Computer-Assisted Discovery and Proof

Bailey, David H.; Borwein, Jonathan M.

doi:10.2172/983782

Cited by 6 publications

(5 citation statements)

References 27 publications

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“…Further introductory information on π can be found in the following excellent papers by Borweins [23], Bailey, Borweins and Plouffe [7], Bailey-Borwein [5,6], Baruah, Berndt and Chan [12] and Guillera [40].…”

Section: Introductionmentioning

confidence: 99%

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Accelerating Dougall’s $_5F_4$-sum and infinite series involving $\pi $

Chu

Zhang

2013

Math. Comp.

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The modified Abel lemma on summation by parts is employed to investigate the partial sum of Dougall's 5 H 5 -series. Several unusual transformation formulae into fast convergent series are established. They lead surprisingly to numerous infinite series identities involving π, ζ(3) and the Catalan constant, including several important ones discovered by Ramanujan (1914) and recently by Guillera.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 476 WENCHANG CHU AND WENLONG ZHANGfor convergence. This is not an isolated case. In fact, all 31 principal series for 1/π and 1/π 2 , including Bauer's, collected by Glaisher [32] can be derived from this formula published by Dougall [31] in 1907, two years after Glaisher's paper. Following the same approach of Bauer, Levrie [45] recently derived two further infinite series expressions for 1/π and 1/π 2 ; but both of them result again from special cases of Dougall's summation theorem. Wilf and Zeilberger [53] introduced a new and powerful method, based on Gosper's indefinite summation algorithm [33], for proving identities for hypergeometric series (cf. Petkovšek, Wilf and Zeilberger [46] also). This approach has further been developed intensively by Guillera [39], who reviewed systematically the formulae discovered by Ramanujan [49], found further π-formulae of Ramanujantype and conjectured experimentally the following beautiful and challenging series

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Section: Introductionmentioning

confidence: 99%

“…(−1) n 80 3n 1 2 , 1 3 , 2 3 , 1 6 , 5 6 1, 1, 1, 1, 1 n 29 + 693n + 5418n 2 , For the first four formulae, Guillera [34,38,41,42] himself found a WZ-style proof via Computer Algebra. The other five formulae just displayed remain unproven.…”

Section: Introductionmentioning

confidence: 99%

Accelerating Dougall’s $_5F_4$-sum and infinite series involving $\pi $

Chu

Zhang

2013

Math. Comp.

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“…These relations have been detected with further PSLQ computations [5]. A similar conjecture for integrals I n with increments nπ/60 has also been written [4] (p. 508).…”

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confidence: 57%

“…occurs in a number of contexts and has received significant attention in the last several years [3,4,5,6]. This and related integrals arise in hyperbolic geometry, knot theory, and quantum field theory [6,7,8].…”

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confidence: 99%

Alternative Evaluation of a ln tan Integral Arising in Quantum Field Theory

Coffey

2010

Math Phys Anal Geom

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A certain dilogarithmic integral I 7 turns up in a number of contexts including Feynman diagram calculations, volumes of tetrahedra in hyperbolic geometry, knot theory, and conjectured relations in analytic number theory. We provide an alternative explicit evaluation of a parameterized family of integrals containing this particular case. By invoking the Bloch-Wigner form of the dilogarithm function, we produce an equivalent result, giving a third evaluation of I 7 . We also alternatively formulate some conjectures which we pose in terms of values of the specific Clausen function Cl 2 .

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“…The integral (1.1) has been much discussed of late [4], and in fact it provides the opening equation of a very recent book [3]. The evaluation of I 7 in the specific form of Eq.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%