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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2008, Article IDΒ 794181, 11 pages
http://dx.doi.org/10.1155/2008/794181
Research Article

Sums of Reciprocals of Triple Binomial Coefficients

School of Computer Science and Mathematics, Victoria University, P.O. Box 14428, Melbourne, VIC 8001, Australia

Received 28 August 2007; Accepted 17 December 2007

Academic Editor: GeorgeΒ Andrews

Copyright Β© 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.

Abstract

We investigate the integral representation of infinite sums involving the reciprocals of triple binomial coefficients. We also recover some wellknown properties of 𝜁(3) and extend the range of results given by other authors.

1. Introduction

In this paper, we investigate the summation of the reciprocal of triple products of combinatorial coefficients. In particular, we develop integral representations forβˆžξ“π‘›=01(π‘—π‘Žπ‘›+𝑗)(π‘˜π‘π‘›+π‘˜)(𝑙𝑐𝑛+𝑙),βˆžξ“π‘›=0(𝑛𝑛+π‘šβˆ’1)(π‘—π‘Žπ‘›+𝑗)(π‘˜π‘π‘›+π‘˜)(𝑙𝑐𝑛+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 (π‘Ž,𝑏,𝑐,𝑗,π‘˜,𝑙,π‘š), various particular sums may be expressed in terms of 𝜁(2) and 𝜁(3). 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.

ApΓ©ry's [6], see also Beukers [7], proof of the irrationality of 𝜁(3) uses an elementary and quite complicated construction of the approximants 𝛼𝑛/π›½π‘›βˆˆπ‘„ to this number based on a recurrence relation. The integral representationξ€ž10π‘₯𝑦1βˆ’π‘₯1βˆ’π‘¦π‘§(1βˆ’π‘§)𝑛1βˆ’(1βˆ’π‘₯𝑦)𝑧𝑛+1𝑑π‘₯𝑑𝑦𝑑𝑧=2π›½π‘›πœ(3)βˆ’2𝛼𝑛(1.2)for the sequence {𝛼𝑛,𝛽𝑛} was proposed.

It is important to note that other integral representations of 𝜁(3) are available in terms of both single and double integrals. Guillera and Sondow [8] list a number of them including the classical results10βˆ’ln(π‘₯𝑦)1βˆ’π‘₯𝑦𝑑π‘₯𝑑𝑦=2𝜁(3),10ln(2βˆ’π‘₯𝑦)51βˆ’π‘₯𝑦𝑑π‘₯𝑑𝑦=8𝜁(3).(1.3)In a recent paper, Muzaffar [9] also obtained some results of the combinatorial typeβˆžξ“π‘›=0(𝑛2𝑛)(12𝑛+1)(12𝑛+π‘˜+1)(2𝑛+2π‘˜π‘›+π‘˜)=π›Όπ‘˜πœ‹2+π›½π‘˜(1.4)by utilising the power series expansion of (sinβˆ’1π‘₯)π‘ž, and (π›Όπ‘˜,π›½π‘˜) are constants depending on π‘˜β‰₯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 prominent identity is the Dougall identity, see [10] or [11],1+2βˆžξ“π‘›=1(βˆ’1)𝑛(π‘Žπ‘›)(𝑏𝑛)(𝑐𝑛)(π‘Žπ‘›+π‘Ž)(𝑏𝑛+𝑏)(𝑐𝑛+𝑐)=(π‘π‘Ž+𝑏+𝑐)(π‘π‘Ž+𝑏)(𝑏𝑏+𝑐)(1.5)for 𝑅(π‘Ž+𝑏+𝑐)>βˆ’1.

2. The Main Results

In this section, we develop integral identities for reciprocals of triple products of binomial coefficients.

Theorem 2.1. For π‘Ž,𝑏, and 𝑐 positive real numbers and 𝑗,π‘˜,𝑙β‰₯0, then π‘‹π‘Œπ‘=π‘₯π‘Žπ‘¦π‘π‘§π‘.(2.9)𝑆=π‘Ž,𝑏,𝑐,𝑗,π‘˜,π‘™βˆžξ“π‘›=01(π‘—π‘Žπ‘›+𝑗)(π‘˜π‘π‘›+π‘˜)(𝑙𝑐𝑛+𝑙)=π‘—π‘˜π‘™βˆžξ“π‘›=0Ξ“ξ€·ξ€ΈΞ“ξ€·π‘—ξ€ΈΞ“ξ€·ξ€ΈΞ“ξ€·π‘˜ξ€ΈΞ“ξ€·ξ€ΈΞ“ξ€·π‘™ξ€Έπ‘Žπ‘›+1𝑏𝑛+1𝑐𝑛+1Ξ“ξ€·ξ€ΈΞ“ξ€·ξ€ΈΞ“ξ€·ξ€Έπ‘Žπ‘›+𝑗+1𝑏𝑛+π‘˜+1𝑐𝑛+𝑙+1=π‘—π‘˜π‘™βˆžξ“π‘›=0𝐡𝐡𝐡,π‘Žπ‘›,𝑗+1𝑏𝑛,π‘˜+1𝑐𝑛,𝑙+1(2.10)Ξ“(β‹…)𝐡(β‹…,β‹…)and similarly π‘†ξ€·ξ€Έπ‘Ž,𝑏,𝑐,𝑗,π‘˜,𝑙=π‘—π‘˜π‘™βˆžξ“π‘›=0ξ€œ1π‘₯=0ξ€·ξ€Έ1βˆ’π‘₯π‘—βˆ’1π‘₯π‘Žπ‘›ξ€œπ‘‘π‘₯1𝑦=0ξ€·ξ€Έ1βˆ’π‘¦π‘˜βˆ’1π‘¦π‘π‘›ξ€œπ‘‘π‘¦1𝑧=0ξ€·ξ€Έ1βˆ’π‘§π‘™βˆ’1π‘§π‘π‘›ξ€œπ‘‘π‘§=π‘—π‘˜π‘™1π‘₯=0ξ€œ1𝑦=0ξ€œ1𝑧=0ξ€·ξ€Έ1βˆ’π‘₯π‘—βˆ’1ξ€·ξ€Έ1βˆ’π‘¦π‘˜βˆ’1ξ€·ξ€Έ1βˆ’π‘§βˆžπ‘™βˆ’1𝑛=0ξ‚€π‘₯π‘Žπ‘¦π‘π‘§π‘ξ‚π‘›π‘‘π‘₯𝑑𝑦𝑑𝑧(2.11)ξ€žπ‘†(π‘Ž,𝑏,𝑐,𝑗,π‘˜,𝑙)=π‘—π‘˜π‘™10ξ€·ξ€Έ1βˆ’π‘₯π‘—βˆ’1ξ€·ξ€Έ1βˆ’π‘¦π‘˜βˆ’1ξ€·ξ€Έ1βˆ’π‘§π‘™βˆ’11βˆ’π‘‹π‘Œπ‘π‘‘π‘₯𝑑𝑦𝑑𝑧,(2.12)=𝑆(π‘Ž,𝑏,𝑐,𝑗,π‘˜,𝑙)βˆžξ“π‘›=0π‘Žπ‘π‘π‘›3Ξ“ξ€·ξ€ΈΞ“ξ€·ξ€ΈΞ“ξ€·ξ€ΈΞ“ξ€·ξ€ΈΞ“ξ€·ξ€ΈΞ“ξ€·ξ€Έπ‘Žπ‘›π‘—+1π‘π‘›π‘˜+1𝑐𝑛𝑙+1Ξ“ξ€·ξ€ΈΞ“ξ€·ξ€ΈΞ“ξ€·ξ€Έ=π‘Žπ‘›+𝑗+1𝑏𝑛+π‘˜+1𝑐𝑛+𝑙+1βˆžξ“π‘›=0π‘Žπ‘π‘π‘›3π΅ξ€·ξ€Έπ΅ξ€·ξ€Έπ΅ξ€·ξ€Έξ€žπ‘—+1,π‘Žπ‘›π‘˜+1,π‘π‘›π‘˜+1,𝑐𝑛=1+π‘Žπ‘π‘10ξ€·ξ€Έ1βˆ’π‘₯𝑗1βˆ’π‘¦π‘˜ξ€·ξ€Έ1βˆ’π‘§π‘™π‘₯π‘¦π‘§βˆžξ“π‘›=1𝑛3ξ‚€π‘₯π‘Žπ‘¦π‘π‘§π‘ξ‚π‘›ξ€žπ‘‘π‘₯𝑑𝑦𝑑𝑧=1+π‘Žπ‘π‘10ξ€·ξ€Έ1βˆ’π‘₯𝑗1βˆ’π‘¦π‘˜ξ€·ξ€Έ1βˆ’π‘§π‘™ξ€·ξ€Έπ‘₯𝑦𝑧1βˆ’π‘‹π‘Œπ‘4ξ‚€ξ€·ξ€Έπ‘‹π‘Œπ‘π‘‹π‘Œπ‘2+4π‘‹π‘Œπ‘+1𝑑π‘₯𝑑𝑦𝑑𝑧(2.13)𝑗+π‘˜+𝑙+1𝐹𝑗+π‘˜+𝑙[11,π‘Ž,2π‘Žπ‘—,…,π‘Ž,1𝑏,2π‘π‘˜,…,𝑏,1𝑐,2𝑐𝑙,…,π‘π‘Ž+1π‘Ž,π‘Ž+2π‘Ž,…,π‘Ž+π‘—π‘Ž,𝑏+1𝑏,𝑏+2𝑏,…,𝑏+π‘˜π‘,𝑐+1𝑐,𝑐+2𝑐,…,𝑐+𝑙𝑐=|1]π‘Ž+𝑏+𝑐+1πΉπ‘Ž+𝑏+𝑐[11,1,1,1,π‘Ž,2π‘Ž,…,π‘Žβˆ’1π‘Ž,1𝑏,2𝑏,…,π‘βˆ’1𝑏,1𝑐,2𝑐,…,π‘βˆ’1𝑐𝑗+1π‘Ž,𝑗+2π‘Ž,…,𝑗+π‘Žπ‘Ž,π‘˜+1𝑏,π‘˜+2𝑏,…,π‘˜+𝑏𝑏,𝑙+1𝑐,𝑙+2𝑐,…,𝑙+𝑐𝑐|1],𝑗+π‘˜+𝑙+1𝐹𝑗+π‘˜+𝑙[11,π‘Ž,2π‘Žπ‘—,…,π‘Ž,1𝑏,2π‘π‘˜,…,𝑏,1𝑐,2𝑐𝑙,…,π‘π‘Ž+1π‘Ž,π‘Ž+2π‘Ž,…,π‘Ž+π‘—π‘Ž,𝑏+1𝑏,𝑏+2𝑏,…,𝑏+π‘˜π‘,𝑐+1𝑐,𝑐+2𝑐,…,𝑐+𝑙𝑐=|βˆ’1]π‘Ž+𝑏+𝑐+1πΉπ‘Ž+𝑏+𝑐[11,1,1,1,π‘Ž,2π‘Ž,…,π‘Žβˆ’1π‘Ž,1𝑏,2𝑏,…,π‘βˆ’1𝑏,1𝑐,2𝑐,…,π‘βˆ’1𝑐𝑗+1π‘Ž,𝑗+2π‘Ž,…,𝑗+π‘Žπ‘Ž,π‘˜+1𝑏,π‘˜+2𝑏,…,π‘˜+𝑏𝑏,𝑙+1𝑐,𝑙+2𝑐,…,𝑙+𝑐𝑐|βˆ’1](2.14)where 𝑆=1,1,1,1,1,1βˆžξ“π‘›=01𝑛+13ξ€·3ξ€Έ=ξ€ž=𝜁10𝑑π‘₯𝑑𝑦𝑑𝑧=1βˆ’π‘₯𝑦𝑧4𝐹3[ξ€ž1,1,1,12,2,2|1]=1+10ξ€·ξ€Έξ‚€1βˆ’π‘₯ξ€Έξ€·1βˆ’π‘¦(1βˆ’π‘§)(π‘₯𝑦𝑧)2+4π‘₯𝑦𝑧+1ξ€·ξ€Έ1βˆ’π‘₯𝑦𝑧4𝑑π‘₯𝑑𝑦𝑑𝑧.(3.1)

Proof. Consider (2.1):𝜁(3)where (βˆ’1)𝑛=ξ€œπ‘›!πœπ‘›+11π‘₯=0π‘₯lnπ‘›ξ€œ1βˆ’π‘₯𝑑π‘₯=1π‘₯=0ln1βˆ’π‘₯𝑛π‘₯𝑑π‘₯.(3.2) is the classical Gamma function and 𝑆=2,2,2,1,1,1βˆžξ“π‘›=01ξ€·ξ€Έ2𝑛+13=78πœξ€·3ξ€Έ=ξ€œπœ‹/4π‘₯=0lncos(π‘₯)lnsin(π‘₯)ξ€žcos(π‘₯)sin(π‘₯)𝑑π‘₯=10𝑑π‘₯𝑑𝑦𝑑𝑧1βˆ’π‘₯2𝑦2𝑧2ξ€ž=1+810ξ€·ξ€Έξ‚€1βˆ’π‘₯ξ€Έξ€·1βˆ’π‘¦(1βˆ’π‘§)π‘₯𝑦𝑧(π‘₯𝑦𝑧)4+4(π‘₯𝑦𝑧)2+1ξ‚€1βˆ’π‘₯2𝑦2𝑧24=𝑑π‘₯𝑑𝑦𝑑𝑧4𝐹3[12,12,123,12,32,32|1],(3.3) is the Beta function. It holds that πœξ€·3ξ€Έξ€œ=βˆ’5ln(πœ™2)π‘₯=0ξ‚€ξ‚€π‘₯π‘₯ln2sinh2𝑑π‘₯,(3.4)by an allowable change of integral and sum, and hence we have πœ™which is the result (2.2).
To prove identity (2.3), consider (2.1) and expand as follows:√(1+5)/2which 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),𝑆=4,2,3,𝑗,π‘˜,π‘™βˆžξ“π‘›=01(𝑗4𝑛+𝑗)(π‘˜2𝑛+π‘˜)(𝑙3𝑛+𝑙)=βˆžξ“π‘›=0𝑗!π‘˜!𝑙!βˆπ‘—π‘Ÿ=1ξ€·ξ€Έβˆ4𝑛+π‘Ÿπ‘˜π‘Ÿ=1ξ€·ξ€Έβˆ2𝑛+π‘Ÿπ‘™π‘Ÿ=1ξ€·ξ€Έξ€ž3𝑛+π‘Ÿ=π‘—π‘˜π‘™10(1βˆ’π‘₯)π‘—βˆ’1(1βˆ’π‘¦)π‘˜βˆ’1(1βˆ’π‘§)π‘™βˆ’1𝑑π‘₯𝑑𝑦𝑑𝑧1βˆ’π‘₯4𝑦2𝑧3=10𝐹9[31,1,1,1,4,23,12,12,13,14𝑗+14,𝑗+24,𝑗+34𝑗+44,π‘˜+12,π‘˜+22,𝑙+13,𝑙+23,𝑙+33|1]=𝛼1+𝛼2πœ‹+𝛼3πœξ€·2ξ€Έ+𝛼4ln(2)+𝛼5ln(3)+𝛼6𝜁(3).(3.5)Examples
Example 3.1. It holds that 𝑗=5,π‘˜=6
Other integral representations of 𝑙=6 do exist, some of which are as follows.
Finch [12] gave the expression𝛼1=49576279909317β‹…11β‹…72β‹…32β‹…24,𝛼2=ξ‚€167β‹…22517β‹…13β‹…11β‹…72β‹…32βˆ’23β‹…5β‹…316√317β‹…13β‹…11β‹…72β‹…24,𝛼3=1709β‹…5β‹…27,𝛼4=βˆ’755357β‹…21917β‹…13β‹…11β‹…72β‹…32,𝛼5=43β‹…23β‹…5β‹…31617β‹…13β‹…11β‹…72β‹…24,𝛼6=53β‹…3β‹…22.(3.6)
Lord [13] posed the problem to show that𝑇=1,1,1,1,1,1βˆžξ“π‘›=0ξ€·ξ€Έβˆ’1𝑛𝑛+13=34=ξ€žπœ(3)10𝑑π‘₯𝑑𝑦𝑑𝑧=1+π‘₯𝑦𝑧4𝐹3[ξ€ž1,1,1,12,2,2|βˆ’1]=1βˆ’10ξ€·ξ€Έξ‚€1βˆ’π‘₯ξ€Έξ€·1βˆ’π‘¦(1βˆ’π‘§)(π‘₯𝑦𝑧)2ξ‚βˆ’4π‘₯𝑦𝑧+1ξ€·ξ€Έ1+π‘₯𝑦𝑧4𝑑π‘₯𝑑𝑦𝑑𝑧.(3.7)the last three expressions are directly from (2.2), (2.3), and (2.4), respectively.
Nan-Yue and Williams [14] also gave𝑇=2/3,2/3,5/6,2,4,3βˆžξ“π‘›=0ξ€·ξ€Έβˆ’1𝑛(22𝑛/3+2)(42𝑛/3+4)(35𝑛/6+3)=βˆžξ“π‘›=0ξ€·ξ€Έβˆ’1𝑛25311𝑛+32ξ€·ξ€Έ(𝑛+6)2𝑛+32ξ€ž(2𝑛+9)(5𝑛+6)(5𝑛+12)(5𝑛+18)=2410ξ€·ξ€Έ1βˆ’π‘₯ξ€Έξ€·1βˆ’π‘¦3ξ€·ξ€Έ1βˆ’π‘§2𝑑π‘₯𝑑𝑦𝑑𝑧1+π‘₯2/3𝑦2/3𝑧5/6=10𝐹9[31,2,329,3,3,26,6,5,125,18552,52,4,4,112,115,175,235=,7|βˆ’1]70663β‹…166913β‹…11β‹…7β‹…5β‹…3β‹…2+33β‹…243𝜁(2)+2β‹…2107𝐺+109β‹…5β‹…3β‹…2911β‹…722ln2βˆ’8β‹…3411β‹…72πœ‹+ξ‚€113β‹…56β‹…βˆš511β‹…72β‹…22βˆ’271β‹…5511β‹…72β‹…23ξ‚ξ€·π›Όξ€Έβˆ’ξ‚€ln113β‹…56β‹…βˆš511β‹…72β‹…22+271β‹…5511β‹…72β‹…23ξ‚ξ€·πœ™ξ€Έβˆ’ξ‚€5ln4βˆšβ‹…3⋅5β‹…πœ™π›Όβˆš5+11β‹…237β‹…54β‹…βˆšξ”5β‹…π›Όπœ™βˆš572ξ‚πœ‹,(3.8)where 𝐺 = golden ratio = πœ™.

Example 3.2. It holds that 𝛼For (√5βˆ’1)/2., and π‘Ž,𝑏,𝑐, we have the values π‘š
Example 3.3. For the alternating case,𝑗,π‘˜,𝑙β‰₯0
Example 3.4. It holds that 𝑗+π‘˜+𝑙β‰₯π‘š,where π‘‹π‘Œπ‘ is Catalan's constant, 𝑝𝛼⋯=Ξ“ξ€·ξ€Έ=𝑝𝑝+1𝑝+π›Όβˆ’1𝑝+𝛼Γ𝑝(3.19) is the golden ratio, and 𝑅=π‘Ž,𝑏,𝑐,𝑗,π‘˜,𝑙,π‘šβˆžξ“π‘›=0ξ€·ξ€Έβˆ’1𝑛(𝑛𝑛+π‘šβˆ’1)(π‘—π‘Žπ‘›+𝑗)(π‘˜π‘π‘›+π‘˜)(𝑙𝑐𝑛+𝑙)=βˆžξ“π‘›=0ξ€·ξ€Έβˆ’1𝑛𝑛ξƒͺ𝑛+π‘šβˆ’1π‘Žπ‘π‘π‘›3Ξ“ξ€·ξ€ΈΞ“ξ€·ξ€ΈΞ“ξ€·ξ€ΈΞ“ξ€·ξ€ΈΞ“ξ€·ξ€ΈΞ“ξ€·ξ€Έπ‘Žπ‘›π‘—+1π‘π‘›π‘˜+1𝑐𝑛𝑙+1Ξ“ξ€·ξ€ΈΞ“ξ€·ξ€ΈΞ“ξ€·ξ€Έ=π‘Žπ‘›+𝑗+1𝑏𝑛+π‘˜+1𝑐𝑛+𝑙+1βˆžξ“π‘›=0ξ€·ξ€Έβˆ’1π‘›π‘Žπ‘π‘π‘›3𝑛ξƒͺ𝐡𝐡𝐡𝑛+π‘šβˆ’1π‘Žπ‘›,𝑗+1𝑏𝑛,π‘˜+1𝑐𝑛,𝑙+1=1+π‘Žπ‘π‘βˆžξ“π‘›=1ξ€·ξ€Έβˆ’1𝑛𝑛3𝑛ξƒͺξ€œπ‘›+π‘šβˆ’110ξ€·ξ€Έ1βˆ’π‘₯π‘—βˆ’1π‘₯π‘Žπ‘›ξ€œπ‘‘π‘₯10ξ€·ξ€Έ1βˆ’π‘¦π‘˜βˆ’1π‘¦π‘π‘›ξ€œπ‘‘π‘¦10ξ€·ξ€Έ1βˆ’π‘§π‘™βˆ’1𝑧𝑐𝑛𝑑𝑧.(3.20) = silver ratio = π‘…ξ€·ξ€Έξ€žπ‘Ž,𝑏,𝑐,𝑗,π‘˜,𝑙,π‘š=1+π‘Žπ‘π‘10ξ€·ξ€Έ1βˆ’π‘₯𝑗1βˆ’π‘¦π‘˜ξ€·ξ€Έ1βˆ’π‘§π‘™π‘₯π‘¦π‘§βˆžξ“π‘›=1(βˆ’1)𝑛ξƒͺ𝑛𝑛+π‘šβˆ’1π‘šβˆ’13ξ‚€π‘₯π‘Žπ‘¦π‘π‘§π‘ξ‚π‘›ξ€žπ‘‘π‘₯𝑑𝑦𝑑𝑧=1βˆ’π‘šπ‘Žπ‘π‘10ξ€·ξ€Έ1βˆ’π‘₯𝑗1βˆ’π‘¦π‘˜ξ€·ξ€Έ1βˆ’π‘§π‘™ξ‚€ξ€·ξ€Έπ‘‹π‘Œπ‘π‘‹π‘Œπ‘2ξ‚βˆ’(3π‘š+1)π‘‹π‘Œπ‘+1ξ€·ξ€Έπ‘₯𝑦𝑧1+π‘‹π‘Œπ‘π‘š+3𝑑π‘₯𝑑𝑦𝑑𝑧(3.21)

Now consider the following theorem, which is a generalisation of Theorem 2.1.

Theorem 3.5. For π‘…ξ€·ξ€Έπ‘Ž,𝑏,𝑗,𝑐,π‘˜,𝑙,π‘š=π‘—π‘˜π‘™βˆžξ“π‘›=0ξ€·ξ€Έβˆ’1𝑛𝑛ξƒͺΞ“ξ€·ξ€ΈΞ“ξ€·π‘—ξ€ΈΞ“ξ€·ξ€ΈΞ“ξ€·π‘˜ξ€ΈΞ“ξ€·ξ€ΈΞ“ξ€·π‘™ξ€Έπ‘›+π‘šβˆ’1π‘Žπ‘›+1𝑏𝑛+1𝑐𝑛+1Ξ“ξ€·ξ€ΈΞ“ξ€·ξ€ΈΞ“ξ€·ξ€Έπ‘Žπ‘›+𝑗+1𝑏𝑛+π‘˜+1𝑐𝑛+𝑙+1=π‘—π‘˜π‘™βˆžξ“π‘›=0ξ€·ξ€Έβˆ’1𝑛𝑛ξƒͺ𝐡𝐡𝐡𝑛+π‘šβˆ’1π‘Žπ‘›+1,𝑗𝑏𝑛+1,π‘˜π‘π‘›+1,𝑙=π‘—π‘˜π‘™βˆžξ“π‘›=0(βˆ’1)𝑛𝑛ξƒͺξ€œπ‘›+π‘šβˆ’110ξ€·ξ€Έ1βˆ’π‘₯π‘—βˆ’1π‘₯π‘Žπ‘›ξ€œπ‘‘π‘₯10ξ€·ξ€Έ1βˆ’π‘¦π‘˜βˆ’1π‘¦π‘π‘›ξ€œπ‘‘π‘¦10ξ€·ξ€Έ1βˆ’π‘§π‘™βˆ’1π‘§π‘π‘›ξ€žπ‘‘π‘§=π‘—π‘˜π‘™10ξ€·ξ€Έ1βˆ’π‘₯π‘—βˆ’1ξ€·ξ€Έ1βˆ’π‘¦π‘˜βˆ’1ξ€·ξ€Έ1βˆ’π‘§βˆžπ‘™βˆ’1𝑛=0(βˆ’1)𝑛𝑛ξƒͺξ‚€π‘₯𝑛+π‘šβˆ’1π‘Žπ‘¦π‘π‘§π‘ξ‚π‘›π‘‘π‘₯𝑑𝑦𝑑𝑧(3.22), and π‘…ξ€·ξ€Έξ€žπ‘Ž,𝑏,𝑗,𝑐,π‘˜,𝑙,π‘š=π‘—π‘˜π‘™10ξ€·ξ€Έ1βˆ’π‘₯π‘—βˆ’1ξ€·ξ€Έ1βˆ’π‘¦π‘˜βˆ’1ξ€·ξ€Έ1βˆ’π‘§π‘™βˆ’1ξ€·ξ€Έ1+π‘‹π‘Œπ‘π‘šπ‘‘π‘₯𝑑𝑦𝑑𝑧(3.23) positive real numbers and π‘š=1, with 𝑄=4,3,2,5,3,6,11βˆžξ“π‘›=0(𝑛𝑛+10)(54𝑛+5)(33𝑛+3)(62𝑛+6)ξ€ž=9010ξ€·ξ€Έ1βˆ’π‘₯4ξ€·ξ€Έ1βˆ’π‘¦2ξ€·ξ€Έ1βˆ’π‘§5ξ‚€1βˆ’π‘₯4𝑦3𝑧211ξ€žπ‘‘π‘₯𝑑𝑦𝑑𝑧=1+26410ξ€·ξ€Έ1βˆ’π‘₯5ξ€·ξ€Έ1βˆ’π‘¦3ξ€·ξ€Έ1βˆ’π‘§6π‘₯3𝑦2𝑧π‘₯8𝑦6𝑧4+34π‘₯4𝑦3𝑧2+1𝑑π‘₯𝑑𝑦𝑑𝑧1βˆ’π‘₯4𝑦3𝑧214=10𝐹9[311,1,1,1,4,23,12,12,13,1474,2,945,2,2,3,74,32,43=|1]3413β‹…43β‹…5β‹…3214πœξ€·2ξ€Έ+ξ‚€1931β‹…509212βˆ’βˆš9037937β‹…3β‹…25ξ‚πœ‹βˆ’1459β‹…12317β‹…32β‹…211βˆ’3798807797β‹…3β‹…210ln2+22567β‹…327β‹…25ln3.(4.1) then 𝑅1/4,1/6,1/2,5,3,7,14∢=βˆžξ“π‘›=0ξ€·ξ€Έβˆ’1𝑛(𝑛𝑛+13)(5𝑛/4+5)(3𝑛/6+3)(7𝑛/2+7)=βˆžξ“π‘›=0(βˆ’1)𝑛5!3!7!𝑛+11313!𝑛/4+15𝑛/6+13𝑛/2+17ξ€ž=10510ξ€·ξ€Έ1βˆ’π‘₯4ξ€·ξ€Έ1βˆ’π‘¦2ξ€·ξ€Έ1βˆ’π‘§6ξ‚€1+π‘₯1/4𝑦1/6𝑧1/2147𝑑π‘₯𝑑𝑦𝑑𝑧=1+ξ€ž2410ξ€·ξ€Έ1βˆ’π‘₯5ξ€·ξ€Έ1βˆ’π‘¦3ξ€·ξ€Έ1βˆ’π‘§7Γ—ξ‚€π‘₯1/2𝑦1/3𝑧+43π‘₯1/4𝑦1/6𝑧1/2+1π‘₯3/4𝑦5/6𝑧1/2ξ‚€1βˆ’π‘₯1/4𝑦1/6𝑧1/217=𝑑π‘₯𝑑𝑦𝑑𝑧16𝐹15[=2,4,4,6,6,8,8,10,12,12,12,14,14,16,18,203,5,5,7,7,9,9,11,13,13,13,15,17,19,21|βˆ’1]7β‹…5β‹…33β‹…24πœξ€·2ξ€Έβˆ’1316231β‹…11β‹…37β‹…5β‹…23.(4.2)𝜁(3)βˆžξ“π‘›=0𝑛𝑠(𝑛𝑛+π‘šβˆ’1)(π‘—π‘Žπ‘›+𝑗)(π‘˜π‘π‘›+π‘˜)(𝑙𝑐𝑛+𝑙),βˆžξ“π‘›=0(𝑛𝑛+π‘šβˆ’1)(π‘Žπ‘›+𝑗𝑏𝑛)(𝑐𝑛+π‘˜π‘‘π‘›)(𝑝𝑛+π‘—π‘žπ‘›).(5.1)𝜁𝜁𝜁𝜁(sinβˆ’1π‘₯)π‘žlog(2sinh(πœƒ/2))log(2sin(πœƒ/2))where is given by (2.9) and is Pochhammer's symbol.

Proof. Consider (3.14):By an allowable change of integral and sum, we havewhich is the result (3.17).
To arrive at the result (3.16), considerby an allowable change of sum and integral, hencewhich 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 Theorem 3.5 reduces to Theorem 2.1.

Examples
Example 4.1. It holds that
Example 4.2. It holds that

3. 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 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

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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