Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 794181, 11 pages
doi:10.1155/2008/794181
Research Article
Sums of Reciprocals of Triple Binomial Coefficients
A. Sofo
School of Computer Science and Mathematics, Victoria University, P.O. Box 14428, Melbourne,
VIC 8001, Australia
Correspondence should be addressed to A. Sofo, anthony.sofo@vu.edu.au
Received 28 August 2007; Accepted 17 December 2007
Recommended by George Andrews
We investigate the integral representation of infinite sums involving the reciprocals of triple binomial coefficients. We also recover some wellknown properties of ζ3 and extend the range of results
given by other authors.
Copyright q 2008 A. 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.
1. Introduction
In this paper, we investigate the summation of the reciprocal of triple products of combinatorial
coefficients. In particular, we develop integral representations for
∞
n0
anj
j
1
bnk
k
cnl
l
,
∞
n0
anj
j
nm−1
n
bnk
k
cnl
l
,
1.1
and their alternating series counterparts.
For the representation of sums of reciprocals of single and double binomial coefficients,
one may refer to some results in the papers 1–3, see also 4.
For designated cases of the parameter values a, b, c, j, k, l, m, various particular sums
may be expressed in terms of ζ2 and ζ3. For many interesting properties of the Zeta function, the reader is referred to 5.
The representation of sums in terms of integrals is extremely useful because it allows
one to estimate bounds on the sums in cases they cannot be written in closed form. Convexity
properties for sums may also be investigated.
Apéry’s 6, see also Beukers 7, proof of the irrationality of ζ3 uses an elementary
and quite complicated construction of the approximants αn /βn ∈ Q to this number based on
2
International Journal of Mathematics and Mathematical Sciences
a recurrence relation. The integral representation
1 0
x1 − xy1 − yz1 − z
n1
1 − 1 − xyz
n
dx dy dz 2βn ζ3 − 2αn
1.2
for the sequence {αn , βn } was proposed.
It is important to note that other integral representations of ζ3 are available in terms of
both single and double integrals. Guillera and Sondow 8 list a number of them including the
classical results
1
−lnxy
dx dy 2ζ3,
0 1 − xy
1.3
1
ln2 − xy
5
dx dy ζ3.
1
−
xy
8
0
In a recent paper, Muzaffar 9 also obtained some results of the combinatorial type
∞
n0
2n1
1
2n
n
2nk1
1
2n2k
nk
αk π 2 βk
1.4
q
by utilising the power series expansion of sin−1 x , and αk , βk are constants depending on
k ≥ 0. In this paper, we complement and extend some of the results given by Muzaffar.
There are some identities in the literature involving reciprocals of triple products of combinatorial coefficients, one prominentidentity is the Dougall identity, see 10 or 11,
n a
c
abc
b
∞ −1
n
n
n
b
12 n1
na
a
nc
c
nb
b
ab
b
bc
b
1.5
for Ra b c > −1.
2. The main results
In this section, we develop integral identities for reciprocals of triple products of binomial
coefficients.
Theorem 2.1. For a, b, and c positive real numbers and j, k, l ≥ 0, then
Sa, b, c, j, k, l ∞
n0
jkl
anj
j
1
1
bnk
k
cnl
l
1 − xj−1 1 − yk−1 1 − zl−1
dx dy dz
1 − XY Z
0
2.1
2.2
A. Sofo
3
1 abc
1
1 − xj 1 − yk 1 − zl XY Z XY Z2 4XY Z 1 dx dy dz
xyz1 − XY Z4
0
2.3
⎤
⎢
⎥
⎢
⎥
⎢
⎥
jkl1 Fjkl ⎢
1⎥ ,
⎢
⎥
⎣a1 a2
aj b1 b2
bk c1 c2
c l ⎦
,
,...,
,
,
,...,
,
,
,...,
a
a
a
b
b
b
c
c
c
2.4
⎡
j 1 2
1 2
k 1 2
l
1, , , . . . , , , , . . . , , , , . . . ,
a a
a b b
b c c
c
and similarly
T a, b, c, j, k, l
∞
n0
jkl
anj
j
−1n
bnk
k
cnl
l
2.5
1
1 − abc
1 − xj−1 1 − yk−1 1 − zl−1
dx dy dz
1 XY Z
0
1
2.6
1 − xj 1 − yk 1 − zl XY Z XY Z2 − 4XY Z 1 dx dy dz
xyz1 XY Z4
0
2.7
⎤
⎥
⎢
⎥
⎢
− 1⎥ ,
jkl1 Fjkl ⎢
⎥
⎢
⎦
⎣a1 a2
aj b1 b2
bk c1 c2
c l ,
,...,
,
,
,...,
,
,
,...,
a
a
a
b
b
b
c
c
c
2.8
⎡
j 1 2
1 2
k 1 2
l
1, , , . . . , , , , . . . , , , , . . . ,
a a
a b b
b c c
c
where
XY Z xa y b zc .
2.9
Proof. Consider 2.1:
Sa, b, c, j, k, l ∞
n0
jkl
anj
j
1
bnk
k
cnl
l
∞
Γan 1ΓjΓbn 1ΓkΓcn 1Γl
n0
jkl
∞
n0
Γan j 1Γbn k 1Γcn l 1
Ban, j 1Bbn, k 1Bcn, l 1,
2.10
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International Journal of Mathematics and Mathematical Sciences
where Γ· is the classical Gamma function and B·, · is the Beta function. It holds that
Sa, b, c, j, k, l jkl
∞ 1
1 − x
y0
1 − y
k−1 bn
1
y dy
y0
1 1 1
x0
1
x dx
x0
n0
jkl
j−1 an
1 − xj−1 1 − yk−1 1 − zl−1
z0
1 − zl−1 zcn dz
z0
∞
x a y b zc
n
dx dy dz
n0
2.11
by an allowable change of integral and sum, and hence we have
Sa, b, c, j, k, l jkl
1
1 − xj−1 1 − yk−1 1 − zl−1
dx dy dz,
1 − XY Z
0
2.12
which is the result 2.2.
To prove identity 2.3, consider 2.1 and expand as follows:
Sa, b, c, j, k, l
∞
abcn3 ΓanΓj 1ΓbnΓk 1ΓcnΓl 1
n0
∞
Γan j 1Γbn k 1Γcn l 1
abcn3 Bj 1, anBk 1, bnBk 1, cn
n0
1 abc
1 abc
1
∞
n
1 − xj 1 − yk 1 − zl n3 xa y b zc dx dy dz
xyz
0
n1
1
0
1 − xj 1 − yk 1 − zl
xyz1 − XY Z
4
XY Z XY Z2 4XY Z 1 dx dy dz
2.13
which is the result 2.3. The results 2.6 and 2.7 may be obtained in a similar fashion and
therefore will not be pursued here.
The hypergeometric representation 2.4 and 2.8 can be obtained by the consideration
of the ratio of successive terms 2.1 and 2.5, respectively.
We may also note that from known properties of the hypergeometric function, we may
write, from 2.4 and 2.8,
⎤
⎢
⎥
⎢
⎥
1⎥
jkl1 Fjkl ⎢
⎣a1 a2
aj b1 b2
bk c1 c2
c l ⎦
,
,...,
,
,
,...,
,
,
,...,
a
a
a
b
b
b
c
c
c
⎡
j 1 2
1 2
k 1 2
l
1, , , . . . , , , , . . . , , , , . . . ,
a a
a b b
b c c
c
A. Sofo
5
⎤
⎢
⎥
⎢
⎥
abc1 Fabc ⎢
1⎥ ,
⎣j 1 j 2
j a k1 k2
kb l1 l2
l c ⎦
,
,...,
,
,
,...,
,
,
,...,
a
a
a
b
b
b
c
c
c
⎡
1 2
a−1 1 2
b−1 1 2
c−1
1, 1, 1, 1, , , . . . ,
, , ,...,
, , ,...,
a a
a b b
b c c
c
⎤
⎥
⎢
⎥
⎢
− 1⎥
jkl1 Fjkl ⎢
⎦
⎣a1 a2
aj b1 b2
bk c1 c2
c l
,
,...,
,
,
,...,
,
,
,...,
a
a
a
b
b
b
c
c
c
⎡
j 1 2
k 1 2
l
1 2
1, , , . . . , , , , . . . , , , , . . . ,
a a
a b b
b c c
c
⎤
1 2
a−1 1 2
b−1 1 2
c−1
1, 1, 1, 1, , , . . . ,
, , ,...,
, , ,...,
⎢
⎥
a a
a b b
b c c
c
− 1⎥
abc1 Fabc ⎢
⎣j 1 j 2
⎦
j a k1 k2
kb l1 l2
l c
,
,...,
,
,
,...,
,
,
,...,
a
a
a
b
b
b
c
c
c
2.14
⎡
3. Examples
Example 3.1. It holds that
S1, 1, 1, 1, 1, 1 ∞
1
n0 n
13
1
1
ζ3 dx dy dz
1, 1, 1, 1
4 F3
1
2, 2, 2 0 1 − xyz
1
1 − x1 − y1 − z xyz2 4xyz 1
1 − xyz4
0
3.1
dx dy dz.
Other integral representations of ζ3 do exist, some of which are as follows.
Finch 12 gave the expression
n
n
1 1 ln x
ln 1 − x
−1n n!ζn 1 dx dx.
1−x
x
x0
x0
3.2
Lord 13 posed the problem to show that
S2, 2, 2, 1, 1, 1
∞
1
7
ζ3 3
8
n0 2n 1
18
π/4
x0
1
ln cosx ln sinx
dx dy dz
dx 2 2 2
cosxsinx
1
0 −x y z
1 − x1 − y1 − zxyz xyz4 4xyz2 1
dx dy dz
4
0
1 − x 2 y 2 z2
1
⎡1 1 1 ⎤
, , , 1
⎢2 2 2 ⎥
⎥
4 F3 ⎢
⎣ 3 3 3 1⎦ ,
, , 2 2 2
the last three expressions are directly from 2.2, 2.3, and 2.4, respectively.
3.3
6
International Journal of Mathematics and Mathematical Sciences
Nan-Yue and Williams 14 also gave
ζ3 −5
lnφ2 x0
where φ golden ratio 1 x
dx,
x ln 2 sinh
2
3.4
√
5/2.
Example 3.2. It holds that
S4, 2, 3, j, k, l ∞
n0
∞
n0
jkl
4nj
j
j
r1 4n
1
2nk
k
k
r
3nl
l
j!k!l!
r1 2n
l
r
r1 3n
r
1
1 − xj−1 1 − yk−1 1 − zl−1 dx dy dz
1 − x 4 y 2 z3
0
3.5
⎤
⎢
⎥
⎢
⎥
10 F9 ⎢
1⎥
⎣ j 1 j 2 j 3 j 4 k 1 k 2 l 1 l 2 l 3 ⎦
,
,
,
,
,
,
,
4
4
4
4
2
2
3
3
3
⎡
3 2 1 1 1 1
1, 1, 1, 1, , , , , ,
4 3 2 2 3 4
α1 α2 π α3 ζ2 α4 ln2 α5 ln3 α6 ζ3.
For j 5, k 6, and l 6, we have the values
495762799093
α1 ,
17 · 11 · 72 · 32 · 24
α3 1709 · 5 · 2
,
7
α2 α4 −
√
167 · 225
23 · 5 · 316 3
,
−
17 · 13 · 11 · 72 · 32 17 · 13 · 11 · 72 · 24
755357 · 219
,
17 · 13 · 11 · 72 · 32
α5 43 · 23 · 5 · 316
,
17 · 13 · 11 · 72 · 24
α6 53 · 3 · 22 .
3.6
Example 3.3. For the alternating case,
T 1, 1, 1, 1, 1, 1 ∞
−1n
3
ζ3
4
n0 n 1
3
dx dy dz
1, 1, 1, 1
4 F3
−1
2, 2, 2 0 1 xyz
1
1−
1
0
1 − x1 − y1 − z xyz2 − 4xyz 1
1 xyz4
3.7
dx dy dz.
A. Sofo
7
Example 3.4. It holds that
T 2/3, 2/3, 5/6, 2, 4, 3
∞
n0
2n/32
2
−1n
2n/34
4
5n/63
3
∞
−1n 25 311
n0 n
24
32 n 62n 32 2n 95n 65n 125n 18
1
1 − x1 − y3 1 − z2 dx dy dz
1 x2/3 y 2/3 z5/6
0
⎤
3 3
9
6 12 18 1,
,
,
3,
3,
,
6,
,
,
⎢ 2 2
2
5 5 5 − 1⎥
⎥
10 F9 ⎢
⎦
⎣5 5
11 11 17 23 , , 4, 4, , , , , 7
2 2
2 5 5 5
⎡
3.8
70663 · 1669
32 · 210
28 · 34
109 · 5 · 3 · 29
ln
2
−
π
33 · 24 ζ2 G
13 · 11 · 7 · 5 · 3 · 2
7
11 · 72
11 · 72
√
√
113 · 56 · 5
113 · 56 · 5
271 · 55
271 · 55
−
lnα −
lnφ
11 · 72 · 22
11 · 72 · 23
11 · 72 · 22
11 · 72 · 23
−
√ √
√
√
4
·
5
·
α
φ 5
37
·
5
5 ·3· 5·φ α 5
π,
2
11 · 2
7
4
√
where G is Catalan’s constant, φ is the golden ratio, and α silver ratio 5 − 1/2.
Now consider the following theorem, which is a generalisation of Theorem 2.1.
Theorem 3.5. For a, b, c, and m positive real numbers and j, k, l ≥ 0 with j k l ≥ m, then
Qa, b, c, j, k, l, m
nm−1
∞
n
n0
anj
j
bnk
k
cnl
l
∞
j!k!l!n 1m−1
m
−
1!an
1j bn 1k cn 1l
n0
jkl
1
1 − xj−1 1 − yk−1 1 − zl−1
dx dy dz
1 − XY Zm
0
3.9
3.10
3.11
8
1 mabc
International Journal of Mathematics and Mathematical Sciences
1
1 − xj 1 − yk 1 − zl XY Z XY Z2 3m 1XY Z 1 dx dy dz
xyz1 − XY Zm3
0
3.12
⎤
⎢
⎥
⎢
⎥
⎢
1⎥ ,
jkl1 Fjkl ⎢
⎥
⎣a1 a2
aj b1 b2
bk c1 c2
c l ⎦
,
,...,
,
,
,...,
,
,
,...,
a
a
a
b
b
b
c
c
c
⎡
j 1 2
k 1 2
l
1 2
m, , , . . . , , , , . . . , , , , . . . ,
a a
a b b
b c c
c
Ra, b, c, j, k, l, m
∞
−1n nm−1
n
n0
anj
j
bnk
k
3.14
cnl
l
∞
−1n j!k!l!n 1m−1
m − 1!an 1j bn 1k cn 1l
n0
jkl
3.15
1
1 − xj−1 1 − yk−1 1 − zl−1
dx dy dz
1 XY Zm
0
1 − mabc
1
3.13
3.16
1 − xj 1 − yk 1 − zl XY Z XY Z2 − 3m 1XY Z 1 dx dy dz
xyz1 XY Zm3
0
3.17
⎤
⎥
⎢
⎥
⎢
jkl1 Fjkl ⎢
− 1⎥ ,
⎦
⎣a1 a2
aj b1 b2
bk c1 c2
c l
,
,...,
,
,
,...,
,
,
,...,
a
a
a
b
b
b
c
c
c
3.18
⎡
j 1 2
1 2
k 1 2
l
m, , , . . . , , , , . . . , , , , . . . ,
a a
a b b
b c c
c
where XY Z is given by 2.9 and
pα pp 1 · · · p α − 1 is Pochhammer’s symbol.
Γp α
Γp
3.19
A. Sofo
9
Proof. Consider 3.14:
Ra, b, c, j, k, l, m
∞
n0
−1n
anj
j
∞
−1n
nm−1
n
bnk
k
n0
cnl
l
nm−1
n
∞
−1n abcn3
n0
abcn3 ΓanΓj 1ΓbnΓk 1ΓcnΓl 1
Γan j 1Γbn k 1Γcn l 1
nm−1
Ban, j 1Bbn, k 1Bcn, l 1
n
1
1
1
nm−1
1 − xj−1 xan dx 1 − yk−1 y bn dy 1 − zl−1 zcn dz.
1 abc −1 n
n
0
0
0
n1
∞
n 3
3.20
By an allowable change of integral and sum, we have
Ra, b, c, j, k, l, m
1 abc
1
∞
1 − xj 1 − yk 1 − zl −1n
xyz
0
n1
1 − mabc
1
1 − xj 1 − yk 1 − zl
n
nm−1
n3 xa y b zc dx dy dz
m−1
XY Z XY Z2 − 3m 1XY Z 1
0
xyz1 XY Zm3
dx dy dz
3.21
which is the result 3.17.
To arrive at the result 3.16, consider
Ra, b, j, c, k, l, m
∞
jkl −1n
n0
∞
jkl −1n
n0
∞
jkl −1n
n0
jkl
1
0
n m − 1 Γan 1ΓjΓbn 1ΓkΓcn 1Γl
n
Γan j 1Γbn k 1Γcn l 1
nm−1
Ban 1, jBbn 1, kBcn 1, l
n
1
1
1
nm−1
1 − xj−1 xan dx 1 − yk−1 y bn dy 1 − zl−1 zcn dz
n
0
0
0
1 − x
j−1
1 − y
k−1
1 − z
l−1
∞
−1n
n0
nm−1
n
x a y b zc
n
dx dy dz
3.22
10
International Journal of Mathematics and Mathematical Sciences
by an allowable change of sum and integral, hence
1
1 − xj−1 1 − yk−1 1 − zl−1
Ra, b, j, c, k, l, m jkl
dx dy dz
1 XY Zm
0
3.23
which is the result 3.16.
The hypergeometric representations 3.13 and 3.18 can be obtained by the consideration of the ratio of successive terms 3.9 and 3.14, respectively.
In the case when m 1, Theorem 3.5 reduces to Theorem 2.1.
4. Examples
Example 4.1. It holds that
Q4, 3, 2, 5, 3, 6, 11
n10
∞
n
n0
90
3n3
3
4n5
5
2n6
6
1
1 − x4 1 − y2 1 − z5
dx dy dz
11
0
1 − x 4 y 3 z2
1 264
1 − x5 1 − y3 1 − z6 x3 y 2 z x8 y 6 z4 34x4 y 3 z2 1 dx dy dz
14
0
1 − x 4 y 3 z2
1
3 2 1 1 1 1 ⎤
11, 1, 1, 1, , , , , , ⎢
4 3 2 2 3 4 ⎥
⎥
10 F9 ⎢
1
⎣
7 9
5 7 3 4 ⎦
4, , , 2, 2, , , ,
2 4
3 4 2 3
⎡
3413 · 43 · 5 · 3
ζ2 214
−
√ 1459 · 1231
1931 · 509 90379 3
π−
−
7 · 3 · 25
212
7 · 32 · 211
379880779
22567 · 32
ln 2 ln 3.
10
7 · 25
7·3·2
Example 4.2. It holds that
∞
R 1/4, 1/6, 1/2, 5, 3, 7, 14 :
n0
−1n
n/45
5
n13
n
n/63
3
n/27
7
∞
−1n 5!3!7!n 113
n0 13! n/4 1 5 n/6 1 3 n/2 1 7
105
1
1 − x4 1 − y2 1 − z6
14 dx dy dz
0
1 x1/4 y 1/6 z1/2
4.1
A. Sofo
11
1
7
24
1
16 F15
1 − x5 1 − y3 1 − z7
0
x1/2 y 1/3 z 43x1/4 y 1/6 z1/2 1
17 dx dy dz
x3/4 y 5/6 z1/2 1 − x1/4 y 1/6 z1/2
2, 4, 4, 6, 6, 8, 8, 10, 12, 12, 12, 14, 14, 16, 18, 20
−1
3, 5, 5, 7, 7, 9, 9, 11, 13, 13, 13, 15, 17, 19, 21 7 · 5 · 33 · 24
16231 · 11 · 3
.
ζ2 −
13
7 · 5 · 23
4.2
5. Conclusion
We have provided triple integral identities for sums of the reciprocal of triple binomial coefficients. In doing so, we have recovered the standard representation for ζ3 and have generalised and extended some results published previously by other authors.
In another forum, we will extend our results to consider binomial coefficients of the form
nm−1
∞
∞
ns nm−1
n
n
5.1
,
.
n0
anj
j
bnk
k
cnl
l
n0
anj
bn
cnk
dn
pnj
qn
Acknowledgment
This paper was completed while the author was a Visiting Professor at the Dipartimento di
Sistemi e Informatica, Universita di Firenze. I wish to express my sincere thanks to Professor
Sprugnoli for his hospitality.
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q
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