Abstract
By using the method of iterated integral representations of series, we establish some explicit relationships between multiple zeta values and integrals of logarithmic functions. As applications of these relations, we show that multiple zeta values of the form
for m = 1 or k = 1, and
for p = 1 and 2, satisfy certain recurrence relations which allow us to write them in terms of zeta values, polylogarithms and ln 2. Furthermore, we also obtain reductions for certain multiple polylogarithmic values at
(Communicated by Filippo Nuccio)
Acknowledgement
The author would like to thank the anonymous referee for his/her helpful comments, which improve the presentation of the paper.
References
[1] Alzer, H.—Karayannakis, D.—Srivastava, H. M.: Series representations for some mathematical constants, J. Math. Anal. Appl. 320 (2006), 145–162.10.1016/j.jmaa.2005.06.059Search in Google Scholar
[2] Andrews, G. E.—Askey, R.—Roy, R.: Special Functions, Cambridge University Press, 2000, 481–532.10.1017/CBO9781107325937Search in Google Scholar
[3] Bailey, D. H.—Borwein, J. M.—Girgensohn, R.: Experimental evaluation of Euler sums, Exp. Math. 3 (1994), 17–30.10.1080/10586458.1994.10504573Search in Google Scholar
[4] Bailey, D. H.—Borwein, J. M.—Crandall, R. E.: Computation and theory of extended Mordell-Tornheim-Witten sums, Math. Comp. 83 (2014), 1795–1821.10.1090/S0025-5718-2014-02768-3Search in Google Scholar
[5] Berndt, B. C.: Ramanujan’s Notebooks, Part I, Springer-Verlag, New York, 1985.10.1007/978-1-4612-1088-7Search in Google Scholar
[6] Berndt, B. C.: Ramanujan’s Notebooks, Part II, Springer-Verlag, New York, 1989.10.1007/978-1-4612-4530-8Search in Google Scholar
[7] Blumlein, J.—Kurth, S.: Harmonic sums and Mellin transforms up to two loop order, Phys. Rev. D. 60 (1999), 14–18.10.1103/PhysRevD.60.014018Search in Google Scholar
[8] Borwein, D.—Borwein, J. M.—Girgensohn, R.: Explicit evaluation of Euler sums, Proc. Edinburgh Math. 38 (1995), 277–294.10.1017/S0013091500019088Search in Google Scholar
[9] Borwein, J.—Borwein, P.—Girgensohn, R.—Parnes, S.: Making sense of experimental mathematics, Math. Intell. 18 (1996), 12–18.10.1007/BF03027288Search in Google Scholar
[10] Borwein, J. M.—Bradley, D. M.—Broadhurst, D. J.: Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k, Electron. J. Combin. 4 (1997), 1–21.10.37236/1320Search in Google Scholar
[11] Borwein, J. M.— Bradley, D. M.—Broadhurst, D. J.—Lisoněk, P.: Special values of multiple polylogarithms, Trans. Amer. Math. Soc. 353 (2001), 907–941.10.1090/S0002-9947-00-02616-7Search in Google Scholar
[12] Borwein, J. M.—Zucker, I. J.—Boersma, J.: The evaluation of character Euler double sums, Ramanujan J. 15 (2008), 377–405.10.1007/s11139-007-9083-zSearch in Google Scholar
[13] Borwein, J. M.—Girgensohn, R.: Evaluation of triple Euler sums, Electron. J. Combin. (1996), 2–7.10.37236/1247Search in Google Scholar
[14] Coffey, M. W.: On some log-cosine integrals related to ζ(3), ζ(4), and ζ(6), J. Comput. Appl. Math. 159 (2003), 205–215.10.1016/S0377-0427(03)00438-2Search in Google Scholar
[15] Coffey, M. W.: On one-dimensional digamma and polygamma series related to the evaluation of Feynman diagrams, J. Comput. Appl. Math. 183 (2005), 84–100.10.1016/j.cam.2005.01.003Search in Google Scholar
[16] Coffey, M. W.: On a three-dimensional symmetric Ising tetrahedron and contributions to the theory of the dilogarithm and Clausen functions, J. Math. Phys. 49 (2008), 542–555.10.1063/1.2902996Search in Google Scholar
[17] Coffey, M. W.—Lubbers, N.: On generalized harmonic number sums, Appl. Math. Comput. 217 (2010), 689–698.10.1016/j.amc.2010.06.006Search in Google Scholar
[18] Comtet, L.: Advanced Combinatorics, Boston D Reidel Publishing Company, 1974.10.1007/978-94-010-2196-8Search in Google Scholar
[19] Dil, A.—Boyadzhiev, K. N.: Euler sums of hyperharmonic numbers, J. Number Theory 147 (2015), 490–498.10.1016/j.jnt.2014.07.018Search in Google Scholar
[20] Flajolet, P.—Salvy, B.: Euler sums and contour integral representations, Exp. Math. 7 (1998), 15–35.10.1080/10586458.1998.10504356Search in Google Scholar
[21] Freitas, P.: Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums, Math. Comput. 74 (2005), 1425–1440.10.1090/S0025-5718-05-01747-3Search in Google Scholar
[22] Hessami Pilehrood, Kh.—Hessami Pilehrood, T.—Tauraso, R.: New properties of multiple harmonic sums modulo p and p-analogues of Leshchiner’s series, Trans. Amer. Math. Soc. 366 (2013), 3131–3159.10.1090/S0002-9947-2013-05980-6Search in Google Scholar
[23] Hoffman, M. E.: Multiple harmonic series, Pacific J. Math. 152 (1992), 275–290.10.2140/pjm.1992.152.275Search in Google Scholar
[24] Li, Z.: Another proof of Zagier’s evaluation formula of the multiple zeta values ζ(2, …, 2, 3, 2, …, 2), Math. Res. Lett. 20 (2012).10.4310/MRL.2013.v20.n5.a10Search in Google Scholar
[25] Li, Z.: On functional relations for the alternating analogues of Tornheim’s double zeta function, Chinese Ann. Math. 36 (2015), 907–918.10.1007/s11401-015-0933-5Search in Google Scholar
[26] Markett, C.: Triple sums and the Riemann zeta function, J. Number Theory 48 (1994), 113–132.10.1006/jnth.1994.1058Search in Google Scholar
[27] Machide, T.: Extended double shuffle relations and the generating function of triple zeta values of any fixed weight, Kyushu J. Math. 67 (2013), 281–307.10.2206/kyushujm.67.281Search in Google Scholar
[28] Mezö, I.—Dil, A.: Hyperharmonic series involving Hurwitz zeta function, J. Number Theory 130 (2010), 360–369.10.1016/j.jnt.2009.08.005Search in Google Scholar
[29] Mezö, I.: Nonlinear Euler sums, Pacific J. Math. 272 (2014), 201–226.10.2140/pjm.2014.272.201Search in Google Scholar
[30] Petojevic, A.—Srivastava, H. M.: Computation of the Mordell-Tornheim zeta values, Proc. Amer. Math. Soc. 136 (2008), 2719–2728.10.1090/S0002-9939-08-09350-7Search in Google Scholar
[31] Rassias, T. M.—Srivastava, H. M.: Some classes of infinite series associated with the Riemann zeta function and polygamma functions and generalized harmonic numbers, Appl. Math. Comput. 131 (2002), 593–605.10.1016/S0096-3003(01)00172-2Search in Google Scholar
[32] Sofo, A.: Integral forms of sums associated with harmonic numbers, Appl. Math. Comput. 207 (2009), 365–372.10.1016/j.amc.2008.10.044Search in Google Scholar
[33] Sofo, A.: Harmonic sums and integral representations, J. Appl. Anal. 16 (2010), 265–277.10.1515/jaa.2010.018Search in Google Scholar
[34] Sofo, A.: Quadratic alternating harmonic number sums, J. Number Theory 154 (2015), 144–159.10.1016/j.jnt.2015.02.013Search in Google Scholar
[35] Sofo, A.—Srivastava, H. M.: Identities for the harmonic numbers and binomial coefficients, Ramanujan J. 25 (2011), 93–113.10.1007/s11139-010-9228-3Search in Google Scholar
[36] Xu, C.: Multiple zeta values and Euler sums, J. Number Theory 177 (2017), 443–478.10.1016/j.jnt.2017.01.018Search in Google Scholar
[37] Xu, C.: Identities for the multiple zeta (star) values. Results Math. 73 (2018), 1–22.10.1007/s00025-018-0761-5Search in Google Scholar
[38] Xu, C.—Cheng, J.: Some results on Euler sums, Funct. et Approx. 54 (2016), 25–37.10.7169/facm/2016.54.1.3Search in Google Scholar
[39] Xu, C.—Yan, Y.—Shi, Z.: Euler sums and integrals of polylogarithm functions, J. Number Theory 165 (2016), 84–108.10.1016/j.jnt.2016.01.025Search in Google Scholar
[40] Zagier, D.: Values of zeta functions and their applications, First European Congress of Mathematics, Volume II, Birkhauser, Boston, 120 (1994), 497–512.10.1007/978-3-0348-9112-7_23Search in Google Scholar
[41] Zagier, D.: Evaluation of the multiple zeta values ζ(2, …, 2, 3, 2, …, 2), Ann. Math. 2 (2012), 977–1000.10.4007/annals.2012.175.2.11Search in Google Scholar
[42] Zhao, J.: On a conjecture of Borwein, Bradley and Broadhurst, J. Reine Angew. Math. 639 (2010), 223–233.10.1515/crelle.2010.016Search in Google Scholar
© 2019 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Regular papers
- RNDr. Miloslav Duchoň, DrSc. passed away ∗ aug. 21, 1932 – †dec. 31, 2018
- Prof. RNDr. Gejza Wimmer, DrSc. – A Septuagenerian?
- Two monads on the category of graphs
- Entropy of MV-algebraic dynamical systems: An example
- Going up and lying over in congruence-modular algebras
- The minimal arity of near unanimity polymorphisms
- On the logarithmic derivative of zeta functions for compact even-dimensional locally symmetric spaces of real rank one
- The generalized Fermat Conjecture
- Evaluation of sums involving products of Gaussian q-binomial coefficients with applications
- Integrals of logarithmic functions and alternating multiple zeta values
- An abelian subextension of the dyadic division field of a hyperelliptic Jacobian
- The class of all semigroups related to semihypergroups of order 2
- Criterion for the weak potency of certain HNN extensions
- Fekete Szegö theorem for a close-to-convex error function
- Bounds for distance between eigenvalues of boundary value problems with retarded argument
- Gradient estimates for a nonlinear heat equation under the Finsler-Ricci flow
- Weak module amenability of triangular banach algebras II
- Some consequences of quasicentral approximate units modulo Hilbert-Schmidt class
- Example of C-rigid polytopes which are not B-rigid
- Some comments on rU-spaces
- On the equivalence of various definitions of mixed poisson processes
- A note on discrete C-embedded subspaces
Articles in the same Issue
- Regular papers
- RNDr. Miloslav Duchoň, DrSc. passed away ∗ aug. 21, 1932 – †dec. 31, 2018
- Prof. RNDr. Gejza Wimmer, DrSc. – A Septuagenerian?
- Two monads on the category of graphs
- Entropy of MV-algebraic dynamical systems: An example
- Going up and lying over in congruence-modular algebras
- The minimal arity of near unanimity polymorphisms
- On the logarithmic derivative of zeta functions for compact even-dimensional locally symmetric spaces of real rank one
- The generalized Fermat Conjecture
- Evaluation of sums involving products of Gaussian q-binomial coefficients with applications
- Integrals of logarithmic functions and alternating multiple zeta values
- An abelian subextension of the dyadic division field of a hyperelliptic Jacobian
- The class of all semigroups related to semihypergroups of order 2
- Criterion for the weak potency of certain HNN extensions
- Fekete Szegö theorem for a close-to-convex error function
- Bounds for distance between eigenvalues of boundary value problems with retarded argument
- Gradient estimates for a nonlinear heat equation under the Finsler-Ricci flow
- Weak module amenability of triangular banach algebras II
- Some consequences of quasicentral approximate units modulo Hilbert-Schmidt class
- Example of C-rigid polytopes which are not B-rigid
- Some comments on rU-spaces
- On the equivalence of various definitions of mixed poisson processes
- A note on discrete C-embedded subspaces