Evaluations of sums involving harmonic numbers and binomial coefficients

References

• D.H. Bailey, J.M. Borwein and D.M. Bradley, Experimental determination of Apéry-like identities for ζ(2n+2), Exp. Math. 15(3) (2006), pp. 281–289. doi: 10.1080/10586458.2006.10128968
   Web of Science ®Google Scholar
• D.H. Bailey, J.M. Borwein and R. Girgensohn, Experimental evaluation of Euler sums, Exp. Math. 3(1) (1994), pp. 17–30. doi: 10.1080/10586458.1994.10504573
   Google Scholar
• B.C. Berndt, Ramanujan's Notebooks. Part I, Springer-Verlag, New York, 1985.
   Google Scholar
• D. Borwein, J.M. Borwein and R. Girgensohn, Explicit evaluation of Euler sums, Proc. Edinburgh Math. Soc. 38(2) (1995), pp. 277–294. doi: 10.1017/S0013091500019088
   Web of Science ®Google Scholar
• J.M. Borwein and D.M. Bradley, Thirty-two Goldbach variations, Int. J. Number Theory 2(1) (2006), pp. 65–103. doi: 10.1142/S1793042106000383
   Web of Science ®Google Scholar
• L. Comtet, Advanced Combinatorics, D. Reidel Publishing Co., Dordrecht, 1974.
   Google Scholar
• P. Flajolet and B. Salvy, Euler sums and contour integral representations, Exp. Math. 7(1) (1998), pp. 15–35. doi: 10.1080/10586458.1998.10504356
   Web of Science ®Google Scholar
• Kh. Hessami Pilehrood, T. Hessami Pilehrood and R. Tauraso, New properties of multiple harmonic sums modulo p and p-analogues of Leshchiner's series, Trans. Am. Math. Soc. 366(6) (2014), pp. 3131–3159. doi: 10.1090/S0002-9947-2013-05980-6
   Web of Science ®Google Scholar
• M.E. Hoffman, Multiple harmonic series, Pacific J. Math. 152(2) (1992), pp. 275–290. doi: 10.2140/pjm.1992.152.275
   Web of Science ®Google Scholar
• M. Kaneko and Y. Ohno, On a kind of duality of multiple zeta-star values, Int. J. Number Theory 6(8) (2010), pp. 1927–1932. doi: 10.1142/S179304211000385X
   Web of Science ®Google Scholar
• I. Mez, Nonlinear Euler sums, Pacific J. Math. 272(1) (2014), pp. 201–226. doi: 10.2140/pjm.2014.272.201
   Web of Science ®Google Scholar
• M. Petitot, Table des Multiple Zeta Values Jusqu'au Poids 16, in http://cristal.univ-lille.fr/∼petitot/recherche/MZV/mzv16.pdf. 2009.
   Google Scholar
• J. Riordan, An Introduction to Combinatorial Analysis, Reprint of the 1958 original, Dover Publications, Inc., Mineola, NY, 2002.
   Google Scholar
• S.M. Ripon, Generalization of harmonic sums involving inverse binomial coefficients, Integral Trans. Spec. Funct. 25(10) (2014), pp. 821–835. doi: 10.1080/10652469.2014.928705
   Web of Science ®Google Scholar
• X. Si, C. Xu and M. Zhang, Quadratic and cubic harmonic number sums, J. Math. Anal. Appl. 447(1) (2017), pp. 419–434. doi: 10.1016/j.jmaa.2016.10.026
   Web of Science ®Google Scholar
• A. Sofo, Harmonic number sums in closed form, Math. Commun. 16(2) (2011), pp. 335–345.
   Web of Science ®Google Scholar
• A. Sofo, Harmonic number sums in higher powers, J. Math. Anal. 2(2) (2011), pp. 15–22.
   Google Scholar
• A. Sofo, Harmonic numbers and cubed binomial coefficients, Int. J. Comb. 2011 (2014), pp. 14. Art. ID 208260
   Google Scholar
• A. Sofo, Harmonic numbers of order two, Miskolc Math. Notes 13(2) (2012), pp. 499–514. doi: 10.18514/MMN.2012.430
   Web of Science ®Google Scholar
• A. Sofo, Harmonic sums and rational multiples of zeta functions, Bol. Soc. Mat. Mexicana (3) 17(1) (2011), pp. 1–12.
   Google Scholar
• A. Sofo, Identities for Reciprocal Binomials, in Analytic Number Theory, Approximation Theory, and Special Functions, Springer, New York, pp. 227–238.2014.
   Google Scholar
• A. Sofo, Integral identities for rational series involving binomial coefficients, Bull. Malays. Math. Sci. Soc. (2) 34(3) (2011), pp. 631–637.
   Web of Science ®Google Scholar
• A. Sofo, Mixed binomial sum identities, Integral Trans. Spec. Funct. 24(3) (2013), pp. 187–200. doi: 10.1080/10652469.2012.685937
   Web of Science ®Google Scholar
• A. Sofo, Quadratic alternating harmonic number sums, J. Number Theory 154 (2015), pp. 144–159. doi: 10.1016/j.jnt.2015.02.013
   Web of Science ®Google Scholar
• A. Sofo, Summation formula involving harmonic numbers, Anal. Math. 37(1) (2011), pp. 51–64. doi: 10.1007/s10476-011-0103-2
   Web of Science ®Google Scholar
• A. Sofo and D. Cvijović, Extensions of Euler harmonic sums, Appl. Anal. Discrete Math. 6(2) (2012), pp. 317–328. doi: 10.2298/AADM120628016S
   Google Scholar
• A. Sofo and H.M. Srivastava, Identities for the harmonic numbers and binomial coefficients, Ramanujan J. 25(1) (2011), pp. 93–113. doi: 10.1007/s11139-010-9228-3
   Web of Science ®Google Scholar
• H.M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Inc., Amsterdam, 2012.
   Google Scholar
• W. Wang and Y. Lyu, Euler sums and Stirling sums, J. Number Theory 185 (2018), pp. 160–193. doi: 10.1016/j.jnt.2017.08.037
   Web of Science ®Google Scholar
• W. Wang and C. Xu, Euler sums of weights 10 and 11, and some special types, Technical Report, https://www.researchgate.net/publication/318504251. 2017.
   Google Scholar
• C. Xu, Multiple zeta values and Euler sums, J. Number Theory 177 (2017), pp. 443–478. doi: 10.1016/j.jnt.2017.01.018
   Web of Science ®Google Scholar
• C. Xu, Some evaluation of parametric Euler sums, J. Math. Anal. Appl. 451(2) (2017), pp. 954–975. doi: 10.1016/j.jmaa.2017.02.047
   Web of Science ®Google Scholar
• C. Xu and Z. Li, Tornheim type series and nonlinear Euler sums, J. Number Theory 174 (2017), pp. 40–67. doi: 10.1016/j.jnt.2016.10.002
   Web of Science ®Google Scholar
• C. Xu and W. Wang, Evaluations of sums involving harmonic numbers and binomial coefficients, Technical Report, https://www.researchgate.net/publication/3226755972018.
   Google Scholar
• C. Xu and W. Wang, Explicit formulas of Euler sums via multiple zeta values, J. Symbolic Comput. (2019). https://doi.org/10.1016/j.jsc.2019.06.009.
   Google Scholar
• C. Xu, Y. Yan and Z. Shi, Euler sums and integrals of polylogarithm functions, J. Number Theory165 (2016), pp. 84–108. doi: 10.1016/j.jnt.2016.01.025
   Web of Science ®Google Scholar
• C. Xu, M. Zhang and W. Zhu, Some evaluation of harmonic number sums, Integral Trans. Spec. Funct.27(12) (2016), pp. 937–955. doi: 10.1080/10652469.2016.1231675
   Web of Science ®Google Scholar
• C. Yamazaki, On the duality for multiple zeta-star values of height one, Kyushu J. Math. 64(1) (2010), pp. 145–152. doi: 10.2206/kyushujm.64.145
   Web of Science ®Google Scholar
• J. Yang and Y. Wang, Summation formulae in relation to Euler sums, Integral Trans. Spec. Funct.28(5) (2017), pp. 336–349. doi: 10.1080/10652469.2017.1293669
   Web of Science ®Google Scholar