Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 074, 20 pages      arXiv:2306.04638      https://doi.org/10.3842/SIGMA.2023.074
Contribution to the Special Issue on Asymptotics and Applications of Special Functions in Memory of Richard Paris

Sun's Series via Cyclotomic Multiple Zeta Values

Yajun Zhou ab
a) Program in Applied and Computational Mathematics (PACM), Princeton University, Princeton, NJ 08544, USA
b) Academy of Advanced Interdisciplinary Studies (AAIS), Peking University, Beijing 100871, P.R. China

Received June 13, 2023, in final form September 29, 2023; Published online October 12, 2023

Abstract
We prove and generalize several recent conjectures of Z.-W. Sun surrounding binomial coefficients and harmonic numbers. We show that Sun's series and their analogs can be represented as cyclotomic multiple zeta values of levels $N\in\{4,8,12,16,24\}$, namely Goncharov's multiple polylogarithms evaluated at $N$-th roots of unity.

Key words: Sun's series; binomial coefficients; harmonic numbers; cyclotomic multiple zeta values.

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References

  1. Ablinger J., Discovering and proving infinite binomial sums identities, Exp. Math. 26 (2017), 62-71, arXiv:1507.01703.
  2. Ablinger J., Discovering and proving infinite Pochhammer sum identities, Exp. Math. 31 (2022), 309-323, arXiv:1902.11001.
  3. Ablinger J., Blümlein J., Raab C.G., Schneider C., Iterated binomial sums and their associated iterated integrals, J. Math. Phys. 55 (2014), 112301, 57 pages, arXiv:1407.1822.
  4. Ablinger J., Blümlein J., Schneider C., Harmonic sums and polylogarithms generated by cyclotomic polynomials, J. Math. Phys. 52 (2011), 102301, 52 pages, arXiv:1105.6063.
  5. Ablinger J., Blümlein J., Schneider C., Analytic and algorithmic aspects of generalized harmonic sums and polylogarithms, J. Math. Phys. 54 (2013), 082301, 74 pages, arXiv:1302.0378.
  6. Au K.C., Evaluation of one-dimensional polylogarithmic integral, with applications to infinite series, arXiv:2007.03957.
  7. Au K.C., Iterated integrals and multiple polylogarithm at algebraic arguments, arXiv:2201.01676, Software available at https://www.researchgate.net/publication/357601353_Mathematica_package_% MultipleZetaValues and https://www.wolframcloud.com/obj/pisco125/MultipleZetaValues-1.2.0.paclet.
  8. Charlton S., Gangl H., Lai L., Xu C., Zhao J., On two conjectures of Sun concerning Apéry-like series, Forum Math., to appear, arXiv:2210.14704.
  9. Davydychev A.I., Kalmykov M.Yu., New results for the $\varepsilon$-expansion of certain one-, two- and three-loop Feynman diagrams, Nuclear Phys. B 605 (2001), 266-318, arXiv:hep-th/0012189.
  10. Davydychev A.I., Kalmykov M.Yu., Massive Feynman diagrams and inverse binomial sums, Nuclear Phys. B 699 (2004), 3-64, arXiv:hep-th/0303162.
  11. Frellesvig H., Tommasini D., Wever C., On the reduction of generalized polylogarithms to $\mathrm{Li}_ n$ and $\mathrm{Li}_{2, 2}$ and on the evaluation thereof, J. High Energy Phys. 2016 (2016), no. 3, 189, 35 pages, arXiv:1601.02649v3.
  12. Goncharov A.B., The double logarithm and Manin's complex for modular curves, Math. Res. Lett. 4 (1997), 617-636.
  13. Goncharov A.B., Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998), 497-516, arXiv:1105.2076.
  14. Gradshteyn I.S., Ryzhik I.M., Table of integrals, series, and products, Elsevier, Amsterdam, 2007.
  15. Kalmykov M.Yu., Veretin O., Single-scale diagrams and multiple binomial sums, Phys. Lett. B 483 (2000), 315-323, arXiv:hep-th/0004010.
  16. Kalmykov M.Yu., Ward B.F.L., Yost S.A., Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order $\varepsilon$-expansion of generalized hypergeometric functions, J. High Energy Phys. 2007 (2007), no. 10, 048, 26 pages, arXiv:0707.3654.
  17. Mező I., Nonlinear Euler sums, Pacific J. Math. 272 (2014), 201-226.
  18. Panzer E., Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals, Comput. Phys. Commun. 188 (2015), 148-166, arXiv:1403.3385.
  19. Sun Z.-W., Series with summands involving harmonic numbers, in Combinatorial and Additive Number Theory, to appear, arXiv:2210.07238v8.
  20. Weinzierl S., Expansion around half-integer values, binomial sums, and inverse binomial sums, J. Math. Phys. 45 (2004), 2656-2673, arXiv:hep-ph/0402131.
  21. Xu C., Zhao J., Sun's three conjectures on Apéry-like sums involving harmonic numbers, 2022, arXiv:2203.04184.
  22. Xu C., Zhao J., A note on Sun's conjectures on Apéry-like sums involving Lucas sequences and harmonic numbers, arXiv:2204.08277.
  23. Zhou Y., Hyper-Mahler measures via Goncharov-Deligne cyclotomy, arXiv:2210.17243v3.

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