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EXTRACTA

MATHEMATICAE

Volumen 33, Número 2, 2018

instituto de investigación de matemáticas de la

universidad de extremadura

EXTRACTA MATHEMATICAE

Vol. 35, Num. 1 (2020), 21 – 34

doi:10.17398/2605-5686.35.1.21

Available online October 16, 2019

Identities in the spirit of Euler

A. Sofo

College of Engineering and Science, Victoria University

P. O. Box 14428, Melbourne City, Victoria 8001, Australia

anthony.sofo@vu.edu.au

Received March 26, 2019 Presented by Jesús M.F. Castillo
Accepted September 17, 2019

Abstract: In this paper we develop new identities in the spirit of Euler. We shall investigate and
report on new Euler identities of weight p+ 2, for p an odd integer, but with a non unitary argument

of the harmonic numbers. Some examples of these Euler identities will be given in terms of Riemann

zeta values, Dirichlet values and other special functions.

Key words: Polylogarithm function, recurrence relations, Euler sums, Zeta functions, Dirichlet
functions, Multiple zeta values.

AMS Subject Class. (2010): 11M06, 11M32, 33B15.

1. Introduction

In a previous paper, [16] we investigated families of integrals, where the
integrand is the product of an inverse trigonometric or inverse hyperbolic
trigonometric and the polylogarithmic function,

J(a,δ,p,m) =

∫ 1
0
ym−1f(y) Lip

(
δy2am

)
dy ,

for a ∈ R+, δ = ±1, p ∈ N, m ∈ R+ and where f(y) = arctan (ym) or
tanh−1 (ym). It was demonstrated that integrals of products of inverse trigono-
metric and polylogarithmic functions can be associated with Euler sums. It is
well known that integrals with polylogarithmic integrands can be associated
with Euler sums. Therefore in the spirit of Euler we shall investigate integrals
of the type

I (a,δ,p,q) =

∫ 1
0

ln x

x
Lip(x) Liq (δx

a) dx (1.1)

for a ∈ R+, δ = ±1, p ∈ N, q ∈ N. Some examples are highlighted, almost
none of which are amenable to a computer mathematical package. We shall

ISSN: 0213-8743 (print), 2605-5686 (online)

https://doi.org/10.17398/2605-5686.35.1.21
mailto:anthony.sofo@vu.edu.au
https://www.eweb.unex.es/eweb/extracta/
https://creativecommons.org/licenses/by-nc/3.0/


22 a. sofo

also develop new Euler identities for sums of the type

∞∑
n=1

H
(2)
n
2

np
,

∞∑
n=1

(−1)n+1 H(2)n
2

np
,

∞∑
n=1

H
(2)

n−1
2

(2n− 1)p
(1.2)

and again, some examples are highlighted, almost none of which are amenable
to a computer mathematical package. This work also extends the results
given in [7], where the author examined integrals with positive arguments of
the polylogarithm. Devoto and Duke [4] also list many identities of lower
order polylogarithmic integrals and their relations to Euler sums. Some other
important sources of information on polylogarithm functions are the works of
[9], [10], and [16]. The famous Euler identity [5], for unitary argument of the
harmonic numbers, states

EU (m) =

∞∑
n=1

Hn
nm

=
(m

2
+ 1
)
ζ (m + 1)

−
1

2

m−2∑
j=1

ζ(m− j)ζ(j + 1) .

(1.3)

The famous Euler identity was further extended in the work of [1]. Relatively
recently multiple zeta values (MZVs) were studied and developed by [8], [21]
and others, for example, [11]. MZVs are defined by

ζ (i1, i2, . . . , ik) =
∑

n1>n2···>nk≥1

1

ni11 n
i2
2 . . .n

ik
k

for positive integers ik and i1 > 1 with weight
∑
ik and length or depth k.

For arbitrary p ∈ N and q ≥ 2 the Euler sum

Sp,q =

∞∑
n=1

H
(p)
n

nq
(1.4)

is readily expressible in terms of MZVs, that is, Sp,q+Sq,p = ζ(p)ζ(q)+ζ(p+q).
Euler developed many relations, including

ζ(2, 1) = ζ(3) , S2,3 = 3ζ(2)ζ(3) −
9

2
ζ(5) .



integrals of polylogarithms 23

It appears that at weight eight, S2,6 cannot be reduced to zeta values and
their products. We also note that we may define alternating MZVs with signs
in the numerator as

ζ
(
i1, i2, . . . , ik

)
=

∑
n1>n2>···>nk≥1

(−1)n1+n2+···+nk

ni11 n
i2
2 . . .n

ik
k

.

Therefore an alternating MZV converges unless the first entry is an unbanned
one, and we also have ζ

(
1
)

= − ln 2 and ζ (n) = η(n) =
(
1 − 21−n

)
ζ(n) for

n ≥ 2, so that, for example

ζ
(
3, 1, 2

)
=

∑
n1>n2>n3≥1

(−1)n1+n3
n31n2n

2
3

.

For arbitrary integer weight p ≥ 1, q ≥ 1 we shall define alternating Euler
sums as,

S (p,q) :=

∞∑
n =1

(−1)n+1 H(p)n
nq

. (1.5)

There are some special cases where the linear Euler sum (1.4) is reducible to
zeta values. For odd weight w = (p + q) we have,

BW (p,q) =
∞∑
n=1

H
(p)
n

nq
=

1

2

(
1 + (−1)p+1

)
ζ(p)ζ(q)

+ (−1)p
[ p2 ]∑
j=1

(
p + q − 2j − 1

p− 1

)
ζ(p + q − 2j)ζ(2j) (1.6)

+ (−1)p
[ p2 ]∑
j=1

(
p + q − 2j − 1

q − 1

)
ζ(p + q − 2j)ζ(2j)

+
ζ(p + q)

2

(
1 + (−1)p+1

(
p + q − 1

p

)
+ (−1)p+1

(
p + q − 1

q

))
,

where [z] is the integer part of z. For alternating Euler sums and specified
odd weights we have some particular identities. Sitaramachandra Rao, [12]
gave the identity, for S(p,q), when p = 1 and for odd weight 1 + q as,

2S(1,q) = (1 + q)η(1 + q) − ζ(1 + q) − 2

q
2
−1∑
j=1

η(2j) ζ(1 + q − 2j) (1.7)



24 a. sofo

and in another special case, gave the integral

S(1, 1 + 2q) =
1

(2q)!

∫ 1
0

ln2q(x) ln(1 + x)

x(1 + x)
dx.

In the case where p and q are both positive integers and p+q is an odd integer,
Flajolet and Salvy [6] gave the identity:

2S(p,q) = (1 − (−1)p) ζ(p)η(q)

+ 2(−1)p
∑

i+2k=q

(
p + i− 1
p− 1

)
ζ(p + i) η(2k) (1.8)

+ η(p + q) − 2
∑

j+2k=p

(
q + j − 1
q − 1

)
(−1)jη(q + j) η(2k) ,

where η(0) = 1
2
, η(1) = ln 2, ζ(1) = 0, and ζ(0) = −1

2
in accordance with the

analytic continuation of the Riemann zeta function. We define the alternating
zeta function (or Dirichlet eta function) η(z) as

η(z) :=

∞∑
n=1

(−1)n+1

nz
=
(
1 − 21−z

)
ζ(z) .

The following Euler identities for harmonic numbers at half integer values
have been given in [19].

Lemma 1. For δ = ±1, a ∈ R+, r ∈ N and m a positive odd integer,

W(a,δ,m,r) =
∑
n≥1

δn+1H
(r)
an

nm

then

W

(
1

2
, 1,m, 1

)
= EU (m) + (−1)m+1S(1,m)

+
m−1∑
k=2

(−1)m−kζ(k)η(m + 1 −k) ,
(1.9)

and

W

(
1

2
,−1,m, 1

)
=
(
1 − 21−m

)
EU (m) + (−1)m+1S(1,m)

+
m−1∑
k=2

(−1)m−kζ(k)η(m + 1 −k) .
(1.10)



integrals of polylogarithms 25

Therefore the main aim of this paper is to develop new Euler identities
for the sums (1.2) and represent the solution of the integral (1.1), in terms
of special functions, for various values of the parameters (a,δ,p,q). First we
define some special functions that we will encounter in the body of this paper.
The Lerch transcendent,

Φ(z,t,a) =

∞∑
m=0

zm

(m + a)t

is defined for |z| < 1 and <(a) > 0 and satisfies the recurrence

Φ(z,t,a) = z Φ(z,t,a + 1) + a−t.

The Lerch transcendent generalizes the Hurwitz zeta function at z = 1,

Φ(1, t,a) =
∞∑
m=0

1

(m + a)t

and the polylogarithm, or de-Jonquière’s function, when a = 1,

Lit(z) :=
∞∑
m=1

zm

mt
, t ∈ C when |z| < 1 ; <(t) > 1 when |z| = 1 .

Let

Hn =

n∑
r=1

1

r
=

∫ 1
0

1 − tn

1 − t
dt = γ + ψ(n + 1) =

∞∑
j=1

n

j(j + n)
, H0 := 0

be the nth harmonic number, where γ denotes the Euler-Mascheroni constant,

H
(m)
n =

∑n
r=1

1
rm

is the mth order harmonic number and ψ(z) is the digamma
(or psi) function defined by

ψ(z) :=
d

dz
{log Γ(z)} =

Γ′(z)

Γ(z)
and ψ(1 + z) = ψ(z) +

1

z
,

moreover,

ψ(z) = −γ +
∞∑
n=0

(
1

n + 1
−

1

n + z

)
.

The polygamma function

ψ(k)(z) =
dk

dzk
{ψ(z)} = (−1)k+1k!

∞∑
r=0

1

(r + z)k+1



26 a. sofo

and has the recurrence

ψ(k)(z + 1) = ψ(k)(z) +
(−1)kk!
zk+1

.

The connection of the polygamma function with harmonic numbers is,

H(α+1)z = ζ(α + 1) +
(−1)α

α!
ψ(α)(z + 1) , z 6= {−1,−2,−3, . . .} , (1.11)

and the multiplication formula is

ψ(k)(pz) = δm,0 ln p +
1

pk+1

p−1∑
j=0

ψ(k)
(
z +

j

p

)
(1.12)

for p a positive integer and δp,k is the Kronecker delta. The work in this paper
also extends the results of [7], [20]. Other works including, [2], [3], [13], [14],
[15], [17], and [18] cite many identities of polylogarithmic integrals and Euler
sums.

2. Integral identities and Euler sums

Theorem 1. For a ∈ R+, δ = {−1, 1}, p, q, positive integers, then

I(a,δ,p,q) =

∫ 1
0

ln x Lip(x) Liq (δx
a)

x
dx

=
(−1)p+1

ap
ζ(2) Lip+q(δ) (2.1)

+
(−1)pp
ap+1

∞∑
n=1

δn Han
np+q+1

+
(−1)p

ap

∞∑
n=1

δn H
(2)
an

np+q

−
p∑

k=2

(−1)p−k(p + 1 −k)
ap+2−k

ζ(k) Lip+q+2−k(δ) .

where Lip+q (δ) is the polylogarithm, Han and H
(2)
an are the harmonic numbers.

Proof. By the definition of the polylogarithmic function we have

I(a,δ,p,q) =

∞∑
n=1

δn

nq

∞∑
j=1

1

jp

∫ 1
0
xj+an−1 ln x dx

= −
∞∑
n=1

δn

nq

∞∑
j=1

1

jp(j + an)2



integrals of polylogarithms 27

and by partial fraction decomposition,

I(a,δ,p,q) = −
∞∑
n=1

δn

nq

∞∑
j=1




p(−1)p+1
(an)pj(j+an)

+
(−1)p

(an)p(j+an)2

+
∑p

k=2
(−1)p−k(p+1−k)

(an)p+2−kjk


 .

Now

I(a,δ,p,q) = −
∞∑
n=1

δn

nq




p(−1)p+1 Han
(an)p+1

+
(−1)p ψ′(an+1)

(an)p

+
∑p

k=2
(−1)p−k(p+1−k)

(an)p+2−k
ζ(k)




=
(−1)p+1

ap
ζ(2) Lip+q(δ) +

(−1)pp
ap+1

∞∑
n=1

δn Han
np+q+1

+
(−1)p

ap

∞∑
n=1

δn H
(2)
an

np+q

−
p∑

k=2

(−1)p−k(p + 1 −k)
ap+2−k

ζ(k) Lip+q+2−k(δ),

and Theorem 1 is proved.

In the next few corollaries we investigate various special values of the pa-
rameters (a,δ,p,q) which will yield solutions to I (a,δ,p,q) that are express-
ible in terms of the Riemann zeta and other special functions. We shall also
present new Euler type identities for the sums (1.2).

Corollary 1. For a = 1, δ = 1, p, q, positive integers with arbitrary
weight p + q, then

I(1, 1,p,q) =

∫ 1
0

ln x Lip(x) Liq(x)

x
dx

= (−1)p (S2,p+q + pS1,p+q+1) + (−1)p+1ζ(2)ζ(p + q) (2.2)

−
p∑

k=2

(−1)p−k(p + 1 −k)ζ(k)ζ(p + q + 2 −k) ,

where Sa,b is the linear Euler sum (1.4).



28 a. sofo

Proof. Here we note that Lim(1) = ζ(m) and the sums S2,p+q =
∑∞

n=1
H

(2)
n

np+q

and S1,p+q+1 =
∑∞

n=1
Hn

np+q+1
.

Remark 1. For a = 1, δ = 1, p, q, positive integers with p + q an odd
integer, then

I(1, 1,p,q) =

∫ 1
0

ln x Lip(x) Liq(x)

x
dx

= p(−1)p EU (p + q + 1) + (−1)p BW (2,p + q)

+ (−1)p+1ζ(2)ζ(p + q) (2.3)

−
p∑

k=2

(−1)p−k(p + 1 −k)ζ(k)ζ(p + q + 2 −k) ,

where EU (·) is the Euler identity (1.3) and BW (·, ·) is the identity (1.6).

Example 1.

I(1, 1, 4, 2) = I(1, 1, 2, 4) = S2,6 − 2ζ(3)ζ(5) +
7

6
ζ(8),

I(1, 1, 4, 4) = S2,8 − 2ζ(3)ζ(7) +
33

10
ζ(10) − 2ζ2(5),

I(1, 1, 4, 5) = I(1, 1, 5, 4) = ζ(4)ζ(7) + ζ(2)ζ(9) − 3ζ(11).

Corollary 2. For a = 1, δ = −1, p, q, positive integers, then

I(1,−1,p,q) =
∫ 1

0

ln x Lip(x) Liq(−x)
x

dx

= p(−1)p+1S(1,p + q + 1) (2.4)

+ (−1)p+1S(2,p + q) + (−1)pζ(2)η(p + q)

+

p∑
k=2

(−1)p−k(p + 1 −k)ζ(k)η(p + q + 2 −k),

where η(·) is the Dirichlet Eta function and S(·, ·) is the alternating linear
Euler sum. We note that when we have odd weight (p + q), S(·, ·) may be
replaced with the identity (1.8).



integrals of polylogarithms 29

Proof. Here we note that Lim(−1) = −η(m) and the sum S(m,p + q) =∑∞
n=1

(−1)n+1H(m)n
np+q

and may be replaced with the identity (1.8) in the case
when we have odd weight (p + q).

Example 2.

I(1,−1, 2, 4) = 2ζ(2)η(6) −S(2, 6) − 2S(1, 7),

I(1,−1, 4, 2) = −
359

48
ζ(8) − 2ζ(3)η(5) −S(2, 6) − 4S(1, 7),

I(1,−1, 2, 3) =
43

32
ζ(2)ζ(5) − 2ζ(7),

I(1,−1, 3, 4) =
7

8
ζ(4)ζ(5) +

5

2
ζ(9) −

249

128
ζ(2)ζ(7).

Corollary 3. For a = 2, δ = 1, p, q, positive integers, then

I(2, 1,p,q) =

∫ 1
0

ln x Lip(x) Liq
(
x2
)

x
dx

=
(−1)p+1

2p
ζ(2)ζ(p + q) (2.5)

+ p(−1)p2q−1 (S1,p+q+1 −S(1,p + q + 1))

+ (−1)p2q−1 (S2,p+q −S(2,p + q))

− (−1)p
p∑

k=2

(−1)k(p + 1 −k)
2p+2−k

ζ(k)ζ(p + q + 2 −k),

where S·,· and S(·, ·) are the linear Euler and alternating Euler sums (1.4)
and (1.5) respectively. In the case when we have odd weight (p + q) then we
may utilize S1,p+q+1 = EU (p + q + 1) is the Euler identity (1.3), S2,p+q =
BW (2,p + q) is the identity (1.6) and S(·, ·) is obtained from the identity
(1.8).

Proof. Here we note that Lim(1) = ζ(m) and the sums S·,· and S(·, ·) are
the linear Euler and alternating Euler sums (1.4) and (1.5) respectively. In
the case of odd weight (p + q) the sums S1,p+q+1 =

∑∞
n=1

Hn
np+q+1

, S2,p+q =

BW (2,p + q) =
∑∞

n=1
H

(2)
n

np+q
and S(a,b) =

∑∞
n=1

(−1)n+1H(a)n
nb

.



30 a. sofo

Example 3.

I(2, 1, 3, 3) =
47

4
ζ(3)ζ(5) −

211

8
ζ(8) − 4S2,6 + 4S(2, 6) + 12S(1, 7),

I(2, 1, 4, 2) =
415

24
ζ(8) −

31

4
ζ(3)ζ(5) + 2S2,6 − 2S(2, 6) − 8S(1, 7),

I(2, 1, 3, 2) =
11

32
ζ(5) −

7

16
ζ(2)ζ(3),

I(2, 1, 0,q) = 2q−1 (EU (q) −S(1,q)) ,

I(2, 1, 5, 0) =
107

256
ζ(7) +

1

4
ζ(4)ζ(3) −

37

64
ζ(5)ζ(2).

The aim now, is to obtain the new Euler identities for the sums (1.2),
hence consider the following corollary.

Corollary 4. From Corollary 3, let q = 0, δ = ±1, and p, a positive odd
integer, then

W

(
1

2
, 1,p, 2

)
=
∑
n≥1

H
(2)
n
2

np
(2.6)

= p(−1)p+122−pζ(2)ζ(p) + 2p(−1)p (EU (p + 1) −S(1,p + 1))

+ 2(−1)p (BW (2,p) −S(2,p))

+ (−1)p+1
p−1∑
k=3

(−1)k(p + 1 −k)
2p−k

ζ(k)ζ(p + 2 −k),

where EU (·), S(·, ·) and BW (·, ·) are the same as in Corollary 3.

Proof. Proceeding as in Theorem 1,

I(a,δ,p, 0) =

∫ 1
0

ln x Lip(x) Li0 (δx
a)

x
dx

= −
δ

a2

∑
n≥1

1

np

∑
j≥1

δj

(n + aj)2
= −

δ

a2

∑
n≥1

1

np
Φ
(
δ, 2, 1 +

n

a

)
,

where Φ
(
δ, 2, 1 + n

a

)
is the Lerch transcendent. We have,

I(a,δ,p, 0) =



− 1
a2
∑

n≥1
ψ′( na +1)

np
for δ = 1 ,

1
4a2
∑

n≥1
ψ′( n

a
+1)

np

(
ψ′(a+n

2a
) −ψ′( 2q+n

2a
)
)

for δ = −1 .
(2.7)



integrals of polylogarithms 31

Now for a = 2 and δ = 1, we have

I(2, 1,p, 0) = −
1

4

∑
n≥1

ψ′(n
2

+ 1)

np
=

1

4

∑
n≥1

H
(2)
n
2

np
−

1

4
ζ(2)ζ(p)

and equating with (2.5) we obtain the desired result (2.6). Also, since

∑
n≥1

H
(2)
n

np
= 21−p

∑
n≥1

H
(2)
n
2

np
(
1 − (−1)n+1

)
we obtain the second Euler identity

W

(
1

2
,−1,p, 2

)
=
∑
n≥1

(−1)n+1H(2)n
2

np

= W

(
1

2
, 1,p, 2

)
− 21−p BW (2,p)

=
(
1 + (−2)p+1

)
ζ(2)ζ(p) − 2pW

(
1

2
,−1,p + 1, 1

)
+ (−2)2−pW(2, 1,p, 2) + (−2)1−p BW (2,p)

+

p∑
k=2

(−2)2−k(p + 1 −k)ζ(k)η(p + 2 −k) .

Similarly, for the third Euler sum identity in (1.2) we have

∞∑
n =1

H
(2)

n−1
2

(2n− 1)p
=

1

2

(
W

(
1

2
, 1,p, 2

)
+ W

(
1

2
,−1,p, 2

))
.

Example 4.

∑
n≥1

Hn
2

n6
=

135

128
ζ(7) −

1

16
ζ(2)ζ(5) −

1

4
ζ(3)ζ(4),

∑
n≥1

(−1)n+1Hn
2

n6
=

119

128
ζ(7) −

1

32
ζ(2)ζ(5) −

7

32
ζ(3)ζ(4),



32 a. sofo

∑
n≥1

H
(2)
n
2

n5
= ζ(3)ζ(4) −

21

16
ζ(2)ζ(5) +

107

64
ζ(7),

∑
n≥1

(−1)n+1H(2)n
2

n5
=

7

8
ζ(3)ζ(4) −

13

8
ζ(2)ζ(5) +

147

64
ζ(7),

∞∑
n=1

H
(2)

n−1
2

(2n− 1)5
=

15

16
ζ(3)ζ(4) −

47

16
ζ(2)ζ(5) +

127

64
ζ(7).

Corollary 5. For a = 1
2
, δ = ±1, p, q, positive integers with p + q an

odd integer, then

I
(1

2
,δ,p,q

)
=

∫ 1
0

ln x Lip(x) Liq

(
δx

1
2

)
x

dx

=




2p(−2)pW
(

1
2
, 1,p + q + 1, 1

)
− (−2)pζ(2)ζ(p + q)

+(−2)pW
(

1
2
, 1,p + q, 2

)
−
∑p

k=2(−1)
p−k(p + 1 −k)ζ(k)ζ(p + q + 2 −k) for δ = 1 ,

(−2)p+1W
(

1
2
,−1,p + q + 1, 1

)
+ (−2)pζ(2)η(p + q)

+(−2)pW
(

1
2
,−1,p + q, 2

)
+
∑p

k=2(−1)
p−k(p + 1 −k)ζ(k)η(p + q + 2 −k) for δ = −1 ,

where W(·, ·, ·, ·) is evaluated from Corollary (4).

Proof. The proof follows from (2.1).

Example 5. In these examples we utilize some results from Example 4:

I(
1

2
, 1, 5, 0) = 16ζ(3)ζ(4) + 218ζ(2)ζ(5) − 391ζ(7),



integrals of polylogarithms 33

I

(
1

2
,−1, 5, 0

)
= 371ζ(7) − 12ζ(3)ζ(4) − 210ζ(2)ζ(5),

I

(
1

2
, 1, 0, 5

)
=

107

16
ζ(7) + ζ(3)ζ(4) −

37

16
ζ(2)ζ(5),

I

(
1

2
,−1, 0, 5

)
= −

147

16
ζ(7) −

7

8
ζ(3)ζ(4) +

41

16
ζ(2)ζ(5),

I

(
1

2
, 1, 3, 2

)
=

75

2
ζ(2)ζ(5) − 64ζ(7),

I

(
1

2
,−1, 3, 2

)
= 63ζ(7) − 37ζ(2)ζ(5).

Remark 2. The integral I(a,δ,p,q) has been represented in terms of spe-
cial functions. For particular values of the constants (a,δ,p,q) the integral
(1.1) has been expressed in closed form in terms of Riemann zeta and Dirich-
let Eta functions. Some examples are given for the solution of the integral
(1.1), most of which are not amenable to a mathematical computer package.
Finally we have developed new identities for the Euler sums (1.2) in the spirit
of Euler (1.3).

Acknowledgements

The author is thankful to a referee for the careful reading and con-
sidered suggestions leading to a better presented paper.

References

[1] Borwein, D.; Borwein, J. M..; Girgensohn, R. Explicit evaluation of Euler
sums. Proc. Edinburgh Math. Soc. (2) 38 (1995), no. 2, 277–294.

[2] Bailey, D. H. Borwein, J. M. Computation and structure of charac-
ter polylogarithms with applications to character Mordell-Tornheim-
Witten sums. Math. Comp. 85 (2016), no. 297, 295–324.

[3] Choi, Junesang Log-sine and log-cosine integrals. Honam Math. J. 35
(2013), no. 2, 137–146.

[4] Devoto, A. Duke, D. W. Table of integrals and formulae for Feynman
diagram calculations. Riv. Nuovo Cimento (3) 7 (1984), no. 6, 1–39.

[5] Euler, L. Meditationes circa singulare serierum genus, Novi Comm. Acad.
Sci. Petropol. 20 (1776), 140-186; reprinted in Opera Omnia, Ser. I,
Vol. 15, B. G.Teubner, Berlin, 1927, pp. 217-267.



34 a. sofo

[6] Flajolet, P. Salvy, B. Euler sums and contour integral representations.
Experiment. Math. 7 (1998), no. 1, 15–35.

[7] Freitas, P. Integrals of polylogarithmic functions, recurrence relations,
and associated Euler sums. Math. Comp. 74 (2005), no. 251, 1425–
1440.

[8] Hoffman, M. E., Multiple harmonic series, Pacific J. Math. 152 (1992),
275-290.

[9] Kölbig, K. S. Nielsen’s generalized polylogarithms. SIAM J. Math. Anal.
17 (1986), no. 5, 1232–1258.

[10] Lewin, R. Polylogarithms and Associated Functions. North Holland, New
York, 1981.

[11] Markett, C. Triple sums and the Riemann zeta function. J. Number The-
ory 48 (1994), no. 2, 113–132.

[12] Sitaramachandra Rao, R. A formula of S. Ramanujan. J. Number Theory
25 (1987), no. 1, 1–19.

[13] Sofo, A. Polylogarithmic connections with Euler sums. Sarajevo J. Math.
12(24) (2016), no. 1, 17–32.

[14] Sofo, A. Integrals of logarithmic and hypergeometric functions. Commun.
Math. 24 (2016), no. 1, 7–22.

[15] Sofo, A. and Cvijović, D. Extensions of Euler Harmonic Sums, Appl.
Anal. Discrete Math. 6 (2012), 317–328.

[16] Sofo, A. Integrals of inverse trigonometric and polylogarithmic functions.
Submitted, 2019.

[17] Sofo, A.; Srivastava, H. M. A family of shifted harmonic sums. Ramanu-
jan J. 37 (2015), no. 1, 89–108.

[18] Sofo, A. New classes of harmonic number identities. J. Integer Seq. 15
(2012), no. 7, Article 12.7.4, 12 pp.

[19] Sofo, A. Families of Integrals of Polylogarithmic Functions, Special Func-
tions and Applications. Editors Choi, J. and Shilin, I. Mathematics
(2019), 7, 143; doi:10.3390/math7020143, Published by MDPI AG,
Basel, Switzerland.

[20] Xu, Ce. Yan, Yuhuan. Shi, Zhijuan. Euler sums and integrals of polylog-
arithm functions. J. Number Theory 165 (2016), 84–108.

[21] Zagier, D. Values of zeta functions and their applications, in First Eu-
ropean Congress of Mathematicians, Vol II (Paris, 1992), Birkhauser,
Boston, 1994, pp. 497-512.


	Introduction
	Integral identities and Euler sums