Sums of Reciprocals of Triple Binomial Coefficients

Sums of Reciprocals of Triple Binomial Coefficients Sums of Reciprocals of Triple Binomial Coefficients //// Hindawi Publishing Corporation Home Journals About Us About this Journal Submit a Manuscript Table of Contents Journal Menu Abstracting and Indexing Aims and Scope Annual Issues Article Processing Charges Articles in Press Author Guidelines Bibliographic Information Contact Information Editorial Board Editorial Workflow Free eTOC Alerts Reviewers Acknowledgment Subscription Information Open Focus Issues Published Focus Issues Focus Issue Guidelines Open Special Issues Published Special Issues Special Issue Guidelines Abstract Full-Text PDF Full-Text HTML Linked References How to Cite this Article International Journal of Mathematics and Mathematical Sciences Volume 2008 (2008), Article ID 794181, 11 pages doi:10.1155/2008/794181 Research Article <h2>Sums of Reciprocals of Triple Binomial Coefficients</h2> A. Sofo 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: George Andrews Copyright © 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 for ∞  𝑛 = 0 1 ( 𝑗 π‘Ž 𝑛 + 𝑗 ) ( π‘˜ 𝑏 𝑛 + π‘˜ ) ( 𝑙 𝑐 𝑛 + 𝑙 ) , ∞  𝑛 = 0 ( 𝑛 𝑛 + π‘š − 1 ) ( 𝑗 π‘Ž 𝑛 + 𝑗 ) ( π‘˜ 𝑏 𝑛 + π‘˜ ) ( 𝑙 𝑐 𝑛 + 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. Apéry's [ 6 ], see also Beukers [ 7 ], proof of the irrationality of 𝜁 ( 3 ) uses an elementary and quite complicated construction of the approximants 𝛼 𝑛 / 𝛽 𝑛 ∈ 𝑄 to this number based on a recurrence relation. The integral representation ξ€ž 1 0  π‘₯ ξ€· ξ€Έ 𝑦 ξ€· ξ€Έ  1 − π‘₯ 1 − 𝑦 𝑧 ( 1 − 𝑧 ) 𝑛 ξ‚€  1 − ( 1 − π‘₯ 𝑦 ) 𝑧 𝑛 + 1 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 = 2 𝛽 𝑛 𝜁 ( 3 ) − 2 𝛼 𝑛 ( 1 . 2 ) for the sequence { 𝛼 𝑛 , 𝛽 𝑛 } 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  1 0 − l n ( π‘₯ 𝑦 )  1 − π‘₯ 𝑦 𝑑 π‘₯ 𝑑 𝑦 = 2 𝜁 ( 3 ) , 1 0 l n ( 2 − π‘₯ 𝑦 ) 5 1 − π‘₯ 𝑦 𝑑 π‘₯ 𝑑 𝑦 = 8 𝜁 ( 3 ) . ( 1 . 3 ) In a recent paper, Muzaffar [ 9 ] also obtained some results of the combinatorial type ∞  𝑛 = 0 ( 𝑛 2 𝑛 ) ( 1 2 𝑛 + 1 ) ( 1 2 𝑛 + π‘˜ + 1 ) ( 2 𝑛 + 2 π‘˜ 𝑛 + π‘˜ ) = 𝛼 π‘˜ πœ‹ 2 + 𝛽 π‘˜ ( 1 . 4 ) by utilising the power series expansion of ( s i n − 1 π‘₯ ) π‘ž , and ( 𝛼 π‘˜ , 𝛽 π‘˜ ) are constants depending on π‘˜ ≥ 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 + 2 ∞  𝑛 = 1 ( − 1 ) 𝑛 ( π‘Ž 𝑛 ) ( 𝑏 𝑛 ) ( 𝑐 𝑛 ) ( π‘Ž 𝑛 + π‘Ž ) ( 𝑏 𝑛 + 𝑏 ) ( 𝑐 𝑛 + 𝑐 ) = ( 𝑏 π‘Ž + 𝑏 + 𝑐 ) ( 𝑏 π‘Ž + 𝑏 ) ( 𝑏 𝑏 + 𝑐 ) ( 1 . 5 ) for 𝑅 ( π‘Ž + 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 π‘Ž , 𝑏 , and 𝑐 positive real numbers and 𝑗 , π‘˜ , 𝑙 ≥ 0 , then 𝑋 π‘Œ 𝑍 = π‘₯ π‘Ž 𝑦 𝑏 𝑧 𝑐 . ( 2 . 9 ) 𝑆 ξ€· ξ€Έ = π‘Ž , 𝑏 , 𝑐 , 𝑗 , π‘˜ , 𝑙 ∞  𝑛 = 0 1 ( 𝑗 π‘Ž 𝑛 + 𝑗 ) ( π‘˜ 𝑏 𝑛 + π‘˜ ) ( 𝑙 𝑐 𝑛 + 𝑙 ) = 𝑗 π‘˜ 𝑙 ∞  𝑛 = 0 Γ ξ€· ξ€Έ Γ ξ€· 𝑗 ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· π‘˜ ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· 𝑙 ξ€Έ π‘Ž 𝑛 + 1 𝑏 𝑛 + 1 𝑐 𝑛 + 1 Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ π‘Ž 𝑛 + 𝑗 + 1 𝑏 𝑛 + π‘˜ + 1 𝑐 𝑛 + 𝑙 + 1 = 𝑗 π‘˜ 𝑙 ∞  𝑛 = 0 𝐡 ξ€· ξ€Έ 𝐡 ξ€· ξ€Έ 𝐡 ξ€· ξ€Έ , π‘Ž 𝑛 , 𝑗 + 1 𝑏 𝑛 , π‘˜ + 1 𝑐 𝑛 , 𝑙 + 1 ( 2 . 1 0 ) Γ ( ⋅ ) 𝐡 ( ⋅ , ⋅ ) and similarly 𝑆 ξ€· ξ€Έ π‘Ž , 𝑏 , 𝑐 , 𝑗 , π‘˜ , 𝑙 = 𝑗 π‘˜ 𝑙 ∞  𝑛 = 0 ξ€œ 1 π‘₯ = 0 ξ€· ξ€Έ 1 − π‘₯ 𝑗 − 1 π‘₯ π‘Ž 𝑛 ξ€œ 𝑑 π‘₯ 1 𝑦 = 0 ξ€· ξ€Έ 1 − 𝑦 π‘˜ − 1 𝑦 𝑏 𝑛 ξ€œ 𝑑 𝑦 1 𝑧 = 0 ξ€· ξ€Έ 1 − 𝑧 𝑙 − 1 𝑧 𝑐 𝑛 ξ€œ 𝑑 𝑧 = 𝑗 π‘˜ 𝑙 1 π‘₯ = 0 ξ€œ 1 𝑦 = 0 ξ€œ 1 𝑧 = 0 ξ€· ξ€Έ 1 − π‘₯ 𝑗 − 1 ξ€· ξ€Έ 1 − 𝑦 π‘˜ − 1 ξ€· ξ€Έ 1 − 𝑧 ∞ 𝑙 − 1  𝑛 = 0 ξ‚€ π‘₯ π‘Ž 𝑦 𝑏 𝑧 𝑐  𝑛 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 ( 2 . 1 1 ) ξ€ž 𝑆 ( π‘Ž , 𝑏 , 𝑐 , 𝑗 , π‘˜ , 𝑙 ) = 𝑗 π‘˜ 𝑙 1 0 ξ€· ξ€Έ 1 − π‘₯ 𝑗 − 1 ξ€· ξ€Έ 1 − 𝑦 π‘˜ − 1 ξ€· ξ€Έ 1 − 𝑧 𝑙 − 1 1 − 𝑋 π‘Œ 𝑍 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 , ( 2 . 1 2 ) = 𝑆 ( π‘Ž , 𝑏 , 𝑐 , 𝑗 , π‘˜ , 𝑙 ) ∞  𝑛 = 0 π‘Ž 𝑏 𝑐 𝑛 3 Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ π‘Ž 𝑛 𝑗 + 1 𝑏 𝑛 π‘˜ + 1 𝑐 𝑛 𝑙 + 1 Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ = π‘Ž 𝑛 + 𝑗 + 1 𝑏 𝑛 + π‘˜ + 1 𝑐 𝑛 + 𝑙 + 1 ∞  𝑛 = 0 π‘Ž 𝑏 𝑐 𝑛 3 𝐡 ξ€· ξ€Έ 𝐡 ξ€· ξ€Έ 𝐡 ξ€· ξ€Έ ξ€ž 𝑗 + 1 , π‘Ž 𝑛 π‘˜ + 1 , 𝑏 𝑛 π‘˜ + 1 , 𝑐 𝑛 = 1 + π‘Ž 𝑏 𝑐 1 0 ξ€· ξ€Έ 1 − π‘₯ 𝑗 ξ€· ξ€Έ 1 − 𝑦 π‘˜ ξ€· ξ€Έ 1 − 𝑧 𝑙 π‘₯ 𝑦 𝑧 ∞  𝑛 = 1 𝑛 3 ξ‚€ π‘₯ π‘Ž 𝑦 𝑏 𝑧 𝑐  𝑛 ξ€ž 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 = 1 + π‘Ž 𝑏 𝑐 1 0 ξ€· ξ€Έ 1 − π‘₯ 𝑗 ξ€· ξ€Έ 1 − 𝑦 π‘˜ ξ€· ξ€Έ 1 − 𝑧 𝑙 ξ€· ξ€Έ π‘₯ 𝑦 𝑧 1 − 𝑋 π‘Œ 𝑍 4 ξ‚€ ξ€· ξ€Έ 𝑋 π‘Œ 𝑍 𝑋 π‘Œ 𝑍 2  + 4 𝑋 π‘Œ 𝑍 + 1 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 ( 2 . 1 3 ) 𝑗 + π‘˜ + 𝑙 + 1 𝐹 𝑗 + π‘˜ + 𝑙 [ 1 1 , π‘Ž , 2 π‘Ž 𝑗 , … , π‘Ž , 1 𝑏 , 2 𝑏 π‘˜ , … , 𝑏 , 1 𝑐 , 2 𝑐 𝑙 , … , 𝑐 π‘Ž + 1 π‘Ž , π‘Ž + 2 π‘Ž , … , π‘Ž + 𝑗 π‘Ž , 𝑏 + 1 𝑏 , 𝑏 + 2 𝑏 , … , 𝑏 + π‘˜ 𝑏 , 𝑐 + 1 𝑐 , 𝑐 + 2 𝑐 , … , 𝑐 + 𝑙 𝑐 = | 1 ] π‘Ž + 𝑏 + 𝑐 + 1 𝐹 π‘Ž + 𝑏 + 𝑐 [ 1 1 , 1 , 1 , 1 , π‘Ž , 2 π‘Ž , … , π‘Ž − 1 π‘Ž , 1 𝑏 , 2 𝑏 , … , 𝑏 − 1 𝑏 , 1 𝑐 , 2 𝑐 , … , 𝑐 − 1 𝑐 𝑗 + 1 π‘Ž , 𝑗 + 2 π‘Ž , … , 𝑗 + π‘Ž π‘Ž , π‘˜ + 1 𝑏 , π‘˜ + 2 𝑏 , … , π‘˜ + 𝑏 𝑏 , 𝑙 + 1 𝑐 , 𝑙 + 2 𝑐 , … , 𝑙 + 𝑐 𝑐 | 1 ] , 𝑗 + π‘˜ + 𝑙 + 1 𝐹 𝑗 + π‘˜ + 𝑙 [ 1 1 , π‘Ž , 2 π‘Ž 𝑗 , … , π‘Ž , 1 𝑏 , 2 𝑏 π‘˜ , … , 𝑏 , 1 𝑐 , 2 𝑐 𝑙 , … , 𝑐 π‘Ž + 1 π‘Ž , π‘Ž + 2 π‘Ž , … , π‘Ž + 𝑗 π‘Ž , 𝑏 + 1 𝑏 , 𝑏 + 2 𝑏 , … , 𝑏 + π‘˜ 𝑏 , 𝑐 + 1 𝑐 , 𝑐 + 2 𝑐 , … , 𝑐 + 𝑙 𝑐 = | − 1 ] π‘Ž + 𝑏 + 𝑐 + 1 𝐹 π‘Ž + 𝑏 + 𝑐 [ 1 1 , 1 , 1 , 1 , π‘Ž , 2 π‘Ž , … , π‘Ž − 1 π‘Ž , 1 𝑏 , 2 𝑏 , … , 𝑏 − 1 𝑏 , 1 𝑐 , 2 𝑐 , … , 𝑐 − 1 𝑐 𝑗 + 1 π‘Ž , 𝑗 + 2 π‘Ž , … , 𝑗 + π‘Ž π‘Ž , π‘˜ + 1 𝑏 , π‘˜ + 2 𝑏 , … , π‘˜ + 𝑏 𝑏 , 𝑙 + 1 𝑐 , 𝑙 + 2 𝑐 , … , 𝑙 + 𝑐 𝑐 | − 1 ] ( 2 . 1 4 ) where 𝑆 ξ€· ξ€Έ = 1 , 1 , 1 , 1 , 1 , 1 ∞  𝑛 = 0 1 ξ€· ξ€Έ 𝑛 + 1 3 ξ€· 3 ξ€Έ = ξ€ž = 𝜁 1 0 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 = 1 − π‘₯ 𝑦 𝑧 4 𝐹 3 [ ξ€ž 1 , 1 , 1 , 1 2 , 2 , 2 | 1 ] = 1 + 1 0 ξ€· ξ€Έ ξ‚€ 1 − π‘₯ ξ€Έ ξ€· 1 − 𝑦 ( 1 − 𝑧 ) ( π‘₯ 𝑦 𝑧 ) 2  + 4 π‘₯ 𝑦 𝑧 + 1 ξ€· ξ€Έ 1 − π‘₯ 𝑦 𝑧 4 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 . ( 3 . 1 ) Proof. Consider ( 2.1 ): 𝜁 ( 3 ) where ( − 1 ) 𝑛 ξ€· ξ€Έ = ξ€œ 𝑛 ! 𝜁 𝑛 + 1 1 π‘₯ = 0  ξ€· π‘₯ ξ€Έ  l n 𝑛 ξ€œ 1 − π‘₯ 𝑑 π‘₯ = 1 π‘₯ = 0  ξ€· ξ€Έ  l n 1 − π‘₯ 𝑛 π‘₯ 𝑑 π‘₯ . ( 3 . 2 ) is the classical Gamma function and 𝑆 ξ€· ξ€Έ = 2 , 2 , 2 , 1 , 1 , 1 ∞  𝑛 = 0 1 ξ€· ξ€Έ 2 𝑛 + 1 3 = 7 8 𝜁 ξ€· 3 ξ€Έ = ξ€œ πœ‹ / 4 π‘₯ = 0 ξ‚€  ξ‚€  l n c o s ( π‘₯ ) l n s i n ( π‘₯ ) ξ€ž c o s ( π‘₯ ) s i n ( π‘₯ ) 𝑑 π‘₯ = 1 0 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 1 − π‘₯ 2 𝑦 2 𝑧 2 ξ€ž = 1 + 8 1 0 ξ€· ξ€Έ ξ‚€ 1 − π‘₯ ξ€Έ ξ€· 1 − 𝑦 ( 1 − 𝑧 ) π‘₯ 𝑦 𝑧 ( π‘₯ 𝑦 𝑧 ) 4 + 4 ( π‘₯ 𝑦 𝑧 ) 2  + 1 ξ‚€ 1 − π‘₯ 2 𝑦 2 𝑧 2  4 = 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 4 𝐹 3 [ 1 2 , 1 2 , 1 2 3 , 1 2 , 3 2 , 3 2 | 1 ] , ( 3 . 3 ) is the Beta function. It holds that 𝜁 ξ€· 3 ξ€Έ ξ€œ = − 5 l n ( πœ™ 2 ) π‘₯ = 0 ξ‚€ ξ‚€ π‘₯ π‘₯ l n 2 s i n h 2   𝑑 π‘₯ , ( 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 ) / 2 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 ), 𝑆 ξ€· ξ€Έ = 4 , 2 , 3 , 𝑗 , π‘˜ , 𝑙 ∞  𝑛 = 0 1 ( 𝑗 4 𝑛 + 𝑗 ) ( π‘˜ 2 𝑛 + π‘˜ ) ( 𝑙 3 𝑛 + 𝑙 ) = ∞  𝑛 = 0 𝑗 ! π‘˜ ! 𝑙 ! ∏ 𝑗 π‘Ÿ = 1 ξ€· ξ€Έ ∏ 4 𝑛 + π‘Ÿ π‘˜ π‘Ÿ = 1 ξ€· ξ€Έ ∏ 2 𝑛 + π‘Ÿ 𝑙 π‘Ÿ = 1 ξ€· ξ€Έ ξ€ž 3 𝑛 + π‘Ÿ = 𝑗 π‘˜ 𝑙 1 0 ( 1 − π‘₯ ) 𝑗 − 1 ( 1 − 𝑦 ) π‘˜ − 1 ( 1 − 𝑧 ) 𝑙 − 1 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 1 − π‘₯ 4 𝑦 2 𝑧 3 = 1 0 𝐹 9 [ 3 1 , 1 , 1 , 1 , 4 , 2 3 , 1 2 , 1 2 , 1 3 , 1 4 𝑗 + 1 4 , 𝑗 + 2 4 , 𝑗 + 3 4 𝑗 + 4 4 , π‘˜ + 1 2 , π‘˜ + 2 2 , 𝑙 + 1 3 , 𝑙 + 2 3 , 𝑙 + 3 3 | 1 ] = 𝛼 1 + 𝛼 2 πœ‹ + 𝛼 3 𝜁 ξ€· 2 ξ€Έ + 𝛼 4 l n ( 2 ) + 𝛼 5 l n ( 3 ) + 𝛼 6 𝜁 ( 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 expression 𝛼 1 = 4 9 5 7 6 2 7 9 9 0 9 3 1 7 ⋅ 1 1 ⋅ 7 2 ⋅ 3 2 ⋅ 2 4 , 𝛼 2 = ξ‚€ 1 6 7 ⋅ 2 2 5 1 7 ⋅ 1 3 ⋅ 1 1 ⋅ 7 2 ⋅ 3 2 − 2 3 ⋅ 5 ⋅ 3 1 6 √ 3 1 7 ⋅ 1 3 ⋅ 1 1 ⋅ 7 2 ⋅ 2 4  , 𝛼 3 = 1 7 0 9 ⋅ 5 ⋅ 2 7 , 𝛼 4 = − 7 5 5 3 5 7 ⋅ 2 1 9 1 7 ⋅ 1 3 ⋅ 1 1 ⋅ 7 2 ⋅ 3 2 , 𝛼 5 = 4 3 ⋅ 2 3 ⋅ 5 ⋅ 3 1 6 1 7 ⋅ 1 3 ⋅ 1 1 ⋅ 7 2 ⋅ 2 4 , 𝛼 6 = 5 3 ⋅ 3 ⋅ 2 2 . ( 3 . 6 ) Lord [ 13 ] posed the problem to show that 𝑇 ξ€· ξ€Έ = 1 , 1 , 1 , 1 , 1 , 1 ∞  𝑛 = 0 ξ€· ξ€Έ − 1 𝑛 ξ€· ξ€Έ 𝑛 + 1 3 = 3 4 = ξ€ž 𝜁 ( 3 ) 1 0 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 = 1 + π‘₯ 𝑦 𝑧 4 𝐹 3 [ ξ€ž 1 , 1 , 1 , 1 2 , 2 , 2 | − 1 ] = 1 − 1 0 ξ€· ξ€Έ ξ‚€ 1 − π‘₯ ξ€Έ ξ€· 1 − 𝑦 ( 1 − 𝑧 ) ( π‘₯ 𝑦 𝑧 ) 2  − 4 π‘₯ 𝑦 𝑧 + 1 ξ€· ξ€Έ 1 + π‘₯ 𝑦 𝑧 4 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 . ( 3 . 7 ) the last three expressions are directly from ( 2.2 ), ( 2.3 ), and ( 2.4 ), respectively. Nan-Yue and Williams [ 14 ] also gave 𝑇 ξ€· ξ€Έ = 2 / 3 , 2 / 3 , 5 / 6 , 2 , 4 , 3 ∞  𝑛 = 0 ξ€· ξ€Έ − 1 𝑛 ( 2 2 𝑛 / 3 + 2 ) ( 4 2 𝑛 / 3 + 4 ) ( 3 5 𝑛 / 6 + 3 ) = ∞  𝑛 = 0 ξ€· ξ€Έ − 1 𝑛 2 5 3 1 1 ξ€· ξ€Έ 𝑛 + 3 2 ξ€· ξ€Έ ( 𝑛 + 6 ) 2 𝑛 + 3 2 ξ€ž ( 2 𝑛 + 9 ) ( 5 𝑛 + 6 ) ( 5 𝑛 + 1 2 ) ( 5 𝑛 + 1 8 ) = 2 4 1 0 ξ€· ξ€Έ 1 − π‘₯ ξ€Έ ξ€· 1 − 𝑦 3 ξ€· ξ€Έ 1 − 𝑧 2 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 1 + π‘₯ 2 / 3 𝑦 2 / 3 𝑧 5 / 6 = 1 0 𝐹 9 [ 3 1 , 2 , 3 2 9 , 3 , 3 , 2 6 , 6 , 5 , 1 2 5 , 1 8 5 5 2 , 5 2 , 4 , 4 , 1 1 2 , 1 1 5 , 1 7 5 , 2 3 5 = , 7 | − 1 ] 7 0 6 6 3 ⋅ 1 6 6 9 1 3 ⋅ 1 1 ⋅ 7 ⋅ 5 ⋅ 3 ⋅ 2 + 3 3 ⋅ 2 4 3 𝜁 ( 2 ) + 2 ⋅ 2 1 0 7 𝐺 + 1 0 9 ⋅ 5 ⋅ 3 ⋅ 2 9 1 1 ⋅ 7 2 2 l n 2 − 8 ⋅ 3 4 1 1 ⋅ 7 2 πœ‹ + ξ‚€ 1 1 3 ⋅ 5 6 ⋅ √ 5 1 1 ⋅ 7 2 ⋅ 2 2 − 2 7 1 ⋅ 5 5 1 1 ⋅ 7 2 ⋅ 2 3  ξ€· 𝛼 ξ€Έ − ξ‚€ l n 1 1 3 ⋅ 5 6 ⋅ √ 5 1 1 ⋅ 7 2 ⋅ 2 2 + 2 7 1 ⋅ 5 5 1 1 ⋅ 7 2 ⋅ 2 3  ξ€· πœ™ ξ€Έ − ξ‚€ 5 l n 4 √ ⋅ 3 ⋅  5 ⋅ πœ™ 𝛼 √ 5 + 1 1 ⋅ 2 3 7 ⋅ 5 4 ⋅ √  5 ⋅ 𝛼 πœ™ √ 5 7 2  πœ‹ , ( 3 . 8 ) where 𝐺 = golden ratio = πœ™ . Example 3.2. It holds that 𝛼 For ( √ 5 − 1 ) / 2 . , and π‘Ž , 𝑏 , 𝑐 , we have the values π‘š Example 3.3. For the alternating case, 𝑗 , π‘˜ , 𝑙 ≥ 0 Example 3.4. It holds that 𝑗 + π‘˜ + 𝑙 ≥ π‘š , where 𝑋 π‘Œ 𝑍 is Catalan's constant, ξ€· 𝑝 ξ€Έ 𝛼 ξ€· ξ€Έ β‹― ξ€· ξ€Έ = Γ ξ€· ξ€Έ = 𝑝 𝑝 + 1 𝑝 + 𝛼 − 1 𝑝 + 𝛼 Γ ξ€· 𝑝 ξ€Έ ( 3 . 1 9 ) is the golden ratio, and 𝑅 ξ€· ξ€Έ = π‘Ž , 𝑏 , 𝑐 , 𝑗 , π‘˜ , 𝑙 , π‘š ∞  𝑛 = 0 ξ€· ξ€Έ − 1 𝑛 ( 𝑛 𝑛 + π‘š − 1 ) ( 𝑗 π‘Ž 𝑛 + 𝑗 ) ( π‘˜ 𝑏 𝑛 + π‘˜ ) ( 𝑙 𝑐 𝑛 + 𝑙 ) = ∞  𝑛 = 0 ξ€· ξ€Έ − 1 𝑛  𝑛 ξƒͺ 𝑛 + π‘š − 1 π‘Ž 𝑏 𝑐 𝑛 3 Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ π‘Ž 𝑛 𝑗 + 1 𝑏 𝑛 π‘˜ + 1 𝑐 𝑛 𝑙 + 1 Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ = π‘Ž 𝑛 + 𝑗 + 1 𝑏 𝑛 + π‘˜ + 1 𝑐 𝑛 + 𝑙 + 1 ∞  𝑛 = 0 ξ€· ξ€Έ − 1 𝑛 π‘Ž 𝑏 𝑐 𝑛 3  𝑛 ξƒͺ 𝐡 ξ€· ξ€Έ 𝐡 ξ€· ξ€Έ 𝐡 ξ€· ξ€Έ 𝑛 + π‘š − 1 π‘Ž 𝑛 , 𝑗 + 1 𝑏 𝑛 , π‘˜ + 1 𝑐 𝑛 , 𝑙 + 1 = 1 + π‘Ž 𝑏 𝑐 ∞  𝑛 = 1 ξ€· ξ€Έ − 1 𝑛 𝑛 3  𝑛 ξƒͺ ξ€œ 𝑛 + π‘š − 1 1 0 ξ€· ξ€Έ 1 − π‘₯ 𝑗 − 1 π‘₯ π‘Ž 𝑛 ξ€œ 𝑑 π‘₯ 1 0 ξ€· ξ€Έ 1 − 𝑦 π‘˜ − 1 𝑦 𝑏 𝑛 ξ€œ 𝑑 𝑦 1 0 ξ€· ξ€Έ 1 − 𝑧 𝑙 − 1 𝑧 𝑐 𝑛 𝑑 𝑧 . ( 3 . 2 0 ) = silver ratio = 𝑅 ξ€· ξ€Έ ξ€ž π‘Ž , 𝑏 , 𝑐 , 𝑗 , π‘˜ , 𝑙 , π‘š = 1 + π‘Ž 𝑏 𝑐 1 0 ξ€· ξ€Έ 1 − π‘₯ 𝑗 ξ€· ξ€Έ 1 − 𝑦 π‘˜ ξ€· ξ€Έ 1 − 𝑧 𝑙 π‘₯ 𝑦 𝑧 ∞  𝑛 = 1 ( − 1 ) 𝑛  ξƒͺ 𝑛 𝑛 + π‘š − 1 π‘š − 1 3 ξ‚€ π‘₯ π‘Ž 𝑦 𝑏 𝑧 𝑐  𝑛 ξ€ž 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 = 1 − π‘š π‘Ž 𝑏 𝑐 1 0 ξ€· ξ€Έ 1 − π‘₯ 𝑗 ξ€· ξ€Έ 1 − 𝑦 π‘˜ ξ€· ξ€Έ 1 − 𝑧 𝑙 ξ‚€ ξ€· ξ€Έ 𝑋 π‘Œ 𝑍 𝑋 π‘Œ 𝑍 2  − ( 3 π‘š + 1 ) 𝑋 π‘Œ 𝑍 + 1 ξ€· ξ€Έ π‘₯ 𝑦 𝑧 1 + 𝑋 π‘Œ 𝑍 π‘š + 3 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 ( 3 . 2 1 ) Now consider the following theorem, which is a generalisation of Theorem 2.1 . Theorem 3.5. For 𝑅 ξ€· ξ€Έ π‘Ž , 𝑏 , 𝑗 , 𝑐 , π‘˜ , 𝑙 , π‘š = 𝑗 π‘˜ 𝑙 ∞  𝑛 = 0 ξ€· ξ€Έ − 1 𝑛  𝑛 ξƒͺ Γ ξ€· ξ€Έ Γ ξ€· 𝑗 ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· π‘˜ ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· 𝑙 ξ€Έ 𝑛 + π‘š − 1 π‘Ž 𝑛 + 1 𝑏 𝑛 + 1 𝑐 𝑛 + 1 Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ π‘Ž 𝑛 + 𝑗 + 1 𝑏 𝑛 + π‘˜ + 1 𝑐 𝑛 + 𝑙 + 1 = 𝑗 π‘˜ 𝑙 ∞  𝑛 = 0 ξ€· ξ€Έ − 1 𝑛  𝑛 ξƒͺ 𝐡 ξ€· ξ€Έ 𝐡 ξ€· ξ€Έ 𝐡 ξ€· ξ€Έ 𝑛 + π‘š − 1 π‘Ž 𝑛 + 1 , 𝑗 𝑏 𝑛 + 1 , π‘˜ 𝑐 𝑛 + 1 , 𝑙 = 𝑗 π‘˜ 𝑙 ∞  𝑛 = 0 ( − 1 ) 𝑛  𝑛 ξƒͺ ξ€œ 𝑛 + π‘š − 1 1 0 ξ€· ξ€Έ 1 − π‘₯ 𝑗 − 1 π‘₯ π‘Ž 𝑛 ξ€œ 𝑑 π‘₯ 1 0 ξ€· ξ€Έ 1 − 𝑦 π‘˜ − 1 𝑦 𝑏 𝑛 ξ€œ 𝑑 𝑦 1 0 ξ€· ξ€Έ 1 − 𝑧 𝑙 − 1 𝑧 𝑐 𝑛 ξ€ž 𝑑 𝑧 = 𝑗 π‘˜ 𝑙 1 0 ξ€· ξ€Έ 1 − π‘₯ 𝑗 − 1 ξ€· ξ€Έ 1 − 𝑦 π‘˜ − 1 ξ€· ξ€Έ 1 − 𝑧 ∞ 𝑙 − 1  𝑛 = 0 ( − 1 ) 𝑛  𝑛 ξƒͺ ξ‚€ π‘₯ 𝑛 + π‘š − 1 π‘Ž 𝑦 𝑏 𝑧 𝑐  𝑛 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 ( 3 . 2 2 ) , and 𝑅 ξ€· ξ€Έ ξ€ž π‘Ž , 𝑏 , 𝑗 , 𝑐 , π‘˜ , 𝑙 , π‘š = 𝑗 π‘˜ 𝑙 1 0 ξ€· ξ€Έ 1 − π‘₯ 𝑗 − 1 ξ€· ξ€Έ 1 − 𝑦 π‘˜ − 1 ξ€· ξ€Έ 1 − 𝑧 𝑙 − 1 ξ€· ξ€Έ 1 + 𝑋 π‘Œ 𝑍 π‘š 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 ( 3 . 2 3 ) positive real numbers and π‘š = 1 , with 𝑄 ξ€· ξ€Έ = 4 , 3 , 2 , 5 , 3 , 6 , 1 1 ∞  𝑛 = 0 ( 𝑛 𝑛 + 1 0 ) ( 5 4 𝑛 + 5 ) ( 3 3 𝑛 + 3 ) ( 6 2 𝑛 + 6 ) ξ€ž = 9 0 1 0 ξ€· ξ€Έ 1 − π‘₯ 4 ξ€· ξ€Έ 1 − 𝑦 2 ξ€· ξ€Έ 1 − 𝑧 5 ξ‚€ 1 − π‘₯ 4 𝑦 3 𝑧 2  1 1 ξ€ž 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 = 1 + 2 6 4 1 0 ξ€· ξ€Έ 1 − π‘₯ 5 ξ€· ξ€Έ 1 − 𝑦 3 ξ€· ξ€Έ 1 − 𝑧 6 π‘₯ 3 𝑦 2 𝑧 ξ‚€ π‘₯ 8 𝑦 6 𝑧 4 + 3 4 π‘₯ 4 𝑦 3 𝑧 2  + 1 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 ξ‚€ 1 − π‘₯ 4 𝑦 3 𝑧 2  1 4 = 1 0 𝐹 9 [ 3 1 1 , 1 , 1 , 1 , 4 , 2 3 , 1 2 , 1 2 , 1 3 , 1 4 7 4 , 2 , 9 4 5 , 2 , 2 , 3 , 7 4 , 3 2 , 4 3 = | 1 ] 3 4 1 3 ⋅ 4 3 ⋅ 5 ⋅ 3 2 1 4 𝜁 ξ€· 2 ξ€Έ + ξ‚€ 1 9 3 1 ⋅ 5 0 9 2 1 2 − √ 9 0 3 7 9 3 7 ⋅ 3 ⋅ 2 5  πœ‹ − 1 4 5 9 ⋅ 1 2 3 1 7 ⋅ 3 2 ⋅ 2 1 1 − 3 7 9 8 8 0 7 7 9 7 ⋅ 3 ⋅ 2 1 0 l n 2 + 2 2 5 6 7 ⋅ 3 2 7 ⋅ 2 5 l n 3 . ( 4 . 1 ) then 𝑅 ξ€· ξ€Έ 1 / 4 , 1 / 6 , 1 / 2 , 5 , 3 , 7 , 1 4 ∢ = ∞  𝑛 = 0 ξ€· ξ€Έ − 1 𝑛 ( 𝑛 𝑛 + 1 3 ) ( 5 𝑛 / 4 + 5 ) ( 3 𝑛 / 6 + 3 ) ( 7 𝑛 / 2 + 7 ) = ∞  𝑛 = 0 ( − 1 ) 𝑛 ξ€· ξ€Έ 5 ! 3 ! 7 ! 𝑛 + 1 1 3 ξ‚€  1 3 ! 𝑛 / 4 + 1 5 ξ‚€  𝑛 / 6 + 1 3 ξ‚€  𝑛 / 2 + 1 7 ξ€ž = 1 0 5 1 0 ξ€· ξ€Έ 1 − π‘₯ 4 ξ€· ξ€Έ 1 − 𝑦 2 ξ€· ξ€Έ 1 − 𝑧 6 ξ‚€ 1 + π‘₯ 1 / 4 𝑦 1 / 6 𝑧 1 / 2  1 4 7 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 = 1 + ξ€ž 2 4 1 0 ξ€· ξ€Έ 1 − π‘₯ 5 ξ€· ξ€Έ 1 − 𝑦 3 ξ€· ξ€Έ 1 − 𝑧 7 × ξ‚€ π‘₯ 1 / 2 𝑦 1 / 3 𝑧 + 4 3 π‘₯ 1 / 4 𝑦 1 / 6 𝑧 1 / 2  + 1 π‘₯ 3 / 4 𝑦 5 / 6 𝑧 1 / 2 ξ‚€ 1 − π‘₯ 1 / 4 𝑦 1 / 6 𝑧 1 / 2  1 7 = 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 1 6 𝐹 1 5 [ = 2 , 4 , 4 , 6 , 6 , 8 , 8 , 1 0 , 1 2 , 1 2 , 1 2 , 1 4 , 1 4 , 1 6 , 1 8 , 2 0 3 , 5 , 5 , 7 , 7 , 9 , 9 , 1 1 , 1 3 , 1 3 , 1 3 , 1 5 , 1 7 , 1 9 , 2 1 | − 1 ] 7 ⋅ 5 ⋅ 3 3 ⋅ 2 4 𝜁 ξ€· 2 ξ€Έ − 1 3 1 6 2 3 1 ⋅ 1 1 ⋅ 3 7 ⋅ 5 ⋅ 2 3 . ( 4 . 2 ) 𝜁 ( 3 ) ∞  𝑛 = 0 𝑛 𝑠 ( 𝑛 𝑛 + π‘š − 1 ) ( 𝑗 π‘Ž 𝑛 + 𝑗 ) ( π‘˜ 𝑏 𝑛 + π‘˜ ) ( 𝑙 𝑐 𝑛 + 𝑙 ) , ∞  𝑛 = 0 ( 𝑛 𝑛 + π‘š − 1 ) ( π‘Ž 𝑛 + 𝑗 𝑏 𝑛 ) ( 𝑐 𝑛 + π‘˜ 𝑑 𝑛 ) ( 𝑝 𝑛 + 𝑗 π‘ž 𝑛 ) . ( 5 . 1 ) 𝜁 𝜁 𝜁 𝜁 ( s i n − 1 π‘₯ ) π‘ž l o g ( 2 s i n h ( πœƒ / 2 ) ) l o g ( 2 s i n ( 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 have which is the result ( 3.17 ). To arrive at the result ( 3.16 ), consider by an allowable change of sum and integral, hence 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 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. <h4>References</h4> A. Sofo, “General properties involving reciprocals of binomial coefficients,” Journal of Integer Sequences , vol. 9, no. 4, Article ID 06.4.5, 13 pages, 2006. A. Sofo, Computational Techniques for the Summation of Series , Kluwer Academic/Plenum Publishers, New York, NY, USA, 2003. A. Sofo, “Integral representations of ratios of binomial coefficients,” International Journal of Pure and Applied Mathematics , vol. 31, no. 1, pp. 29–46, 2006. A. Sofo, “Some properties of reciprocals of double binomial coefficients,” accepted. http://mathworld.wolfram.com/RiemannZetaFunction.html . R. Apéry, “Irrationalitè ζ (2) and ζ (3),” Astérisque , vol. 61, pp. 11–13, 1979. F. Beukers, “ A note on the irrationality of ζ (2) and ζ (3) ,” Bulletin of the London Mathematical Society , vol. 11, no. 3, pp. 268–272, 1979. J. Guillera and J. Sondow, “Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent,” to appear in The Ramanujan Journal . H. Muzaffar, “ Some interesting series arising from the power series expansion of ( sin − 1 x ) q ,” International Journal of Mathematics and Mathematical Sciences , vol. 2005, no. 14, pp. 2329–2336, 2005. S. B. Ekhad and D. Zeilberger, “ A 21st century proof of Dougall's hypergeometric sum identity ,” Journal of Mathematical Analysis and Applications , vol. 147, no. 2, pp. 610–611, 1990. L. J. Slater, Generalized Hypergeometric Functions , Cambridge University Press, Cambridge, UK, 1966. S. R. Finch, Mathematical Constants , vol. 94 of Encyclopedia of Mathematics and Its Applications , Cambridge University Press, Cambridge, UK, 2003. N. Lord, “Problem corner,” The Mathematical Gazette , vol. 89, no. 514, pp. 115–119, 2005. Z. Nan-Yue and K. S. Williams, “Values of the Riemann zeta function and integrals involving log ( 2 sinh ( θ /2 ) ) and log ( 2 sin ( θ /2 ) ) ,” Pacific Journal of Mathematics , vol. 168, no. 2, pp. 271–289, 1995. // http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

Sums of Reciprocals of Triple Binomial Coefficients

Jan 28, 2009

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Sums of Reciprocals of Triple Binomial Coefficients //// Hindawi Publishing Corporation Home Journals About Us About this Journal Submit a Manuscript Table of Contents Journal Menu Abstracting and Indexing Aims and Scope Annual Issues Article Processing Charges Articles in Press Author Guidelines Bibliographic Information Contact Information Editorial Board Editorial Workflow Free eTOC Alerts Reviewers Acknowledgment Subscription Information Open Focus Issues Published Focus Issues Focus Issue Guidelines Open Special Issues Published Special Issues Special Issue Guidelines Abstract Full-Text PDF Full-Text HTML Linked References How to Cite this Article International Journal of Mathematics and Mathematical Sciences Volume 2008 (2008), Article ID 794181, 11 pages doi:10.1155/2008/794181 Research Article <h2>Sums of Reciprocals of Triple Binomial Coefficients</h2> A. Sofo 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: George Andrews Copyright © 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 for ∞  𝑛 = 0 1 ( 𝑗 π‘Ž 𝑛 + 𝑗 ) ( π‘˜ 𝑏 𝑛 + π‘˜ ) ( 𝑙 𝑐 𝑛 + 𝑙 ) , ∞  𝑛 = 0 ( 𝑛 𝑛 + π‘š − 1 ) ( 𝑗 π‘Ž 𝑛 + 𝑗 ) ( π‘˜ 𝑏 𝑛 + π‘˜ ) ( 𝑙 𝑐 𝑛 + 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. Apéry's [ 6 ], see also Beukers [ 7 ], proof of the irrationality of 𝜁 ( 3 ) uses an elementary and quite complicated construction of the approximants 𝛼 𝑛 / 𝛽 𝑛 ∈ 𝑄 to this number based on a recurrence relation. The integral representation ξ€ž 1 0  π‘₯ ξ€· ξ€Έ 𝑦 ξ€· ξ€Έ  1 − π‘₯ 1 − 𝑦 𝑧 ( 1 − 𝑧 ) 𝑛 ξ‚€  1 − ( 1 − π‘₯ 𝑦 ) 𝑧 𝑛 + 1 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 = 2 𝛽 𝑛 𝜁 ( 3 ) − 2 𝛼 𝑛 ( 1 . 2 ) for the sequence { 𝛼 𝑛 , 𝛽 𝑛 } 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  1 0 − l n ( π‘₯ 𝑦 )  1 − π‘₯ 𝑦 𝑑 π‘₯ 𝑑 𝑦 = 2 𝜁 ( 3 ) , 1 0 l n ( 2 − π‘₯ 𝑦 ) 5 1 − π‘₯ 𝑦 𝑑 π‘₯ 𝑑 𝑦 = 8 𝜁 ( 3 ) . ( 1 . 3 ) In a recent paper, Muzaffar [ 9 ] also obtained some results of the combinatorial type ∞  𝑛 = 0 ( 𝑛 2 𝑛 ) ( 1 2 𝑛 + 1 ) ( 1 2 𝑛 + π‘˜ + 1 ) ( 2 𝑛 + 2 π‘˜ 𝑛 + π‘˜ ) = 𝛼 π‘˜ πœ‹ 2 + 𝛽 π‘˜ ( 1 . 4 ) by utilising the power series expansion of ( s i n − 1 π‘₯ ) π‘ž , and ( 𝛼 π‘˜ , 𝛽 π‘˜ ) are constants depending on π‘˜ ≥ 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 + 2 ∞  𝑛 = 1 ( − 1 ) 𝑛 ( π‘Ž 𝑛 ) ( 𝑏 𝑛 ) ( 𝑐 𝑛 ) ( π‘Ž 𝑛 + π‘Ž ) ( 𝑏 𝑛 + 𝑏 ) ( 𝑐 𝑛 + 𝑐 ) = ( 𝑏 π‘Ž + 𝑏 + 𝑐 ) ( 𝑏 π‘Ž + 𝑏 ) ( 𝑏 𝑏 + 𝑐 ) ( 1 . 5 ) for 𝑅 ( π‘Ž + 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 π‘Ž , 𝑏 , and 𝑐 positive real numbers and 𝑗 , π‘˜ , 𝑙 ≥ 0 , then 𝑋 π‘Œ 𝑍 = π‘₯ π‘Ž 𝑦 𝑏 𝑧 𝑐 . ( 2 . 9 ) 𝑆 ξ€· ξ€Έ = π‘Ž , 𝑏 , 𝑐 , 𝑗 , π‘˜ , 𝑙 ∞  𝑛 = 0 1 ( 𝑗 π‘Ž 𝑛 + 𝑗 ) ( π‘˜ 𝑏 𝑛 + π‘˜ ) ( 𝑙 𝑐 𝑛 + 𝑙 ) = 𝑗 π‘˜ 𝑙 ∞  𝑛 = 0 Γ ξ€· ξ€Έ Γ ξ€· 𝑗 ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· π‘˜ ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· 𝑙 ξ€Έ π‘Ž 𝑛 + 1 𝑏 𝑛 + 1 𝑐 𝑛 + 1 Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ π‘Ž 𝑛 + 𝑗 + 1 𝑏 𝑛 + π‘˜ + 1 𝑐 𝑛 + 𝑙 + 1 = 𝑗 π‘˜ 𝑙 ∞  𝑛 = 0 𝐡 ξ€· ξ€Έ 𝐡 ξ€· ξ€Έ 𝐡 ξ€· ξ€Έ , π‘Ž 𝑛 , 𝑗 + 1 𝑏 𝑛 , π‘˜ + 1 𝑐 𝑛 , 𝑙 + 1 ( 2 . 1 0 ) Γ ( ⋅ ) 𝐡 ( ⋅ , ⋅ ) and similarly 𝑆 ξ€· ξ€Έ π‘Ž , 𝑏 , 𝑐 , 𝑗 , π‘˜ , 𝑙 = 𝑗 π‘˜ 𝑙 ∞  𝑛 = 0 ξ€œ 1 π‘₯ = 0 ξ€· ξ€Έ 1 − π‘₯ 𝑗 − 1 π‘₯ π‘Ž 𝑛 ξ€œ 𝑑 π‘₯ 1 𝑦 = 0 ξ€· ξ€Έ 1 − 𝑦 π‘˜ − 1 𝑦 𝑏 𝑛 ξ€œ 𝑑 𝑦 1 𝑧 = 0 ξ€· ξ€Έ 1 − 𝑧 𝑙 − 1 𝑧 𝑐 𝑛 ξ€œ 𝑑 𝑧 = 𝑗 π‘˜ 𝑙 1 π‘₯ = 0 ξ€œ 1 𝑦 = 0 ξ€œ 1 𝑧 = 0 ξ€· ξ€Έ 1 − π‘₯ 𝑗 − 1 ξ€· ξ€Έ 1 − 𝑦 π‘˜ − 1 ξ€· ξ€Έ 1 − 𝑧 ∞ 𝑙 − 1  𝑛 = 0 ξ‚€ π‘₯ π‘Ž 𝑦 𝑏 𝑧 𝑐  𝑛 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 ( 2 . 1 1 ) ξ€ž 𝑆 ( π‘Ž , 𝑏 , 𝑐 , 𝑗 , π‘˜ , 𝑙 ) = 𝑗 π‘˜ 𝑙 1 0 ξ€· ξ€Έ 1 − π‘₯ 𝑗 − 1 ξ€· ξ€Έ 1 − 𝑦 π‘˜ − 1 ξ€· ξ€Έ 1 − 𝑧 𝑙 − 1 1 − 𝑋 π‘Œ 𝑍 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 , ( 2 . 1 2 ) = 𝑆 ( π‘Ž , 𝑏 , 𝑐 , 𝑗 , π‘˜ , 𝑙 ) ∞  𝑛 = 0 π‘Ž 𝑏 𝑐 𝑛 3 Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ π‘Ž 𝑛 𝑗 + 1 𝑏 𝑛 π‘˜ + 1 𝑐 𝑛 𝑙 + 1 Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ = π‘Ž 𝑛 + 𝑗 + 1 𝑏 𝑛 + π‘˜ + 1 𝑐 𝑛 + 𝑙 + 1 ∞  𝑛 = 0 π‘Ž 𝑏 𝑐 𝑛 3 𝐡 ξ€· ξ€Έ 𝐡 ξ€· ξ€Έ 𝐡 ξ€· ξ€Έ ξ€ž 𝑗 + 1 , π‘Ž 𝑛 π‘˜ + 1 , 𝑏 𝑛 π‘˜ + 1 , 𝑐 𝑛 = 1 + π‘Ž 𝑏 𝑐 1 0 ξ€· ξ€Έ 1 − π‘₯ 𝑗 ξ€· ξ€Έ 1 − 𝑦 π‘˜ ξ€· ξ€Έ 1 − 𝑧 𝑙 π‘₯ 𝑦 𝑧 ∞  𝑛 = 1 𝑛 3 ξ‚€ π‘₯ π‘Ž 𝑦 𝑏 𝑧 𝑐  𝑛 ξ€ž 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 = 1 + π‘Ž 𝑏 𝑐 1 0 ξ€· ξ€Έ 1 − π‘₯ 𝑗 ξ€· ξ€Έ 1 − 𝑦 π‘˜ ξ€· ξ€Έ 1 − 𝑧 𝑙 ξ€· ξ€Έ π‘₯ 𝑦 𝑧 1 − 𝑋 π‘Œ 𝑍 4 ξ‚€ ξ€· ξ€Έ 𝑋 π‘Œ 𝑍 𝑋 π‘Œ 𝑍 2  + 4 𝑋 π‘Œ 𝑍 + 1 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 ( 2 . 1 3 ) 𝑗 + π‘˜ + 𝑙 + 1 𝐹 𝑗 + π‘˜ + 𝑙 [ 1 1 , π‘Ž , 2 π‘Ž 𝑗 , … , π‘Ž , 1 𝑏 , 2 𝑏 π‘˜ , … , 𝑏 , 1 𝑐 , 2 𝑐 𝑙 , … , 𝑐 π‘Ž + 1 π‘Ž , π‘Ž + 2 π‘Ž , … , π‘Ž + 𝑗 π‘Ž , 𝑏 + 1 𝑏 , 𝑏 + 2 𝑏 , … , 𝑏 + π‘˜ 𝑏 , 𝑐 + 1 𝑐 , 𝑐 + 2 𝑐 , … , 𝑐 + 𝑙 𝑐 = | 1 ] π‘Ž + 𝑏 + 𝑐 + 1 𝐹 π‘Ž + 𝑏 + 𝑐 [ 1 1 , 1 , 1 , 1 , π‘Ž , 2 π‘Ž , … , π‘Ž − 1 π‘Ž , 1 𝑏 , 2 𝑏 , … , 𝑏 − 1 𝑏 , 1 𝑐 , 2 𝑐 , … , 𝑐 − 1 𝑐 𝑗 + 1 π‘Ž , 𝑗 + 2 π‘Ž , … , 𝑗 + π‘Ž π‘Ž , π‘˜ + 1 𝑏 , π‘˜ + 2 𝑏 , … , π‘˜ + 𝑏 𝑏 , 𝑙 + 1 𝑐 , 𝑙 + 2 𝑐 , … , 𝑙 + 𝑐 𝑐 | 1 ] , 𝑗 + π‘˜ + 𝑙 + 1 𝐹 𝑗 + π‘˜ + 𝑙 [ 1 1 , π‘Ž , 2 π‘Ž 𝑗 , … , π‘Ž , 1 𝑏 , 2 𝑏 π‘˜ , … , 𝑏 , 1 𝑐 , 2 𝑐 𝑙 , … , 𝑐 π‘Ž + 1 π‘Ž , π‘Ž + 2 π‘Ž , … , π‘Ž + 𝑗 π‘Ž , 𝑏 + 1 𝑏 , 𝑏 + 2 𝑏 , … , 𝑏 + π‘˜ 𝑏 , 𝑐 + 1 𝑐 , 𝑐 + 2 𝑐 , … , 𝑐 + 𝑙 𝑐 = | − 1 ] π‘Ž + 𝑏 + 𝑐 + 1 𝐹 π‘Ž + 𝑏 + 𝑐 [ 1 1 , 1 , 1 , 1 , π‘Ž , 2 π‘Ž , … , π‘Ž − 1 π‘Ž , 1 𝑏 , 2 𝑏 , … , 𝑏 − 1 𝑏 , 1 𝑐 , 2 𝑐 , … , 𝑐 − 1 𝑐 𝑗 + 1 π‘Ž , 𝑗 + 2 π‘Ž , … , 𝑗 + π‘Ž π‘Ž , π‘˜ + 1 𝑏 , π‘˜ + 2 𝑏 , … , π‘˜ + 𝑏 𝑏 , 𝑙 + 1 𝑐 , 𝑙 + 2 𝑐 , … , 𝑙 + 𝑐 𝑐 | − 1 ] ( 2 . 1 4 ) where 𝑆 ξ€· ξ€Έ = 1 , 1 , 1 , 1 , 1 , 1 ∞  𝑛 = 0 1 ξ€· ξ€Έ 𝑛 + 1 3 ξ€· 3 ξ€Έ = ξ€ž = 𝜁 1 0 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 = 1 − π‘₯ 𝑦 𝑧 4 𝐹 3 [ ξ€ž 1 , 1 , 1 , 1 2 , 2 , 2 | 1 ] = 1 + 1 0 ξ€· ξ€Έ ξ‚€ 1 − π‘₯ ξ€Έ ξ€· 1 − 𝑦 ( 1 − 𝑧 ) ( π‘₯ 𝑦 𝑧 ) 2  + 4 π‘₯ 𝑦 𝑧 + 1 ξ€· ξ€Έ 1 − π‘₯ 𝑦 𝑧 4 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 . ( 3 . 1 ) Proof. Consider ( 2.1 ): 𝜁 ( 3 ) where ( − 1 ) 𝑛 ξ€· ξ€Έ = ξ€œ 𝑛 ! 𝜁 𝑛 + 1 1 π‘₯ = 0  ξ€· π‘₯ ξ€Έ  l n 𝑛 ξ€œ 1 − π‘₯ 𝑑 π‘₯ = 1 π‘₯ = 0  ξ€· ξ€Έ  l n 1 − π‘₯ 𝑛 π‘₯ 𝑑 π‘₯ . ( 3 . 2 ) is the classical Gamma function and 𝑆 ξ€· ξ€Έ = 2 , 2 , 2 , 1 , 1 , 1 ∞  𝑛 = 0 1 ξ€· ξ€Έ 2 𝑛 + 1 3 = 7 8 𝜁 ξ€· 3 ξ€Έ = ξ€œ πœ‹ / 4 π‘₯ = 0 ξ‚€  ξ‚€  l n c o s ( π‘₯ ) l n s i n ( π‘₯ ) ξ€ž c o s ( π‘₯ ) s i n ( π‘₯ ) 𝑑 π‘₯ = 1 0 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 1 − π‘₯ 2 𝑦 2 𝑧 2 ξ€ž = 1 + 8 1 0 ξ€· ξ€Έ ξ‚€ 1 − π‘₯ ξ€Έ ξ€· 1 − 𝑦 ( 1 − 𝑧 ) π‘₯ 𝑦 𝑧 ( π‘₯ 𝑦 𝑧 ) 4 + 4 ( π‘₯ 𝑦 𝑧 ) 2  + 1 ξ‚€ 1 − π‘₯ 2 𝑦 2 𝑧 2  4 = 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 4 𝐹 3 [ 1 2 , 1 2 , 1 2 3 , 1 2 , 3 2 , 3 2 | 1 ] , ( 3 . 3 ) is the Beta function. It holds that 𝜁 ξ€· 3 ξ€Έ ξ€œ = − 5 l n ( πœ™ 2 ) π‘₯ = 0 ξ‚€ ξ‚€ π‘₯ π‘₯ l n 2 s i n h 2   𝑑 π‘₯ , ( 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 ) / 2 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 ), 𝑆 ξ€· ξ€Έ = 4 , 2 , 3 , 𝑗 , π‘˜ , 𝑙 ∞  𝑛 = 0 1 ( 𝑗 4 𝑛 + 𝑗 ) ( π‘˜ 2 𝑛 + π‘˜ ) ( 𝑙 3 𝑛 + 𝑙 ) = ∞  𝑛 = 0 𝑗 ! π‘˜ ! 𝑙 ! ∏ 𝑗 π‘Ÿ = 1 ξ€· ξ€Έ ∏ 4 𝑛 + π‘Ÿ π‘˜ π‘Ÿ = 1 ξ€· ξ€Έ ∏ 2 𝑛 + π‘Ÿ 𝑙 π‘Ÿ = 1 ξ€· ξ€Έ ξ€ž 3 𝑛 + π‘Ÿ = 𝑗 π‘˜ 𝑙 1 0 ( 1 − π‘₯ ) 𝑗 − 1 ( 1 − 𝑦 ) π‘˜ − 1 ( 1 − 𝑧 ) 𝑙 − 1 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 1 − π‘₯ 4 𝑦 2 𝑧 3 = 1 0 𝐹 9 [ 3 1 , 1 , 1 , 1 , 4 , 2 3 , 1 2 , 1 2 , 1 3 , 1 4 𝑗 + 1 4 , 𝑗 + 2 4 , 𝑗 + 3 4 𝑗 + 4 4 , π‘˜ + 1 2 , π‘˜ + 2 2 , 𝑙 + 1 3 , 𝑙 + 2 3 , 𝑙 + 3 3 | 1 ] = 𝛼 1 + 𝛼 2 πœ‹ + 𝛼 3 𝜁 ξ€· 2 ξ€Έ + 𝛼 4 l n ( 2 ) + 𝛼 5 l n ( 3 ) + 𝛼 6 𝜁 ( 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 expression 𝛼 1 = 4 9 5 7 6 2 7 9 9 0 9 3 1 7 ⋅ 1 1 ⋅ 7 2 ⋅ 3 2 ⋅ 2 4 , 𝛼 2 = ξ‚€ 1 6 7 ⋅ 2 2 5 1 7 ⋅ 1 3 ⋅ 1 1 ⋅ 7 2 ⋅ 3 2 − 2 3 ⋅ 5 ⋅ 3 1 6 √ 3 1 7 ⋅ 1 3 ⋅ 1 1 ⋅ 7 2 ⋅ 2 4  , 𝛼 3 = 1 7 0 9 ⋅ 5 ⋅ 2 7 , 𝛼 4 = − 7 5 5 3 5 7 ⋅ 2 1 9 1 7 ⋅ 1 3 ⋅ 1 1 ⋅ 7 2 ⋅ 3 2 , 𝛼 5 = 4 3 ⋅ 2 3 ⋅ 5 ⋅ 3 1 6 1 7 ⋅ 1 3 ⋅ 1 1 ⋅ 7 2 ⋅ 2 4 , 𝛼 6 = 5 3 ⋅ 3 ⋅ 2 2 . ( 3 . 6 ) Lord [ 13 ] posed the problem to show that 𝑇 ξ€· ξ€Έ = 1 , 1 , 1 , 1 , 1 , 1 ∞  𝑛 = 0 ξ€· ξ€Έ − 1 𝑛 ξ€· ξ€Έ 𝑛 + 1 3 = 3 4 = ξ€ž 𝜁 ( 3 ) 1 0 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 = 1 + π‘₯ 𝑦 𝑧 4 𝐹 3 [ ξ€ž 1 , 1 , 1 , 1 2 , 2 , 2 | − 1 ] = 1 − 1 0 ξ€· ξ€Έ ξ‚€ 1 − π‘₯ ξ€Έ ξ€· 1 − 𝑦 ( 1 − 𝑧 ) ( π‘₯ 𝑦 𝑧 ) 2  − 4 π‘₯ 𝑦 𝑧 + 1 ξ€· ξ€Έ 1 + π‘₯ 𝑦 𝑧 4 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 . ( 3 . 7 ) the last three expressions are directly from ( 2.2 ), ( 2.3 ), and ( 2.4 ), respectively. Nan-Yue and Williams [ 14 ] also gave 𝑇 ξ€· ξ€Έ = 2 / 3 , 2 / 3 , 5 / 6 , 2 , 4 , 3 ∞  𝑛 = 0 ξ€· ξ€Έ − 1 𝑛 ( 2 2 𝑛 / 3 + 2 ) ( 4 2 𝑛 / 3 + 4 ) ( 3 5 𝑛 / 6 + 3 ) = ∞  𝑛 = 0 ξ€· ξ€Έ − 1 𝑛 2 5 3 1 1 ξ€· ξ€Έ 𝑛 + 3 2 ξ€· ξ€Έ ( 𝑛 + 6 ) 2 𝑛 + 3 2 ξ€ž ( 2 𝑛 + 9 ) ( 5 𝑛 + 6 ) ( 5 𝑛 + 1 2 ) ( 5 𝑛 + 1 8 ) = 2 4 1 0 ξ€· ξ€Έ 1 − π‘₯ ξ€Έ ξ€· 1 − 𝑦 3 ξ€· ξ€Έ 1 − 𝑧 2 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 1 + π‘₯ 2 / 3 𝑦 2 / 3 𝑧 5 / 6 = 1 0 𝐹 9 [ 3 1 , 2 , 3 2 9 , 3 , 3 , 2 6 , 6 , 5 , 1 2 5 , 1 8 5 5 2 , 5 2 , 4 , 4 , 1 1 2 , 1 1 5 , 1 7 5 , 2 3 5 = , 7 | − 1 ] 7 0 6 6 3 ⋅ 1 6 6 9 1 3 ⋅ 1 1 ⋅ 7 ⋅ 5 ⋅ 3 ⋅ 2 + 3 3 ⋅ 2 4 3 𝜁 ( 2 ) + 2 ⋅ 2 1 0 7 𝐺 + 1 0 9 ⋅ 5 ⋅ 3 ⋅ 2 9 1 1 ⋅ 7 2 2 l n 2 − 8 ⋅ 3 4 1 1 ⋅ 7 2 πœ‹ + ξ‚€ 1 1 3 ⋅ 5 6 ⋅ √ 5 1 1 ⋅ 7 2 ⋅ 2 2 − 2 7 1 ⋅ 5 5 1 1 ⋅ 7 2 ⋅ 2 3  ξ€· 𝛼 ξ€Έ − ξ‚€ l n 1 1 3 ⋅ 5 6 ⋅ √ 5 1 1 ⋅ 7 2 ⋅ 2 2 + 2 7 1 ⋅ 5 5 1 1 ⋅ 7 2 ⋅ 2 3  ξ€· πœ™ ξ€Έ − ξ‚€ 5 l n 4 √ ⋅ 3 ⋅  5 ⋅ πœ™ 𝛼 √ 5 + 1 1 ⋅ 2 3 7 ⋅ 5 4 ⋅ √  5 ⋅ 𝛼 πœ™ √ 5 7 2  πœ‹ , ( 3 . 8 ) where 𝐺 = golden ratio = πœ™ . Example 3.2. It holds that 𝛼 For ( √ 5 − 1 ) / 2 . , and π‘Ž , 𝑏 , 𝑐 , we have the values π‘š Example 3.3. For the alternating case, 𝑗 , π‘˜ , 𝑙 ≥ 0 Example 3.4. It holds that 𝑗 + π‘˜ + 𝑙 ≥ π‘š , where 𝑋 π‘Œ 𝑍 is Catalan's constant, ξ€· 𝑝 ξ€Έ 𝛼 ξ€· ξ€Έ β‹― ξ€· ξ€Έ = Γ ξ€· ξ€Έ = 𝑝 𝑝 + 1 𝑝 + 𝛼 − 1 𝑝 + 𝛼 Γ ξ€· 𝑝 ξ€Έ ( 3 . 1 9 ) is the golden ratio, and 𝑅 ξ€· ξ€Έ = π‘Ž , 𝑏 , 𝑐 , 𝑗 , π‘˜ , 𝑙 , π‘š ∞  𝑛 = 0 ξ€· ξ€Έ − 1 𝑛 ( 𝑛 𝑛 + π‘š − 1 ) ( 𝑗 π‘Ž 𝑛 + 𝑗 ) ( π‘˜ 𝑏 𝑛 + π‘˜ ) ( 𝑙 𝑐 𝑛 + 𝑙 ) = ∞  𝑛 = 0 ξ€· ξ€Έ − 1 𝑛  𝑛 ξƒͺ 𝑛 + π‘š − 1 π‘Ž 𝑏 𝑐 𝑛 3 Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ π‘Ž 𝑛 𝑗 + 1 𝑏 𝑛 π‘˜ + 1 𝑐 𝑛 𝑙 + 1 Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ = π‘Ž 𝑛 + 𝑗 + 1 𝑏 𝑛 + π‘˜ + 1 𝑐 𝑛 + 𝑙 + 1 ∞  𝑛 = 0 ξ€· ξ€Έ − 1 𝑛 π‘Ž 𝑏 𝑐 𝑛 3  𝑛 ξƒͺ 𝐡 ξ€· ξ€Έ 𝐡 ξ€· ξ€Έ 𝐡 ξ€· ξ€Έ 𝑛 + π‘š − 1 π‘Ž 𝑛 , 𝑗 + 1 𝑏 𝑛 , π‘˜ + 1 𝑐 𝑛 , 𝑙 + 1 = 1 + π‘Ž 𝑏 𝑐 ∞  𝑛 = 1 ξ€· ξ€Έ − 1 𝑛 𝑛 3  𝑛 ξƒͺ ξ€œ 𝑛 + π‘š − 1 1 0 ξ€· ξ€Έ 1 − π‘₯ 𝑗 − 1 π‘₯ π‘Ž 𝑛 ξ€œ 𝑑 π‘₯ 1 0 ξ€· ξ€Έ 1 − 𝑦 π‘˜ − 1 𝑦 𝑏 𝑛 ξ€œ 𝑑 𝑦 1 0 ξ€· ξ€Έ 1 − 𝑧 𝑙 − 1 𝑧 𝑐 𝑛 𝑑 𝑧 . ( 3 . 2 0 ) = silver ratio = 𝑅 ξ€· ξ€Έ ξ€ž π‘Ž , 𝑏 , 𝑐 , 𝑗 , π‘˜ , 𝑙 , π‘š = 1 + π‘Ž 𝑏 𝑐 1 0 ξ€· ξ€Έ 1 − π‘₯ 𝑗 ξ€· ξ€Έ 1 − 𝑦 π‘˜ ξ€· ξ€Έ 1 − 𝑧 𝑙 π‘₯ 𝑦 𝑧 ∞  𝑛 = 1 ( − 1 ) 𝑛  ξƒͺ 𝑛 𝑛 + π‘š − 1 π‘š − 1 3 ξ‚€ π‘₯ π‘Ž 𝑦 𝑏 𝑧 𝑐  𝑛 ξ€ž 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 = 1 − π‘š π‘Ž 𝑏 𝑐 1 0 ξ€· ξ€Έ 1 − π‘₯ 𝑗 ξ€· ξ€Έ 1 − 𝑦 π‘˜ ξ€· ξ€Έ 1 − 𝑧 𝑙 ξ‚€ ξ€· ξ€Έ 𝑋 π‘Œ 𝑍 𝑋 π‘Œ 𝑍 2  − ( 3 π‘š + 1 ) 𝑋 π‘Œ 𝑍 + 1 ξ€· ξ€Έ π‘₯ 𝑦 𝑧 1 + 𝑋 π‘Œ 𝑍 π‘š + 3 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 ( 3 . 2 1 ) Now consider the following theorem, which is a generalisation of Theorem 2.1 . Theorem 3.5. For 𝑅 ξ€· ξ€Έ π‘Ž , 𝑏 , 𝑗 , 𝑐 , π‘˜ , 𝑙 , π‘š = 𝑗 π‘˜ 𝑙 ∞  𝑛 = 0 ξ€· ξ€Έ − 1 𝑛  𝑛 ξƒͺ Γ ξ€· ξ€Έ Γ ξ€· 𝑗 ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· π‘˜ ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· 𝑙 ξ€Έ 𝑛 + π‘š − 1 π‘Ž 𝑛 + 1 𝑏 𝑛 + 1 𝑐 𝑛 + 1 Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ Γ ξ€· ξ€Έ π‘Ž 𝑛 + 𝑗 + 1 𝑏 𝑛 + π‘˜ + 1 𝑐 𝑛 + 𝑙 + 1 = 𝑗 π‘˜ 𝑙 ∞  𝑛 = 0 ξ€· ξ€Έ − 1 𝑛  𝑛 ξƒͺ 𝐡 ξ€· ξ€Έ 𝐡 ξ€· ξ€Έ 𝐡 ξ€· ξ€Έ 𝑛 + π‘š − 1 π‘Ž 𝑛 + 1 , 𝑗 𝑏 𝑛 + 1 , π‘˜ 𝑐 𝑛 + 1 , 𝑙 = 𝑗 π‘˜ 𝑙 ∞  𝑛 = 0 ( − 1 ) 𝑛  𝑛 ξƒͺ ξ€œ 𝑛 + π‘š − 1 1 0 ξ€· ξ€Έ 1 − π‘₯ 𝑗 − 1 π‘₯ π‘Ž 𝑛 ξ€œ 𝑑 π‘₯ 1 0 ξ€· ξ€Έ 1 − 𝑦 π‘˜ − 1 𝑦 𝑏 𝑛 ξ€œ 𝑑 𝑦 1 0 ξ€· ξ€Έ 1 − 𝑧 𝑙 − 1 𝑧 𝑐 𝑛 ξ€ž 𝑑 𝑧 = 𝑗 π‘˜ 𝑙 1 0 ξ€· ξ€Έ 1 − π‘₯ 𝑗 − 1 ξ€· ξ€Έ 1 − 𝑦 π‘˜ − 1 ξ€· ξ€Έ 1 − 𝑧 ∞ 𝑙 − 1  𝑛 = 0 ( − 1 ) 𝑛  𝑛 ξƒͺ ξ‚€ π‘₯ 𝑛 + π‘š − 1 π‘Ž 𝑦 𝑏 𝑧 𝑐  𝑛 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 ( 3 . 2 2 ) , and 𝑅 ξ€· ξ€Έ ξ€ž π‘Ž , 𝑏 , 𝑗 , 𝑐 , π‘˜ , 𝑙 , π‘š = 𝑗 π‘˜ 𝑙 1 0 ξ€· ξ€Έ 1 − π‘₯ 𝑗 − 1 ξ€· ξ€Έ 1 − 𝑦 π‘˜ − 1 ξ€· ξ€Έ 1 − 𝑧 𝑙 − 1 ξ€· ξ€Έ 1 + 𝑋 π‘Œ 𝑍 π‘š 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 ( 3 . 2 3 ) positive real numbers and π‘š = 1 , with 𝑄 ξ€· ξ€Έ = 4 , 3 , 2 , 5 , 3 , 6 , 1 1 ∞  𝑛 = 0 ( 𝑛 𝑛 + 1 0 ) ( 5 4 𝑛 + 5 ) ( 3 3 𝑛 + 3 ) ( 6 2 𝑛 + 6 ) ξ€ž = 9 0 1 0 ξ€· ξ€Έ 1 − π‘₯ 4 ξ€· ξ€Έ 1 − 𝑦 2 ξ€· ξ€Έ 1 − 𝑧 5 ξ‚€ 1 − π‘₯ 4 𝑦 3 𝑧 2  1 1 ξ€ž 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 = 1 + 2 6 4 1 0 ξ€· ξ€Έ 1 − π‘₯ 5 ξ€· ξ€Έ 1 − 𝑦 3 ξ€· ξ€Έ 1 − 𝑧 6 π‘₯ 3 𝑦 2 𝑧 ξ‚€ π‘₯ 8 𝑦 6 𝑧 4 + 3 4 π‘₯ 4 𝑦 3 𝑧 2  + 1 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 ξ‚€ 1 − π‘₯ 4 𝑦 3 𝑧 2  1 4 = 1 0 𝐹 9 [ 3 1 1 , 1 , 1 , 1 , 4 , 2 3 , 1 2 , 1 2 , 1 3 , 1 4 7 4 , 2 , 9 4 5 , 2 , 2 , 3 , 7 4 , 3 2 , 4 3 = | 1 ] 3 4 1 3 ⋅ 4 3 ⋅ 5 ⋅ 3 2 1 4 𝜁 ξ€· 2 ξ€Έ + ξ‚€ 1 9 3 1 ⋅ 5 0 9 2 1 2 − √ 9 0 3 7 9 3 7 ⋅ 3 ⋅ 2 5  πœ‹ − 1 4 5 9 ⋅ 1 2 3 1 7 ⋅ 3 2 ⋅ 2 1 1 − 3 7 9 8 8 0 7 7 9 7 ⋅ 3 ⋅ 2 1 0 l n 2 + 2 2 5 6 7 ⋅ 3 2 7 ⋅ 2 5 l n 3 . ( 4 . 1 ) then 𝑅 ξ€· ξ€Έ 1 / 4 , 1 / 6 , 1 / 2 , 5 , 3 , 7 , 1 4 ∢ = ∞  𝑛 = 0 ξ€· ξ€Έ − 1 𝑛 ( 𝑛 𝑛 + 1 3 ) ( 5 𝑛 / 4 + 5 ) ( 3 𝑛 / 6 + 3 ) ( 7 𝑛 / 2 + 7 ) = ∞  𝑛 = 0 ( − 1 ) 𝑛 ξ€· ξ€Έ 5 ! 3 ! 7 ! 𝑛 + 1 1 3 ξ‚€  1 3 ! 𝑛 / 4 + 1 5 ξ‚€  𝑛 / 6 + 1 3 ξ‚€  𝑛 / 2 + 1 7 ξ€ž = 1 0 5 1 0 ξ€· ξ€Έ 1 − π‘₯ 4 ξ€· ξ€Έ 1 − 𝑦 2 ξ€· ξ€Έ 1 − 𝑧 6 ξ‚€ 1 + π‘₯ 1 / 4 𝑦 1 / 6 𝑧 1 / 2  1 4 7 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 = 1 + ξ€ž 2 4 1 0 ξ€· ξ€Έ 1 − π‘₯ 5 ξ€· ξ€Έ 1 − 𝑦 3 ξ€· ξ€Έ 1 − 𝑧 7 × ξ‚€ π‘₯ 1 / 2 𝑦 1 / 3 𝑧 + 4 3 π‘₯ 1 / 4 𝑦 1 / 6 𝑧 1 / 2  + 1 π‘₯ 3 / 4 𝑦 5 / 6 𝑧 1 / 2 ξ‚€ 1 − π‘₯ 1 / 4 𝑦 1 / 6 𝑧 1 / 2  1 7 = 𝑑 π‘₯ 𝑑 𝑦 𝑑 𝑧 1 6 𝐹 1 5 [ = 2 , 4 , 4 , 6 , 6 , 8 , 8 , 1 0 , 1 2 , 1 2 , 1 2 , 1 4 , 1 4 , 1 6 , 1 8 , 2 0 3 , 5 , 5 , 7 , 7 , 9 , 9 , 1 1 , 1 3 , 1 3 , 1 3 , 1 5 , 1 7 , 1 9 , 2 1 | − 1 ] 7 ⋅ 5 ⋅ 3 3 ⋅ 2 4 𝜁 ξ€· 2 ξ€Έ − 1 3 1 6 2 3 1 ⋅ 1 1 ⋅ 3 7 ⋅ 5 ⋅ 2 3 . ( 4 . 2 ) 𝜁 ( 3 ) ∞  𝑛 = 0 𝑛 𝑠 ( 𝑛 𝑛 + π‘š − 1 ) ( 𝑗 π‘Ž 𝑛 + 𝑗 ) ( π‘˜ 𝑏 𝑛 + π‘˜ ) ( 𝑙 𝑐 𝑛 + 𝑙 ) , ∞  𝑛 = 0 ( 𝑛 𝑛 + π‘š − 1 ) ( π‘Ž 𝑛 + 𝑗 𝑏 𝑛 ) ( 𝑐 𝑛 + π‘˜ 𝑑 𝑛 ) ( 𝑝 𝑛 + 𝑗 π‘ž 𝑛 ) . ( 5 . 1 ) 𝜁 𝜁 𝜁 𝜁 ( s i n − 1 π‘₯ ) π‘ž l o g ( 2 s i n h ( πœƒ / 2 ) ) l o g ( 2 s i n ( 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 have which is the result ( 3.17 ). To arrive at the result ( 3.16 ), consider by an allowable change of sum and integral, hence 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 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. <h4>References</h4> A. Sofo, “General properties involving reciprocals of binomial coefficients,” Journal of Integer Sequences , vol. 9, no. 4, Article ID 06.4.5, 13 pages, 2006. A. Sofo, Computational Techniques for the Summation of Series , Kluwer Academic/Plenum Publishers, New York, NY, USA, 2003. A. Sofo, “Integral representations of ratios of binomial coefficients,” International Journal of Pure and Applied Mathematics , vol. 31, no. 1, pp. 29–46, 2006. A. Sofo, “Some properties of reciprocals of double binomial coefficients,” accepted. http://mathworld.wolfram.com/RiemannZetaFunction.html . R. Apéry, “Irrationalitè ζ (2) and ζ (3),” Astérisque , vol. 61, pp. 11–13, 1979. F. Beukers, “ A note on the irrationality of ζ (2) and ζ (3) ,” Bulletin of the London Mathematical Society , vol. 11, no. 3, pp. 268–272, 1979. J. Guillera and J. Sondow, “Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent,” to appear in The Ramanujan Journal . H. Muzaffar, “ Some interesting series arising from the power series expansion of ( sin − 1 x ) q ,” International Journal of Mathematics and Mathematical Sciences , vol. 2005, no. 14, pp. 2329–2336, 2005. S. B. Ekhad and D. Zeilberger, “ A 21st century proof of Dougall's hypergeometric sum identity ,” Journal of Mathematical Analysis and Applications , vol. 147, no. 2, pp. 610–611, 1990. L. J. Slater, Generalized Hypergeometric Functions , Cambridge University Press, Cambridge, UK, 1966. S. R. Finch, Mathematical Constants , vol. 94 of Encyclopedia of Mathematics and Its Applications , Cambridge University Press, Cambridge, UK, 2003. N. Lord, “Problem corner,” The Mathematical Gazette , vol. 89, no. 514, pp. 115–119, 2005. Z. Nan-Yue and K. S. Williams, “Values of the Riemann zeta function and integrals involving log ( 2 sinh ( θ /2 ) ) and log ( 2 sin ( θ /2 ) ) ,” Pacific Journal of Mathematics , vol. 168, no. 2, pp. 271–289, 1995. //

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