+
1
)
g
n
+
1
(
x
)
−
(
n
−
1
)
g
n
−
1
(
x
)
=
2
x
g
n
(
x
)
{\displaystyle \displaystyle (2)\quad (n+1)g_{n+1}(x)-(n-1)g_{n-1}(x)=2xg_{n}(x)}
(
3
)
x
(
g
n
(
x
+
1
)
−
g
n
(
x
−
1
)
)
=
2
n
g
n
(
x
)
{\displaystyle (3)\quad x{\Bigl (}g_{n}(x+1)-g_{n}(x-1){\Bigr )}=2ng_{n}(x)}
(
4
)
g
n
+
1
(
m
)
=
g
n
(
m
)
+
2
∑
k
=
1
m
−
1
g
n
(
k
)
=
g
n
(
1
)
+
g
n
(
2
)
+
⋯
+
g
n
(
m
)
+
g
n
(
m
−
1
)
+
⋯
+
g
n
(
1
)
{\displaystyle (4)\quad g_{n+1}(m)=g_{n}(m)+2\sum _{k=1}^{m-1}g_{n}(k)=g_{n}(1)+g_{n}(2)+\cdots +g_{n}(m)+g_{n}(m-1)+\cdots +g_{n}(1)}
Concerning recursion formula (3), the polynomial
g
n
(
x
)
{\displaystyle g_{n}(x)}
is the unique polynomial solution of the difference equation
x
(
f
(
x
+
1
)
−
f
(
x
−
1
)
)
=
2
n
f
(
x
)
{\displaystyle x(f(x+1)-f(x-1))=2nf(x)}
, normalized so that
f
(
1
)
=
1
{\displaystyle f(1)=1}
.[ 5] Further note that (2) and (3) are dual to each other in the sense that for
x
∈
N
{\displaystyle x\in \mathbb {N} }
, we can apply the reflection formula to one of the identities and then swap
x
{\displaystyle x}
and
n
{\displaystyle n}
to obtain the other one. (As the
g
n
(
x
)
{\displaystyle g_{n}(x)}
are polynomials, the validity extends from natural to all real values of
x
{\displaystyle x}
.)
The table of the initial values of
g
n
(
m
)
{\displaystyle g_{n}(m)}
(these values are also called the "figurate numbers for the n-dimensional cross polytopes" in the OEIS[ 6] ) may illustrate the recursion formula (1), which can be taken to mean that each entry is the sum of the three neighboring entries: to its left, above and above left, e.g.
g
5
(
3
)
=
51
=
33
+
8
+
10
{\displaystyle g_{5}(3)=51=33+8+10}
. It also illustrates the reflection formula
n
g
n
(
m
)
=
m
g
m
(
n
)
{\displaystyle ng_{n}(m)=mg_{m}(n)}
with respect to the main diagonal, e.g.
3
⋅
44
=
4
⋅
33
{\displaystyle 3\cdot 44=4\cdot 33}
.
n
m
1
2
3
4
5
6
7
8
9
10
1
1
1
1
1
1
1
1
1
1
1
2
2
4
6
8
10
12
14
16
18
3
3
9
19
33
51
73
99
129
4
4
16
44
96
180
304
476
5
5
25
85
225
501
985
6
6
36
146
456
1182
7
7
49
231
833
8
8
64
344
9
9
81
10
10
Orthogonality relations
edit
For
m
,
n
∈
N
{\displaystyle m,n\in \mathbb {N} }
the following orthogonality relation holds:[ 7]
∫
−
∞
∞
g
n
(
−
i
y
)
g
m
(
i
y
)
y
sinh
π
y
d
y
=
1
2
n
δ
m
n
.
{\displaystyle \int _{-\infty }^{\infty }{\frac {g_{n}(-iy)g_{m}(iy)}{y\sinh \pi y}}dy={\frac {1}{2n}}\delta _{mn}.}
(Note that this is not a complex integral. As each
g
n
{\displaystyle g_{n}}
is an even or an odd polynomial, the imaginary arguments just produce alternating signs for their coefficients. Moreover, if
m
{\displaystyle m}
and
n
{\displaystyle n}
have different parity, the integral vanishes trivially.)
Being a Sheffer sequence of binomial type , the Mittag-Leffler polynomials
M
n
(
x
)
{\displaystyle M_{n}(x)}
also satisfy the binomial identity[ 8]
M
n
(
x
+
y
)
=
∑
k
=
0
n
(
n
k
)
M
k
(
x
)
M
n
−
k
(
y
)
{\displaystyle M_{n}(x+y)=\sum _{k=0}^{n}{\binom {n}{k}}M_{k}(x)M_{n-k}(y)}
.
Integral representations
edit
Based on the representation as a hypergeometric function, there are several ways of representing
g
n
(
z
)
{\displaystyle g_{n}(z)}
for
|
z
|
<
1
{\displaystyle |z|<1}
directly as integrals,[ 9] some of them being even valid for complex
z
{\displaystyle z}
, e.g.
(
26
)
g
n
(
z
)
=
sin
(
π
z
)
2
π
∫
−
1
1
t
n
−
1
(
1
+
t
1
−
t
)
z
d
t
{\displaystyle (26)\qquad g_{n}(z)={\frac {\sin(\pi z)}{2\pi }}\int _{-1}^{1}t^{n-1}{\Bigl (}{\frac {1+t}{1-t}}{\Bigr )}^{z}dt}
(
27
)
g
n
(
z
)
=
sin
(
π
z
)
2
π
∫
−
∞
∞
e
u
z
(
tanh
u
2
)
n
sinh
u
d
u
{\displaystyle (27)\qquad g_{n}(z)={\frac {\sin(\pi z)}{2\pi }}\int _{-\infty }^{\infty }e^{uz}{\frac {(\tanh {\frac {u}{2}})^{n}}{\sinh u}}du}
(
32
)
g
n
(
z
)
=
1
π
∫
0
π
cot
z
(
u
2
)
cos
(
π
z
2
)
cos
(
n
u
)
d
u
{\displaystyle (32)\qquad g_{n}(z)={\frac {1}{\pi }}\int _{0}^{\pi }\cot ^{z}({\frac {u}{2}})\cos({\frac {\pi z}{2}})\cos(nu)du}
(
33
)
g
n
(
z
)
=
1
π
∫
0
π
cot
z
(
u
2
)
sin
(
π
z
2
)
sin
(
n
u
)
d
u
{\displaystyle (33)\qquad g_{n}(z)={\frac {1}{\pi }}\int _{0}^{\pi }\cot ^{z}({\frac {u}{2}})\sin({\frac {\pi z}{2}})\sin(nu)du}
(
34
)
g
n
(
z
)
=
1
2
π
∫
0
2
π
(
1
+
e
i
t
)
z
(
2
+
e
i
t
)
n
−
1
e
−
i
n
t
d
t
{\displaystyle (34)\qquad g_{n}(z)={\frac {1}{2\pi }}\int _{0}^{2\pi }(1+e^{it})^{z}(2+e^{it})^{n-1}e^{-int}dt}
.
There are several families of integrals with closed-form expressions in terms of zeta values where the coefficients of the Mittag-Leffler polynomials occur as coefficients. All those integrals can be written in a form containing either a factor
tan
±
n
{\displaystyle \tan ^{\pm n}}
or
tanh
±
n
{\displaystyle \tanh ^{\pm n}}
, and the degree of the Mittag-Leffler polynomial varies with
n
{\displaystyle n}
. One way to work out those integrals is to obtain for them the corresponding recursion formulas as for the Mittag-Leffler polynomials using integration by parts.
1. For instance,[ 10] define for
n
⩾
m
⩾
2
{\displaystyle n\geqslant m\geqslant 2}
I
(
n
,
m
)
:=
∫
0
1
artanh
n
x
x
m
d
x
=
∫
0
1
log
n
/
2
(
1
+
x
1
−
x
)
d
x
x
m
=
∫
0
∞
z
n
coth
m
−
2
z
sinh
2
z
d
z
.
{\displaystyle I(n,m):=\int _{0}^{1}{\dfrac {{\text{artanh}}^{n}x}{x^{m}}}dx=\int _{0}^{1}\log ^{n/2}{\Bigl (}{\dfrac {1+x}{1-x}}{\Bigr )}{\dfrac {dx}{x^{m}}}=\int _{0}^{\infty }z^{n}{\dfrac {\coth ^{m-2}z}{\sinh ^{2}z}}dz.}
These integrals have the closed form
(
1
)
I
(
n
,
m
)
=
n
!
2
n
−
1
ζ
n
+
1
g
m
−
1
(
1
ζ
)
{\displaystyle (1)\quad I(n,m)={\frac {n!}{2^{n-1}}}\zeta ^{n+1}~g_{m-1}({\frac {1}{\zeta }})}
in umbral notation, meaning that after expanding the polynomial in
ζ
{\displaystyle \zeta }
, each power
ζ
k
{\displaystyle \zeta ^{k}}
has to be replaced by the zeta value
ζ
(
k
)
{\displaystyle \zeta (k)}
. E.g. from
g
6
(
x
)
=
1
45
(
23
x
2
+
20
x
4
+
2
x
6
)
{\displaystyle g_{6}(x)={\frac {1}{45}}(23x^{2}+20x^{4}+2x^{6})\ }
we get
I
(
n
,
7
)
=
n
!
2
n
−
1
23
ζ
(
n
−
1
)
+
20
ζ
(
n
−
3
)
+
2
ζ
(
n
−
5
)
45
{\displaystyle \ I(n,7)={\frac {n!}{2^{n-1}}}{\frac {23~\zeta (n-1)+20~\zeta (n-3)+2~\zeta (n-5)}{45}}\ }
for
n
⩾
7
{\displaystyle n\geqslant 7}
.
2. Likewise take for
n
⩾
m
⩾
2
{\displaystyle n\geqslant m\geqslant 2}
J
(
n
,
m
)
:=
∫
1
∞
arcoth
n
x
x
m
d
x
=
∫
1
∞
log
n
/
2
(
x
+
1
x
−
1
)
d
x
x
m
=
∫
0
∞
z
n
tanh
m
−
2
z
cosh
2
z
d
z
.
{\displaystyle J(n,m):=\int _{1}^{\infty }{\dfrac {{\text{arcoth}}^{n}x}{x^{m}}}dx=\int _{1}^{\infty }\log ^{n/2}{\Bigl (}{\dfrac {x+1}{x-1}}{\Bigr )}{\dfrac {dx}{x^{m}}}=\int _{0}^{\infty }z^{n}{\dfrac {\tanh ^{m-2}z}{\cosh ^{2}z}}dz.}
In umbral notation, where after expanding,
η
k
{\displaystyle \eta ^{k}}
has to be replaced by the Dirichlet eta function
η
(
k
)
:=
(
1
−
2
1
−
k
)
ζ
(
k
)
{\displaystyle \eta (k):=\left(1-2^{1-k}\right)\zeta (k)}
, those have the closed form
(
2
)
J
(
n
,
m
)
=
n
!
2
n
−
1
η
n
+
1
g
m
−
1
(
1
η
)
{\displaystyle (2)\quad J(n,m)={\frac {n!}{2^{n-1}}}\eta ^{n+1}~g_{m-1}({\frac {1}{\eta }})}
.
3. The following[ 11] holds for
n
⩾
m
{\displaystyle n\geqslant m}
with the same umbral notation for
ζ
{\displaystyle \zeta }
and
η
{\displaystyle \eta }
, and completing by continuity
η
(
1
)
:=
ln
2
{\displaystyle \eta (1):=\ln 2}
.
(
3
)
∫
0
π
/
2
x
n
tan
m
x
d
x
=
cos
(
m
2
π
)
(
π
/
2
)
n
+
1
n
+
1
+
cos
(
m
−
n
−
1
2
π
)
n
!
m
2
n
ζ
n
+
2
g
m
(
1
ζ
)
+
∑
v
=
0
n
cos
(
m
−
v
−
1
2
π
)
n
!
m
π
n
−
v
(
n
−
v
)
!
2
n
η
n
+
2
g
m
(
1
η
)
.
{\displaystyle (3)\quad \int \limits _{0}^{\pi /2}{\frac {x^{n}}{\tan ^{m}x}}dx=\cos {\Bigl (}{\frac {m}{2}}\pi {\Bigr )}{\frac {(\pi /2)^{n+1}}{n+1}}+\cos {\Bigl (}{\frac {m-n-1}{2}}\pi {\Bigr )}{\frac {n!~m}{2^{n}}}\zeta ^{n+2}g_{m}({\frac {1}{\zeta }})+\sum \limits _{v=0}^{n}\cos {\Bigl (}{\frac {m-v-1}{2}}\pi {\Bigr )}{\frac {n!~m~\pi ^{n-v}}{(n-v)!~2^{n}}}\eta ^{n+2}g_{m}({\frac {1}{\eta }}).}
Note that for
n
⩾
m
⩾
2
{\displaystyle n\geqslant m\geqslant 2}
, this also yields a closed form for the integrals
∫
0
∞
arctan
n
x
x
m
d
x
=
∫
0
π
/
2
x
n
tan
m
x
d
x
+
∫
0
π
/
2
x
n
tan
m
−
2
x
d
x
.
{\displaystyle \int \limits _{0}^{\infty }{\frac {\arctan ^{n}x}{x^{m}}}dx=\int \limits _{0}^{\pi /2}{\frac {x^{n}}{\tan ^{m}x}}dx+\int \limits _{0}^{\pi /2}{\frac {x^{n}}{\tan ^{m-2}x}}dx.}
4. For
n
⩾
m
⩾
2
{\displaystyle n\geqslant m\geqslant 2}
, define[ 12]
K
(
n
,
m
)
:=
∫
0
∞
tanh
n
(
x
)
x
m
d
x
{\displaystyle \quad K(n,m):=\int \limits _{0}^{\infty }{\dfrac {\tanh ^{n}(x)}{x^{m}}}dx}
.
If
n
+
m
{\displaystyle n+m}
is even and we define
h
k
:=
(
−
1
)
k
−
1
2
(
k
−
1
)
!
(
2
k
−
1
)
ζ
(
k
)
2
k
−
1
π
k
−
1
{\displaystyle h_{k}:=(-1)^{\frac {k-1}{2}}{\frac {(k-1)!(2^{k}-1)\zeta (k)}{2^{k-1}\pi ^{k-1}}}}
, we have in umbral notation, i.e. replacing
h
k
{\displaystyle h^{k}}
by
h
k
{\displaystyle h_{k}}
,
(
4
)
K
(
n
,
m
)
:=
∫
0
∞
tanh
n
(
x
)
x
m
d
x
=
n
⋅
2
m
−
1
(
m
−
1
)
!
(
−
h
)
m
−
1
g
n
(
h
)
.
{\displaystyle (4)\quad K(n,m):=\int \limits _{0}^{\infty }{\dfrac {\tanh ^{n}(x)}{x^{m}}}dx={\dfrac {n\cdot 2^{m-1}}{(m-1)!}}(-h)^{m-1}g_{n}(h).}
Note that only odd zeta values (odd
k
{\displaystyle k}
) occur here (unless the denominators are cast as even zeta values), e.g.
K
(
5
,
3
)
=
−
2
3
(
3
h
3
+
10
h
5
+
2
h
7
)
=
−
7
ζ
(
3
)
π
2
+
310
ζ
(
5
)
π
4
−
1905
ζ
(
7
)
π
6
,
{\displaystyle K(5,3)=-{\frac {2}{3}}(3h_{3}+10h_{5}+2h_{7})=-7{\frac {\zeta (3)}{\pi ^{2}}}+310{\frac {\zeta (5)}{\pi ^{4}}}-1905{\frac {\zeta (7)}{\pi ^{6}}},}
K
(
6
,
2
)
=
4
15
(
23
h
3
+
20
h
5
+
2
h
7
)
,
K
(
6
,
4
)
=
4
45
(
23
h
5
+
20
h
7
+
2
h
9
)
.
{\displaystyle K(6,2)={\frac {4}{15}}(23h_{3}+20h_{5}+2h_{7}),\quad K(6,4)={\frac {4}{45}}(23h_{5}+20h_{7}+2h_{9}).}
5. If
n
+
m
{\displaystyle n+m}
is odd, the same integral is much more involved to evaluate, including the initial one
∫
0
∞
tanh
3
(
x
)
x
2
d
x
{\displaystyle \int \limits _{0}^{\infty }{\dfrac {\tanh ^{3}(x)}{x^{2}}}dx}
. Yet it turns out that the pattern subsists if we define[ 13]
s
k
:=
η
′
(
−
k
)
=
2
k
+
1
ζ
(
−
k
)
ln
2
−
(
2
k
+
1
−
1
)
ζ
′
(
−
k
)
{\displaystyle s_{k}:=\eta '(-k)=2^{k+1}\zeta (-k)\ln 2-(2^{k+1}-1)\zeta '(-k)}
, equivalently
s
k
=
ζ
(
−
k
)
ζ
′
(
−
k
)
η
(
−
k
)
+
ζ
(
−
k
)
η
(
1
)
−
η
(
−
k
)
η
(
1
)
{\displaystyle s_{k}={\frac {\zeta (-k)}{\zeta '(-k)}}\eta (-k)+\zeta (-k)\eta (1)-\eta (-k)\eta (1)}
. Then
K
(
n
,
m
)
{\displaystyle K(n,m)}
has the following closed form in umbral notation, replacing
s
k
{\displaystyle s^{k}}
by
s
k
{\displaystyle s_{k}}
:
(
5
)
K
(
n
,
m
)
=
∫
0
∞
tanh
n
(
x
)
x
m
d
x
=
n
⋅
2
m
(
m
−
1
)
!
(
−
s
)
m
−
2
g
n
(
s
)
{\displaystyle (5)\quad K(n,m)=\int \limits _{0}^{\infty }{\dfrac {\tanh ^{n}(x)}{x^{m}}}dx={\frac {n\cdot 2^{m}}{(m-1)!}}(-s)^{m-2}g_{n}(s)}
, e.g.
K
(
5
,
4
)
=
8
9
(
3
s
3
+
10
s
5
+
2
s
7
)
,
K
(
6
,
3
)
=
−
8
15
(
23
s
3
+
20
s
5
+
2
s
7
)
,
K
(
6
,
5
)
=
−
8
45
(
23
s
5
+
20
s
7
+
2
s
9
)
.
{\displaystyle K(5,4)={\frac {8}{9}}(3s_{3}+10s_{5}+2s_{7}),\quad K(6,3)=-{\frac {8}{15}}(23s_{3}+20s_{5}+2s_{7}),\quad K(6,5)=-{\frac {8}{45}}(23s_{5}+20s_{7}+2s_{9}).}
Note that by virtue of the logarithmic derivative
ζ
′
ζ
(
s
)
+
ζ
′
ζ
(
1
−
s
)
=
log
π
−
1
2
Γ
′
Γ
(
s
2
)
−
1
2
Γ
′
Γ
(
1
−
s
2
)
{\displaystyle {\frac {\zeta '}{\zeta }}(s)+{\frac {\zeta '}{\zeta }}(1-s)=\log \pi -{\frac {1}{2}}{\frac {\Gamma '}{\Gamma }}\left({\frac {s}{2}}\right)-{\frac {1}{2}}{\frac {\Gamma '}{\Gamma }}\left({\frac {1-s}{2}}\right)}
of Riemann's functional equation , taken after applying Euler's reflection formula ,[ 14] these expressions in terms of the
s
k
{\displaystyle s_{k}}
can be written in terms of
ζ
′
(
2
j
)
ζ
(
2
j
)
{\displaystyle {\frac {\zeta '(2j)}{\zeta (2j)}}}
, e.g.
K
(
5
,
4
)
=
8
9
(
3
s
3
+
10
s
5
+
2
s
7
)
=
1
9
{
1643
420
−
16
315
ln
2
+
3
ζ
′
(
4
)
ζ
(
4
)
−
20
ζ
′
(
6
)
ζ
(
6
)
+
17
ζ
′
(
8
)
ζ
(
8
)
}
.
{\displaystyle K(5,4)={\frac {8}{9}}(3s_{3}+10s_{5}+2s_{7})={\frac {1}{9}}\left\{{\frac {1643}{420}}-{\frac {16}{315}}\ln 2+3{\frac {\zeta '(4)}{\zeta (4)}}-20{\frac {\zeta '(6)}{\zeta (6)}}+17{\frac {\zeta '(8)}{\zeta (8)}}\right\}.}
6. For
n
<
m
{\displaystyle n<m}
, the same integral
K
(
n
,
m
)
{\displaystyle K(n,m)}
diverges because the integrand behaves like
x
n
−
m
{\displaystyle x^{n-m}}
for
x
↘
0
{\displaystyle x\searrow 0}
. But the difference of two such integrals with corresponding degree differences is well-defined and exhibits very similar patterns, e.g.
(
6
)
K
(
n
−
1
,
n
)
−
K
(
n
,
n
+
1
)
=
∫
0
∞
(
tanh
n
−
1
(
x
)
x
n
−
tanh
n
(
x
)
x
n
+
1
)
d
x
=
−
1
n
+
(
n
+
1
)
⋅
2
n
(
n
−
1
)
!
s
n
−
2
g
n
(
s
)
{\displaystyle (6)\quad K(n-1,n)-K(n,n+1)=\int \limits _{0}^{\infty }\left({\dfrac {\tanh ^{n-1}(x)}{x^{n}}}-{\dfrac {\tanh ^{n}(x)}{x^{n+1}}}\right)dx=-{\frac {1}{n}}+{\frac {(n+1)\cdot 2^{n}}{(n-1)!}}s^{n-2}g_{n}(s)}
.