New Evaluations of Inverse Binomial Series via Cyclotomic Multiple Zeta Values

John M. Campbell ; M. Lawrence Glasser ; Yajun Zhou

Symmetry, integrability and geometry: methods and applications, Tome 20 (2024)

Résumé

Through the application of an evaluation technique based on cyclotomic multiple zeta values recently due to Au, we solve open problems on inverse binomial series that were included in a 2010 analysis textbook by Chen.

Détail

Mots-clés : binomial coefficients; cyclotomic multiple zeta values; multiple polylogarithms.

Bibliographie

[1] Ablinger J., “Discovering and proving infinite binomial sums identities”, Exp. Math., 26 (2017), 62–71, arXiv: 1507.01703 | DOI | MR | Zbl

[2] Ablinger J., Blümlein J., Raab C.G., Schneider C., “Iterated binomial sums and their associated iterated integrals”, J. Math. Phys., 55 (2014), 112301, 57 pp., arXiv: 1407.1822 | DOI | MR | Zbl

[3] Au K.C., Iterated integrals and multiple polylogarithm at algebraic arguments, arXiv: 2201.01676

[4] Blümlein J., “Algebraic relations between harmonic sums and associated quantities”, Comput. Phys. Comm., 159 (2004), 19–54, arXiv: hep-ph/0311046 | DOI | MR | Zbl

[5] Borwein J., Bailey D., Girgensohn R., Experimentation in mathematics. Computational paths to discovery, A K Peters, Ltd., Natick, MA, 2004 | DOI | MR | Zbl

[6] Borwein J.M., Bradley D.M., Broadhurst D.J., Lisoněk P., “Special values of multiple polylogarithms”, Trans. Amer. Math. Soc., 353 (2001), 907–941, arXiv: math.CA/9910045 | DOI | MR | Zbl

[7] Borwein J.M., Broadhurst D.J., Kamnitzer J., “Central binomial sums, multiple Clausen values, and zeta values”, Experiment. Math., 10 (2001), 25–34, arXiv: hep-th/0004153 | DOI | MR | Zbl

[8] Brown F., On the periods of some Feynman integrals, arXiv: 0910.0114

[9] Brown F., “The massless higher-loop two-point function”, Comm. Math. Phys., 287 (2009), 925–958, arXiv: 0804.1660 | DOI | MR | Zbl

[10] Brown F., “Multiple zeta values and periods of moduli spaces $\overline{\mathfrak M}_{0,n}$”, Ann. Sci. Éc. Norm. Supér., 42 (2009), 371–489, arXiv: math.AG/0606419 | DOI | MR | Zbl

[11] Campbell J.M., Chen K.-W., “Explicit identities for infinite families of series involving squared binomial coefficients”, J. Math. Anal. Appl., 513 (2022), 126219, 23 pp. | DOI | MR | Zbl

[12] Campbell J.M., Levrie P., “Proof of a conjecture due to Chu on Gosper-type sums”, Aequationes Math., 98 (2024), 1071–1079 | DOI | MR

[13] Campbell J.M., Levrie P., Xu C., Zhao J., “On a problem involving the squares of odd harmonic numbers”, Ramanujan J., 63 (2024), 387–408, arXiv: 2206.05026 | DOI | MR | Zbl

[14] Chen H., Excursions in classical analysis. Pathways to advanced problem solving and undergraduate research, Classr. Res. Mater. Ser., Mathematical Association of America, Washington, DC, 2010 | DOI | MR | Zbl

[15] Chu W., “Further Apéry-like series for Riemann zeta function”, Math. Notes, 109 (2021), 136–146 | DOI | MR | Zbl

[16] Chudnovsky D.V., Chudnovsky G.V., “Classification of hypergeometric identities for $\pi$ and other logarithms of algebraic numbers”, Proc. Natl. Acad. Sci. USA, 95 (1998), 2744–2749 | DOI | MR | Zbl

[17] Davydychev A.I., Kalmykov M.Yu., “New results for the $\epsilon$-expansion of certain one-, two- and three-loop Feynman diagrams”, Nuclear Phys. B, 605 (2001), 266–318, arXiv: hep-th/0012189 | DOI | MR | Zbl

[18] Davydychev A.I., Kalmykov M.Yu., “Massive Feynman diagrams and inverse binomial sums”, Nuclear Phys. B, 699 (2004), 3–64, arXiv: hep-th/0303162 | DOI | MR | Zbl

[19] Duhr C., Dulat F., “PolyLogTools – polylogs for the masses”, J. High Energy Phys., 2019:8 (2019), 135, 56 pp., arXiv: 1904.07279 | DOI | MR

[20] Dyson F.J., Frankel N.E., Glasser M.L., “Lehmer's interesting series”, Amer. Math. Monthly, 120 (2013), 116–130, arXiv: 1009.4274 | DOI | MR | Zbl

[21] Glasser M.L., “A generalized Apéry series”, J. Integer Seq., 15 (2012), 12.4.3, 7 pp., arXiv: 1204.1078 | MR | Zbl

[22] Goncharov A.B., “Multiple polylogarithms, cyclotomy and modular complexes”, Math. Res. Lett., 5 (1998), 497–516, arXiv: 1105.2076 | DOI | MR | Zbl

[23] Hoffman M.E., “Quasi-shuffle products”, J. Algebraic Combin., 11 (2000), 49–68, arXiv: math.QA/9907173 | DOI | MR | Zbl

[24] Kalmykov M.Yu., Veretin O., “Single-scale diagrams and multiple binomial sums”, Phys. Lett. B, 483 (2000), 315–323, arXiv: hep-th/0004010 | DOI | MR | Zbl

[25] Kalmykov M.Yu., Ward B.F.L., Yost S.A., “On the all-order $\epsilon$-expansion of generalized hypergeometric functions with integer values of parameters”, J. High Energy Phys., 2007:11 (2007), 009, 13 pp., arXiv: 0708.0803 | DOI | MR | Zbl

[26] Kummer E.E., “Ueber die Transcendenten, welche aus wiederholten Integrationen rationaler Formeln entstehen”, J. Reine Angew. Math., 21 (1840), 74–90 | DOI | MR

[27] Kummer E.E., “Ueber die Transcendenten, welche aus wiederholten Integrationen rationaler Formeln entstehen. (Fortsetzung)”, J. Reine Angew. Math., 21 (1840), 193–225 | DOI | MR

[28] Kummer E.E., “Ueber die Transcendenten, welche aus wiederholten Integrationen rationaler Formeln entstehen. (Fortsetzung)”, J. Reine Angew. Math., 21 (1840), 328–371 | DOI | MR

[29] Laporta S., “High-precision calculation of the 4-loop contribution to the electron $g-2$ in QED”, Phys. Lett. B, 772 (2017), 232–238, arXiv: 1704.06996 | DOI

[30] Laporta S., Remiddi E., “The analytical value of the electron $(g-2)$ at order $\alpha^3$ in QED”, Phys. Lett. B, 379 (1996), 283–291, arXiv: hep-ph/9602417 | DOI

[31] Lehmer D.H., “Interesting series involving the central binomial coefficient”, Amer. Math. Monthly, 92 (1985), 449–457 | DOI | MR | Zbl

[32] Maître D., “HPL, a Mathematica implementation of the harmonic polylogarithms”, Comput. Phys. Comm., 174 (2006), 222–240, arXiv: ; software available at hep-ph/0507152https://www.physik.uzh.ch/data/HPL/ | DOI

[33] Maître D., “Extension of HPL to complex arguments”, Comput. Phys. Comm., 183 (2012), 846, arXiv: hep-ph/0703052 | DOI

[34] Panzer E., “Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals”, Comput. Phys. Comm., 188 (2015), 148–166, arXiv: 1403.3385 | DOI | Zbl

[35] Schnetz O., “The Galois coaction on the electron anomalous magnetic moment”, Commun. Number Theory Phys., 12 (2018), 335–354, arXiv: 1711.05118 | DOI | MR | Zbl

[36] Sun Z.-W., Zhou Y., Evaluations of $\sum_{k=1}^\infty \frac{x^k}{k^2\binom{3k}{k}}$ and related series, arXiv: 2401.12083

[37] van der Poorten A.J., “Some wonderful formulae $\ldots \ $footnotes to Apéry's proof of the irrationality of $\zeta (3)$”, Théorie des Nombres (1978/1979), v. 2, Séminaire Delange-Pisot-Poitou, 20, Secrétariat Math., Paris, 1980, 29, 7 pp. | MR

[38] Wang W., Xu C., “Alternating multiple zeta values, and explicit formulas of some Euler–Apéry-type series”, European J. Combin., 93 (2021), 103283, 25 pp., arXiv: 1909.02943 | DOI | MR | Zbl

[39] Weinzierl S., “Expansion around half-integer values, binomial sums, and inverse binomial sums”, J. Math. Phys., 45 (2004), 2656–2673, arXiv: hep-ph/0402131 | DOI | MR | Zbl

[40] Young P.T., “From Madhava–Leibniz to Lehmer's limit”, Amer. Math. Monthly, 129 (2022), 524–538 | DOI | MR | Zbl

[41] Zhou Y., Hyper-Mahler measures via Goncharov–Deligne cyclotomy, arXiv: 2210.17243

[42] Zhou Y., “Sun's series via cyclotomic multiple zeta values”, SIGMA, 19 (2023), 074, 20 pp., arXiv: 2306.04638 | DOI | MR | Zbl

[43] Zucker I.J., “On the series $\sum^\infty_{k=1}(^{2k}_{\;k})^{-1}k^{-n}$ and related sums”, J. Number Theory, 20 (1985), 92–102 | DOI | MR | Zbl