OFFSET
1,1
COMMENTS
From Amiram Eldar, Dec 14 2025: (Start)
This series is called "the quadratic series of Au-Yeung", after Enrico Au-Yeung who discovered numerically the value of the sum on April 1993, when he was an undergraduate student in the Faculty of Mathematics in Waterloo. His conjecture was proved by David Borwein and Jonathan Borwein (1995).
An earlier proof was given by the German mathematician Martin Kneser (1928-2004) (1950) as a solution to the problem proposed by the Irish mathematician Henry Francis (Harry) Sandham (1917-1963) (1948).
This series is called "Sandham-Yeung Series" by Sîntămărian and Furdui (2021). (End)
REFERENCES
Jörg Arndt and Christoph Haenel, Pi - Unleashed, Springer Berlin, Heidelberg, 2001, p. 235, eq. (16.143).
Jonathan Borwein, David Bailey, and Roland Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A K Peters, Natick, MA, 2004, pp. 173-174.
Jonathan Borwein and Keith Devlin, The Computer as Crucible: An Introduction to Experimental Mathematics, A K Peters, 2009, p. 62.
Hongwei Chen, Excursions in Classical Analysis, MAA, 2010, p. 190.
Hongwei Chen, Monthly Problem Gems, CRC Press, 2021, Problem 2.2, "Old wine in a new bottle", pp. 57-70.
Richard E. Crandall, Topics in Advanced Scientific Computation, Springer-Verlag, New York, 1996, p. 80.
Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See section 3.70, pp. 150, 166, and 217.
Paul J. Nahin, In Pursuit of Zeta-3, Princeton University Press, 2021, pp. 258-259.
LINKS
David H. Bailey, Finding new mathematical identities via numerical computations, ACM SIGNUM Newsletter, Vol. 33, No. 1 (1998), pp. 17-22.
David H. Bailey and Jonathan M. Borwein, Experimental Mathematics: Examples, Methods and Implications, Notices of the American Mathematical Society, Vol. 52, No. 5 (2005), pp. 502-514, see Euler's Multi-Zeta Sums, p. 506; preprint.
David H. Bailey, Jonathan M. Borwein and Roland Girgensohn, Experimental evaluation of Euler sums, Experimental Mathematics, Vol. 3, No. 1 (1994), pp. 17-30; alternative link.
Narendra Bhandari and Yogesh Joshi, A new proof of quadratic series of Au-Yeung and explicit evaluation of its alternating sum, Indian J. Pure Appl. Math., Vol. 55 (2024), pp. 1251-1260.
David Borwein and Jonathan M. Borwein, On an intriguing integral and some series related to zeta(4), Proceedings of the American Mathematical Society, Vol. 123, No. 4 (1995), pp. 1191-1198.
Jonathan Michael Borwein, Exploratory Experimentation: Digitally-Assisted Discovery and Proof, in: G. Hanna and M. de Villiers (eds.), Proof and Proving in Mathematics Education, New ICMI Study Series, Vol 15. Springer, Dordrecht, 2012, pp. 69-96, see p. 81.
Joseph Breen, Some Very Challenging Calculus Problems. [Wayback Machine link]
Dario Castellanos, The Ubiquitous Pi, Mathematics Magazine, Vol. 61, No. 2 (1988), pp. 67-98. See p. 86.
Philippe Flajolet and Bruno Salvy, Euler sums and contour integral representations, Experimental Mathematics, Vol. 7, No. 1 (1998), pp. 15-35; alternative link.
Pedro Freitas, Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums, Mathematics of Computation, Vol. 74, No. 251 (2005), pp. 1425-1440. See p. 1439.
Ovidiu Furdui, Series Involving Products of Two Harmonic Numbers, Mathematics Magazine, Vol. 84, No. 5 (2011), pp. 371-377.
Ovidiu Furdui and Alina Sîntămărian, A new proof of the quadratic series of Au-Yeung, Gazeta Matematică, Seria A, Vol. 37 (116), No. 1-2 (2019), pp. 1-6.
Ovidiu Furdui and Alina Sîntămărian, Two New Proofs of Shandam-Yeung Series, Gazeta Matematică, Seria A, Vol. 39 (118), No. 1-2 (2021), pp. 6-15.
J. C. Rainwater, Evaluation of frequency sums for the free energy of superfluid ^3He, Phys. Rev. B, Vol. 18 (1978), pp. 3728-3729.
H. F. Sandham, proposer, Problem 4305, Advanced Problems and Solution, The American Mathematical Monthly, Vol. 55, No. 7 (1948), p. 431; A Summation Problem, Solution to Problem 4305 by Martin Kneser, ibid., Vol. 57, No. 4 (1950), pp. 267-268.
Alina Sîntămărian and Ovidiu Furdui, An Artistry of Quadratic Series: Two New Proofs of Sandham-Yeung Series, in: Sharpening Mathematical Analysis Skills, Problem Books in Mathematics, Springer, Cham, 2021, pp. 245-249.
Seán Mark Stewart, Explicit Evaluation of Some Quadratic Euler-Type Sums Containing Double-Index Harmonic Numbers, Tatra Mountains Mathematical Publications, Vol. 77, No. 1 (2020), pp. 73-98.
Seán M. Stewart, The Shadow of Euler's Greatness: Adventures in the Rediscovery of an Intriguing Sum, Math. Intelligencer, Vol. 43 (2021), pp. 82-91.
Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer International Publishing, 2019, section 6.22, pp. 392-393.
Cornel Ioan Vălean and Ovidiu Furdui, Reviving the quadratic series of Au-Yeung, J. Class. Anal., Vol. 6, No. 2 (2015), pp. 113-118.
Eric Weisstein's World of Mathematics, Euler Sum. See eq. (3).
FORMULA
Equals 17*zeta(4)/4.
Equals 17*Pi^4/360.
Equals (17/4) * Sum_{k>=1} 1/k^4.
Equals (17/(22*Pi)) * Integral_{t=0..Pi} (Pi-t)^2*log(2*sin(t/2))^2 dt.
Equals (17/11)*A241753. - Amiram Eldar, Dec 14 2025
Equals Integral_{x=0..1, y=0..1} log(1-x)*log(1-y)/(1-x*y) dx dy (Freitas, 2005, p. 1439). - Amiram Eldar, Jun 30 2026
EXAMPLE
4.5998737432723373139430157102999635867926915456545893...
MATHEMATICA
17*Pi^4/360 // N[#, 100] & // RealDigits // First
PROG
(PARI) 17*Pi^4/360 \\ Charles R Greathouse IV, Sep 02 2024
CROSSREFS
KEYWORD
AUTHOR
Jean-François Alcover, Mar 28 2013
EXTENSIONS
Offset corrected by Rick L. Shepherd, Jan 01 2014
STATUS
approved