Hindawi Publishing Corporation
International Journal of Combinatorics
Volume 2011, Article ID 208260, 14 pages
doi:10.1155/2011/208260
Research Article
Harmonic Numbers and Cubed Binomial
Coefficients
Anthony Sofo
Victoria University College, Victoria University, P.O. Box 14428, Melbourne City, VIC 8001, Australia
Correspondence should be addressed to Anthony Sofo, anthony.sofo@vu.edu.au
Received 18 January 2011; Accepted 3 April 2011
Academic Editor: Toufik Mansour
Copyright q 2011 Anthony Sofo. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
Euler related results on the sum of the ratio of harmonic numbers and cubed binomial coefficients
are investigated in this paper. Integral and closed-form representation of sums are developed in
terms of zeta and polygamma functions. The given representations are new.
1. Introduction
The well-known Riemann zeta function is defined as
ζz ∞
1
,
z
r
r1
Rez > 1.
1.1
The generalized harmonic numbers of order α are given by
α
Hn n
1
rα
r1
for α, n ∈ Æ × Æ ,
Æ : {1, 2, 3, . . .},
1.2
and for α 1,
1
Hn Hn 1
0
n
1 − tn
1
dt γ ψn 1,
1−t
r
r1
1.3
2
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where γ denotes the Euler-Mascheroni constant defined by
γ lim
n
1
n→∞
r1
r
− logn
−ψ1 ≈ 0.5772156649 . . .,
1.4
and where ψz denotes the Psi, or digamma function defined by
∞ d
Γ z 1
1
ψz log Γz −
− γ,
dz
Γz n0 n 1 n z
1.5
∞
and the Gamma function Γz 0 uz−1 e−u du, for Êz > 0.
Variant Euler sums of the form
1
1
∞ H
x2n
2n − 1/2Hn
np
n1
for p 1 and p 2
1.6
have been considered by Chen 1, and recently Boyadzhiev 2 evaluated various binomial
1
identities involving power sums with harmonic numbers Hn . Other remarkable harmonic
number identities known to Euler are
1
∞
Hn
n1 n
2
1
1
∞
Hn
ζ3,
n1
n3
5
ζ4,
4
1.7
there is also a recurrence formula
2n 1ζ2n 2
n−1
ζ2rζ2n − 2r,
1.8
r1
which shows that in particular, for n 2, 5ζ4 2ζ22 and more generally that ζ2n is a
rational multiple of ζn2 . Another elegant recursion known to Euler 3 is
2
1
∞
Hn
n1
nq
q−2
q 2 ζ q 1 − ζr 1ζ q − r .
1.9
r1
Further work in the summation of harmonic numbers and binomial coefficients has also
been done by Flajolet and Salvy 4 and Basu 5. In this paper it is intended to add,
in a small way, some results related to 1.7 and to extend the result of Cloitre, as
1 nk k/k − 12 for k > 1. Specifically, we investigate
reported in 6, ∞
n1 Hn /
k
integral representations and closed form representations for sums of harmonic numbers and
cubed binomial coefficients. The works of 7–13 also investigate various representations of
binomial sums and zeta functions in simpler form by the use of the Beta function and other
techniques. Some of the material in this paper was inspired by the work of Mansour, 8,
where he used, in part, the Beta function to obtain very general results for finite binomial
sums.
International Journal of Combinatorics
3
2. Integral Representations and Identities
The following Lemma, given by Sofo 11, is stated without proof and deals with the
derivative of a reciprocal binomial coefficient.
−1
Lemma 2.1. Let a be a positive real number, z ≥ 0, n is a positive integer and let Qan, z anz
z be an analytic function of z. Then,
⎧
⎨−Qan, z ψz 1 an − ψz 1
for z > 0,
dQ
Q an, z dz ⎩−H 1 ,
for z 0 and a 1.
n
2.1
Theorem 2.2. Let a, b, c, d ≥ 0 be real positive numbers, |t| ≤ 1, p ≥ 0 and let j, k, l, m ≥ 0 be real
positive numbers. Then
tn
n
n≥1
ψ j 1 an − ψ j 1
anj bnk cnl dnm np−1
n−1
j
k
−aklmt
l
m
k−1
1 − xj 1 − y
1 − zl−1 1 − wm−1
p1
0
1 − txa yb zc wd
1
2.2
× ln1 − x · xa−1 yb zc wd dx dy dz dw.
Proof. Expand
n≥1 n
tn np−1
n−1
anj bnk cnl dnm j
k
m
l
n p − 1 anΓanΓ j 1 Γbn 1k · Γk
n
t
nΓ an j 1 Γbn k 1
n−1
n≥1
Γcn 1l · ΓlΓdn 1m · Γm
×
Γcn l 1Γdn m 1
aklm
1
0
1 − xj an n n p − 1
× Bbn 1, kBcn 1, lBdn 1, mdx,
x t
x
n−1
n≥1
2.3
where
1
ΓαΓ β
α−1 β−1
B α, β y dy
1−y
Γ αβ
0
1
β−1 α−1
1−y
y dy, for α > 0 and β > 0
0
2.4
4
International Journal of Combinatorics
is the classical Beta function. Differentiating with respect to the parameter j, and utilizing
Lemma 2.1 implies the resulting equation is as follows:
t
n
n
n≥1
ψ j 1 an − ψ j 1
anj bnk cnl dnm np−1
n−1
j
k
−aklm
1
0
l
m
np−1
1 − xj
an n
ln1 − x x t
x
n−1
n≥1
× Bbn 1, kBcn 1, lBdn 1, mdx,
1
k−1
1 − xj ln1 − x 1−y
1 − zl−1 1 − wm−1
x
0
n
np−1 txa yb zc wd dx dy dz dw
×
n−1
n≥1
−aklm
−aklmt
2.5
k−1
1 − xj ln1 − x 1 − y
1 − zl−1 1 − wm−1
p1
0
1 − txa yb zc wd
1
× xa−1 yb zc wd dx dy dz dw
for |txa yb zc wd | < 1.
In the following three corollaries we encounter harmonic numbers at possible rational
α
values of the argument, of the form Hr/b−1 where r 1, 2, 3, . . . , k, α 1, 2, 3, . . ., and k ∈ Æ .
The polygamma function ψ α z is defined as
ψ α z dα dα1 log
Γz
ψz ,
dzα
dzα1
z
/ {0, −1, −2, −3, . . .}.
2.6
α
To evaluate Hr/b−1 we have available a relation in terms of the polygamma function ψ α z,
for rational arguments z,
α1
Hr/b−1 ζα 1 −1α α r ψ
,
α!
b
2.7
where ζz is the Riemann zeta function. We also define
1
Hr/b−1 γ ψ
r ,
b
α
H0
0.
2.8
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5
The evaluation of the polygamma function ψ α r/b at rational values of the argument can be
explicitly done via a formula as given by Kölbig 14 see also 15 or Choi and Cvijović 16
in terms of the polylogarithmic or other special functions. Some specific values are given as
1
−1n n! 2n1 − 1 ζn 1
2
√
1 4 1
2π 5 3
2
− 120ζ5,
H1/4 16 − 8G − 5ζ2,
ζ5 − ψ
−
24
3
9
ψ n
5
H−2/3
2.9
and can be confirmed on a mathematical computer package, such as Mathematica 17.
Corollary 2.3. Let a 1, d c b > 0, t 1, p 0, j 0 and let l m k ≥ 1 be a positive
integer. Then
n≥1 n
1
Hn
bnk 3
k
−k3
1 0
k
r1
−1
r1
k−1
1 − y 1 − z1 − w
b
ln1 − x · yzw dx dy dz dw
b 1 − x yzw
3
k
r
r2
× − 2 ζ4 −
4b
2.10
r 2 1
r r2
H
XRk ζ3
b2 r/b−1 b b
r 1
r 2 2
r 1
2
1 − Hr/b−1 − 2 Hr/b−1 r 1 − Hr/b−1 XRk r Y Rk ζ2
b
b
b
2
1 1
r 1
2
2
3
Hr/b−1 Hr/b−1 Hr/b−1 Hr/b−1 Hr/b−1
2
b
2
r 2 2
1
3
4
Hr/b−1 2Hr/b−1 Hr/b−1 3Hr/b−1
2
2b
2
r 2 1
r 1
2
2
3
Hr/b−1 Hr/b−1 Hr/b−1 Hr/b−1 Hr/b−1
XRk
2
b
2
r 2 1
2
Hr/b−1 Hr/b−1 Y Rk ,
2
2.11
6
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where
1
1
XRk : −3 Hk−r − Hr−1 ,
2.12
3 XR2 k
2
2
Hk−r Hr−1 .
Y Rk :
2
3
2.13
Proof. Let
1
1
Hn k!3
Hn
k
bnk 3
3
n≥1 n
n≥1 n
r1 bn r
k
k
Hn1 k!3 Ar
Br
Cr
,
n
bn r bn r2 bn r3
n≥1
r1
2.14
where
Cr k
lim
r1 bn
n → −r/b
d
Br lim
n → −r/b dn
bn r3
r1
r3
−1
bn r3
k
r1 bn
r
k!
3
k
r
,
r3
3
k
1
1
3−1
Hk−r − Hr−1 ,
r
d2
1
bn r3
Ar lim
k
3
2 n → −r/b dn2
r1 bn r
r
3
−1r1
2
r
k!
r
k!
2.15
3 k
1
1 2
2
2
3 Hk−r − Hr−1 Hk−r Hr−1 .
r
Now, by interchanging sums, we have
n≥1 n
1
k
Hn1
Hn
3
A
Br k!3
k!
r
bnk 3
nbn r
k
r1
n≥1
1
Hn
n≥1 nbn
r2
Cr k!
3
1
Hn
n≥1 nbn
.
r3
2.16
International Journal of Combinatorics
7
We can evaluate
1
ψn 1/n
γ
ψn
γ
1
Hn
2
nbn r n≥1
nbn r
nbn r
nbn r n bn r nbn r
n≥1
n≥1
n≥1
ψn
1
2
nbn r rn
γ − b/r 1
1
1
1
−
−
r
n 1 n r/b
n r/b n 1 r/b
n≥0
2
ψr/b
bγ γ 2 3ζ2
ψ r/b γ r − 2 −
−
ψ
γ b ,
2r
2r
2r
2r
r
b
r
2.17
here we have used the result from 18
2
r
ψn
1 r
− γ 2 ζ2 − ψ ψ
1
1 .
nbn r 2r
b
b
n≥1
2.18
Now using 2.7 and 2.8, we may write
1
2
ζ2
1 1
Hn
2
Hr/b−1 Hr/b−1 .
nbn r
r
2r
n≥1
Similarly
1
Hn
n≥1 nbn
r2
1
1
b Hn
1 Hn
−
r n≥1 nbn r r n≥1 bn r2
2
1 1
ζ3 ζ2 ζ2Hr/b−1
2
2 −
2 Hr/b−1 Hr/b−1
br
br
r
2r
1
−
1 1
2
3
Hr/b−1 Hr/b−1 Hr/b−1 ,
br
2.19
8
International Journal of Combinatorics
Hn1
1
b
b
1
−
Hn
−
3
r 2 nbn r r 2 bn r2 rbn r3
n≥1 nbn r
n≥1
1
1
ζ4 ζ3 ζ3Hr/b−1 ζ2 ζ2Hr/b−1
3 −
− 2 −
−
4b r
br 2
b2 r
r
br 2
2
−
ζ2Hr/b−1
b2 r
2
1 1
2
H
H
r/b−1
r/b−1
2r 3
1 1
2
3
H
H
H
r/b−1 r/b−1
r/b−1
br 2
2
1
2
1
3
4
2
Hr/b−1 2Hr/b−1 Hr/b−1 3Hr/b−1 .
2b r
2.20
Substituting 2.19, 2.20 into 2.16 where XRk and Y Rk are given by 2.12 and 2.13,
respectively, on simplifying the identity 2.11 is realized.
For k 1 and b 1 the following identity is valid:
1
Hn
n≥1 nn
3
1
ζ2 − ζ3 −
ζ4
.
4
2.21
Theorem 2.4.
t
n≥1
n
ψ j 1 an − ψ j 1
cnl dnm bnk
2 anj
np−1
n−1
n
j
−ablmt
k
l
m
k
1 − xj 1 − y 1 − zl−1 1 − wm−1
p1
0
1 − txa yb zc wd
1
2.22
× ln1 − x · xa−1 yb−1 zc wd dx dy dz dw.
Proof. The proof of this theorem is very similar to that of Theorem 2.2 and will not be given
here.
International Journal of Combinatorics
9
Corollary 2.5. Let a 1, d c b > 0, t 1, p 0, j 0, and let l m k ≥ 1 be a positive
integer. Then
1
n≥1 n2
Hn
3
bnk
k
1 −bk2
0
k
1 − y 1 − z1 − wk−1
ln1 − x · yb−1 zwb dx dy dz dw
b 1 − x yzw
2.23
3
k
k
r1
−1
r
r1
×
r 1
r
ζ4 4 Hr/b−1 3rXRk 2r 2 Y Rk ζ3
4b
b
3b
r 2
1
1
− 2Hr/b−1 Hr/b−1 rHr/b−1 − 2b XRk − brY Rk ζ2
r
b
−
2
3b 1
2
1
2
3
Hr/b−1 Hr/b−1 − 2 Hr/b−1 Hr/b−1 Hr/b−1
2r
−
2
r 2
1
3
4
Hr/b−1 2Hr/b−1 Hr/b−1 3Hr/b−1
2b
2
1
2
1
2
3
− b Hr/b−1 Hr/b−1 r Hr/b−1 Hr/b−1 Hr/b−1 XRk
−
2
br 1
2
Hr/b−1 Hr/b−1 Y Rk ,
2
2.24
where XRk is given by 2.12 and Y Rk is given by 2.13.
Proof. Following similar steps to Corollary 2.3, we may write
n≥1 n
1
Hn
2 bnk 3
k
k
r1
1
1
1
Hn
Hn
Hn
3
3
Cr k!
Br k!
Ar k!
,
2
3
2
2
n2 bn r
n≥1
n≥1 n bn r
n≥1 n bn r
3
2.25
10
International Journal of Combinatorics
and evaluate
1
1 1
b
Hn
H
−
n
n2 bn r n≥1
rn2 rnbn r
n≥1
2
2ζ3 bζ2
b 1
2
−
H
−
H
r/b−1
r/b−1 ,
r
r2
2r 2
n≥1 n
1
1
Hn
2 bn
r2
3ζ3 2bζ2 ζ2Hr/b−1
−
r2
r3
r2
−
2
1 1
b 1
2
2
3
H
H
r/b−1
r/b−1 − 2 Hr/b−1 Hr/b−1 Hr/b−1 ,
3
r
r
1
1
2
1
ζ4 3bζ2 2ζ2Hr/b−1 ζ2Hr/b−1 4ζ3 ζ3Hr/b−1
−
3
2
4br 2
r4
r3
br 2
r3
br 2
n≥1 n bn r
2
2 1
3b 1
2
2
3
− 4 Hr/b−1 Hr/b−1 − 3 Hr/b−1 Hr/b−1 Hr/b−1
r
2r
2
1
2
1
3
4
−
Hr/b−1 2Hr/b−1 Hr/b−1 3Hr/b−1 .
2br 2
Hn
2.26
By substituting 2.26 into 2.25 and collecting zeta functions, the identity 2.24 is obtained.
For k 1 and b 1 the following identity is valid:
1
n≥1
Hn
n2 n
3
1
ζ4
4ζ3 − 3ζ2.
4
2.27
Theorem 2.6.
t
n≥1
n
ψ j 1 an − ψ j 1
cnl dnm bnk
3 anj
np−1
n−1
n
j
−abcmt
k
l
m
k
1 − xj 1 − y 1 − zl 1 − wm−1
p1
0
1 − txa yb zc wd
1
2.28
× ln1 − x · xa−1 yb−1 zc−1 wd dx dy dz dw.
Proof. The proof of this theorem is very similar to that of Theorem 2.2 and will not be given
here.
International Journal of Combinatorics
11
Corollary 2.7. Let a 1, d c b > 0, t 1, j 0, and let l m k ≥ 1 be a positive integer.
Then
n≥1 n
1
Hn
3 bnk 3
k
−b2 k
1 0
k
r1
−1
×
2.29
3
k
r
r1
k
1 − y 1 − z 1 − wk−1
b−1 b
w dx dy dz dw
ln1 − x · yz
b 1 − x yzw
5
5
1 XRk r 2 Y Rk ζ4
4
4
−
9b
1
Hr/b−1 5bXRk 2brY Rk ζ3
r
⎛
1
6b2 3bHr/b−1
2
⎝ 2 −
− Hr/b−1 −
r
r
⎞
3b2
1
bHr/b−1 XRk b2 Y Rk⎠ζ2
r
2
3b2 1
3b 1
2
2
3
2
Hr/b−1 Hr/b−1 Hr/b−1
Hr/b−1 Hr/b−1 r
r
2
1
2
1
3
4
Hr/b−1 2Hr/b−1 Hr/b−1 3Hr/b−1
2
2
3b2 1
2
1
2
3
Hr/b−1 Hr/b−1 b Hr/b−1 Hr/b−1 Hr/b−1
XRk
2r
2
b2 1
2
Hr/b−1 Hr/b−1 Y Rk ,
2
2.30
where XRk is given by 2.12 and Y Rk is given by 2.13.
Proof. We follow similar steps as the previous corollary so that
1
n≥1 n3
Hn
3
bnk
k
k
r1
1
1
1
Hn
Hn
Hn
3
3
Cr k!
Br k!
Ar k!
.
2
3
3
3
n3 bn r
n≥1
n≥1 n bn r
n≥1 n bn r
3
2.31
12
International Journal of Combinatorics
After much algebraic simplification, the following identity is obtained:
n≥1 n
1
Hn
3 bn
r3
1
Hn
n≥1
3b
b3
3b3
6b2
1
−
−
−
r 3 n3 r 4 n2 r 3 bn r3 r 4 bn r2 r 4 nbn r
1
1
2
ζ4 9bζ3 ζ3Hr/b−1 6b2 ζ2 3bζ2Hr/b−1 ζ2Hr/b−1
3 −
−
−
−
r
r3
r5
r3
r4
r4
2
3b 1
3b2 1
2
2
3
Hr/b−1 Hr/b−1 4 Hr/b−1 Hr/b−1 Hr/b−1
5
r
r
2
1
2
1
3
4
3 Hr/b−1 2Hr/b−1 Hr/b−1 3Hr/b−1 ,
2r
1
1
5ζ4 5bζ3 3b2 ζ2 bζ2Hr/b−1
−
−
2
3
4r 2
r3
r3
r4
n≥1 n bn r
2
b 1
3b2 1
2
2
3
Hr/b−1 Hr/b−1 3 Hr/b−1 Hr/b−1 Hr/b−1 ,
4
r
2r
2
Hn1
b2 1
5ζ4 b2 ζ2 2bζ3
2
H
−
H
r/b−1
r/b−1 .
4r
n3 bn r
r3
r2
2r 3
n≥1
Hn
2.32
Now we can substitute 2.32 into 2.31, collecting zeta functions and using 2.12 and 2.13
for XRk and Y Rk, respectively, the identity 2.30 is obtained.
Some specific examples of Corollary 2.7 are as follows.
For k 1 and b 1 the following identity is valid,
1
Hn
1
ζ4 − 9ζ3 ζ2,
3
6
n≥1 n n 1
3
1
440380018688
Hn
12729073000 2641017400832
−
ln 2 ln 22
4n8 3 3
27
11025
3675
n≥1 n
8
−
16604790784
1576747008
G 11272192G2 G ln 2
315
35
−
1390679293952π 172233728π 3 203992775989
−
−
ζ2
33075
105
2450
−
4069970159
62246737
ζ3 28974848 ln 2ζ3 −
ζ4,
210
2
2.33
International Journal of Combinatorics
13
where G is Catalan’s constant, defined by
1
G
2
1
Ksds 0
∞
−1r
12
r1 2r
≈ 0.915965 . . . ,
2.34
and Ks is the complete elliptic integral of the first kind. The degenerate case k 0, gives
the well-known result
Hn1
n3
n≥1
5
ζ4.
4
2.35
Remark 2.8. Corollaries 2.3, 2.5, and 2.7 are important and can be evaluated as demonstrated
independently of their integral representations. Similarly the proofs of Corollaries 2.3, 2.5,
and 2.7 are not obvious therefore their explicit representations is desired.
Remark 2.9. Theoretically it should be possible to obtain an integral representation for the
general sum
tn
n≥1
ψ j 1 an − ψ j 1
,
cnl dnm bnk
q anj
np−1
n−1
n
j
k
l
for q 1, 2, 3, . . . ,
2.36
m
with its associated corollaries. This work will be investigated in a forthcoming paper.
References
1 H. Chen, “Evaluations of some Variant Euler Sums,” Journal of Integer Sequences, vol. 9, no. 2, article
06.2.3, p. 9, 2006.
2 K. N. Boyadzhiev, “Harmonic number identities via Euler’s transform,” Journal of Integer Sequences,
vol. 12, no. 6, article 09.6.1, p. 8, 2009.
3 L. Euler, Opera Omnia, Series 1, vol. 15, Teubner, Berlin, Germany, 1917.
4 P. Flajolet and B. Salvy, “Euler sums and contour integral representations,” Experimental Mathematics,
vol. 7, no. 1, pp. 15–35, 1998.
5 A. Basu, “A new method in the study of Euler sums,” Ramanujan Journal, vol. 16, no. 1, pp. 7–24, 2008.
6 J. Sondow and E. W. Weisstein, Harmonic number. From MathWorld-A Wolfram Web Rescources,
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