Signed 𝑞-analogs of Tornheim’s double series

Zhou, Xia; Cai, Tianxin; Bradley, David

doi:10.1090/S0002-9939-08-09208-3

Signed $q$-analogs of Tornheim’s double series
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by Xia Zhou, Tianxin Cai and David M. Bradley PDF

Proc. Amer. Math. Soc. 136 (2008), 2689-2698 Request permission

Abstract:

We introduce signed $q$-analogs of Tornheim’s double series and evaluate them in terms of double $q$-Euler sums. As a consequence, we provide explicit evaluations of signed and unsigned Tornheim double series and correct some mistakes in the literature.

References

Bruce C. Berndt, Ramanujan’s notebooks. Part I, Springer-Verlag, New York, 1985. With a foreword by S. Chandrasekhar. MR 781125, DOI 10.1007/978-1-4612-1088-7
David Borwein, Jonathan M. Borwein, and David M. Bradley, Parametric Euler sum identities, J. Math. Anal. Appl. 316 (2006), no. 1, 328–338. MR 2201764, DOI 10.1016/j.jmaa.2005.04.040
Jonathan M. Borwein and David M. Bradley, Thirty-two Goldbach variations, Int. J. Number Theory 2 (2006), no. 1, 65–103. MR 2217795, DOI 10.1142/S1793042106000383
J. M. Borwein, D. M. Bradley, and D. J. Broadhurst, Evaluations of $k$-fold Euler/Zagier sums: a compendium of results for arbitrary $k$, Electron. J. Combin. 4 (1997), no. 2, Research Paper 5, approx. 21. The Wilf Festschrift (Philadelphia, PA, 1996). MR 1444152
Jonathan M. Borwein, David M. Bradley, David J. Broadhurst, and Petr Lisoněk, Combinatorial aspects of multiple zeta values, Electron. J. Combin. 5 (1998), Research Paper 38, 12. MR 1637378
Jonathan M. Borwein, David M. Bradley, David J. Broadhurst, and Petr Lisoněk, Special values of multiple polylogarithms, Trans. Amer. Math. Soc. 353 (2001), no. 3, 907–941. MR 1709772, DOI 10.1090/S0002-9947-00-02616-7
Douglas Bowman and David M. Bradley, Multiple polylogarithms: a brief survey, $q$-series with applications to combinatorics, number theory, and physics (Urbana, IL, 2000) Contemp. Math., vol. 291, Amer. Math. Soc., Providence, RI, 2001, pp. 71–92. MR 1874522, DOI 10.1090/conm/291/04893
Douglas Bowman and David M. Bradley, The algebra and combinatorics of shuffles and multiple zeta values, J. Combin. Theory Ser. A 97 (2002), no. 1, 43–61. MR 1879045, DOI 10.1006/jcta.2001.3194
Douglas Bowman and David M. Bradley, Resolution of some open problems concerning multiple zeta evaluations of arbitrary depth, Compositio Math. 139 (2003), no. 1, 85–100. MR 2024966, DOI 10.1023/B:COMP.0000005036.52387.da
Douglas Bowman, David M. Bradley, and Ji Hoon Ryoo, Some multi-set inclusions associated with shuffle convolutions and multiple zeta values, European J. Combin. 24 (2003), no. 1, 121–127. MR 1957970, DOI 10.1016/S0195-6698(02)00117-8
David M. Bradley, Partition identities for the multiple zeta function, Zeta functions, topology and quantum physics, Dev. Math., vol. 14, Springer, New York, 2005, pp. 19–29. MR 2179270, DOI 10.1007/0-387-24981-8_{2}
David M. Bradley, Multiple $q$-zeta values, J. Algebra 283 (2005), no. 2, 752–798. MR 2111222, DOI 10.1016/j.jalgebra.2004.09.017
David M. Bradley, Duality for finite multiple harmonic $q$-series, Discrete Math. 300 (2005), no. 1-3, 44–56. MR 2170113, DOI 10.1016/j.disc.2005.06.008
—, On the sum formula for multiple $q$-zeta values, Rocky Mountain J. Math., to appear. http://arxiv.org/abs/math.QA/0411274
David M. Bradley, A $q$-analog of Euler’s decomposition formula for the double zeta function, Int. J. Math. Math. Sci. 21 (2005), 3453–3458. MR 2206867, DOI 10.1155/IJMMS.2005.3453
D. J. Broadhurst and D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to $9$ loops, Phys. Lett. B 393 (1997), no. 3-4, 403–412. MR 1435933, DOI 10.1016/S0370-2693(96)01623-1
Francis C. S. Brown, Périodes des espaces des modules $\overline {\mathfrak {M}}_{0,n}$ et valeurs zêtas multiples, C. R. Math. Acad. Sci. Paris 342 (2006), no. 12, 949–954 (French, with English and French summaries). MR 2235616, DOI 10.1016/j.crma.2006.04.016
Pierre Cartier, Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents, Astérisque 282 (2002), Exp. No. 885, viii, 137–173 (French, with French summary). Séminaire Bourbaki, Vol. 2000/2001. MR 1975178
Pierre Cartier, Values of the $\zeta$-function, Surveys in modern mathematics, London Math. Soc. Lecture Note Ser., vol. 321, Cambridge Univ. Press, Cambridge, 2005, pp. 260–273. MR 2166932, DOI 10.1017/CBO9780511614156.013
L. Euler, Meditationes Circa Singulare Serierum Genus, Novi Comm. Acad. Sci. Petropol., 20 (1775), 140–186. Reprinted in “Opera Omnia,” ser. I, 15, B. G. Teubner, Berlin (1927), 217–267.
Leonhardus Eulerus, Commercium epistolicum, Leonhardi Euleri Opera Omnia (Series Quarta A: Commercium Epistolicum), Volumen I, Birkhäuser, Basel, 1975 (German). Descriptio commercii epistolici; Ediderunt Adolf P. Juškevič, Vladimir I. Smirnov et Walter Habicht; Auxilio Johann Jakob Burckhardt, Joachim Otto Fleckenstein et Ašot T. Grigorijan; Sub Auspiciis Societatis Scientiarum Naturalium Helveticae et Academiae Scientiarum URSS edenda curaverunt Charles Blanc, Ašot T. Grigorijan, Walter Habicht, Adolf P. Juškevič, Vladimir I. Smirnov, Ernst Trost. MR 0497632
L. Euler and C. Goldbach, Briefwechsel 1729–1764, Akademie-Verlag, Berlin, 1965.
Philippe Flajolet and Bruno Salvy, Euler sums and contour integral representations, Experiment. Math. 7 (1998), no. 1, 15–35. MR 1618286
Michael E. Hoffman, Periods of mirrors and multiple zeta values, Proc. Amer. Math. Soc. 130 (2002), no. 4, 971–974. MR 1873769, DOI 10.1090/S0002-9939-01-06263-3
James G. Huard, Kenneth S. Williams, and Nan-Yue Zhang, On Tornheim’s double series, Acta Arith. 75 (1996), no. 2, 105–117. MR 1379394, DOI 10.4064/aa-75-2-105-117
Kentaro Ihara and Takashi Takamuki, The quantum $\mathfrak {g}_2$ invariant and relations of multiple zeta values, J. Knot Theory Ramifications 10 (2001), no. 7, 983–997. MR 1867104, DOI 10.1142/S0218216501001293
Tu Quoc Thang Le and Jun Murakami, Kontsevich’s integral for the Homfly polynomial and relations between values of multiple zeta functions, Topology Appl. 62 (1995), no. 2, 193–206. MR 1320252, DOI 10.1016/0166-8641(94)00054-7
M. V. Subbarao and R. Sitaramachandra Rao, On some infinite series of L. J. Mordell and their analogues, Pacific J. Math. 119 (1985), no. 1, 245–255. MR 797027
Tomohide Terasoma, Selberg integrals and multiple zeta values, Compositio Math. 133 (2002), no. 1, 1–24. MR 1918286, DOI 10.1023/A:1016377828316
Tomohide Terasoma, Mixed Tate motives and multiple zeta values, Invent. Math. 149 (2002), no. 2, 339–369. MR 1918675, DOI 10.1007/s002220200218
Tomohide Terasoma, Period integrals and multiple zeta values, Sūgaku 57 (2005), no. 3, 255–266 (Japanese). MR 2163671
Leonard Tornheim, Harmonic double series, Amer. J. Math. 72 (1950), 303–314. MR 34860, DOI 10.2307/2372034
Hirofumi Tsumura, On some combinatorial relations for Tornheim’s double series, Acta Arith. 105 (2002), no. 3, 239–252. MR 1931792, DOI 10.4064/aa105-3-3
Hirofumi Tsumura, On alternating analogues of Tornheim’s double series, Proc. Amer. Math. Soc. 131 (2003), no. 12, 3633–3641. MR 1998168, DOI 10.1090/S0002-9939-03-07186-7
Hirofumi Tsumura, Evaluation formulas for Tornheim’s type of alternating double series, Math. Comp. 73 (2004), no. 245, 251–258. MR 2034120, DOI 10.1090/S0025-5718-03-01572-2
Michel Waldschmidt, Valeurs zêta multiples. Une introduction, J. Théor. Nombres Bordeaux 12 (2000), no. 2, 581–595 (French, with English and French summaries). Colloque International de Théorie des Nombres (Talence, 1999). MR 1823204
M. Waldschmidt, Multiple polylogarithms: an introduction, Number theory and discrete mathematics (Chandigarh, 2000) Trends Math., Birkhäuser, Basel, 2002, pp. 1–12. MR 1952273
V. V. Zudilin, Algebraic relations for multiple zeta values, Uspekhi Mat. Nauk 58 (2003), no. 1(349), 3–32 (Russian, with Russian summary); English transl., Russian Math. Surveys 58 (2003), no. 1, 1–29. MR 1992130, DOI 10.1070/RM2003v058n01ABEH000592