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Ohno-type identities for multiple harmonic sums. (English) Zbl 1469.11332

Summary: We establish Ohno-type identities for multiple harmonic \((q\)-)sums which generalize Hoffman’s identity and Bradley’s identity. Our result leads to a new proof of the Ohno-type relation for \(\mathcal{A}\)-finite multiple zeta values recently proved by M. Hirose et al., [“Ohno type relations for classical and finite multiple zeta-star values”, Preprint, arXiv:1806.09299]. As a further application, we give certain sum formulas for \(\mathcal{A}_2\)- or \(\mathcal{A}_3\)-finite multiple zeta values.

MSC:

11M32 Multiple Dirichlet series and zeta functions and multizeta values
11B65 Binomial coefficients; factorials; \(q\)-identities
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References:

[1] S.-J. Bang, J. E. Dawson, A. N. ’t Woord, O. P. Lossers and V. Hernandez, Problems and solutions: Solutions: A reciprocal summation identity: 10490, Amer. Math. Monthly, 106 (1999), 588-590. · doi:10.2307/2589481
[2] D. M. Bradley, Multiple \(q\)-zeta values, J. Algebra, 283 (2005), 752-798. · Zbl 1114.11075 · doi:10.1016/j.jalgebra.2004.09.017
[3] D. M. Bradley, Duality for finite multiple harmonic \(q\)-series, Discrete Math., 300 (2005), 44-56. · Zbl 1085.11041 · doi:10.1016/j.disc.2005.06.008
[4] K. Dilcher, Some \(q\)-series identities related to divisor functions, Discrete Math., 145 (1995), 83-93. · Zbl 0834.05005 · doi:10.1016/0012-365X(95)00092-B
[5] L. Euler, Demonstratio insignis theorematis numerici circa uncias potestatum binomialium, Nova Acta Academiae Scientiarum Imperialis Petropolitanae, 15 (1806), 33-43, reprinted in Opera Omnia, 16, B. G. Teubner, Leipzig, 1935, 104-116.
[6] A. Granville, A decomposition of Riemann’s zeta-function, In: Analytic Number Theory, London Math. Soc. Lecture Note Ser., 247, Cambridge Univ. Press, Cambridge, 1997, 95-101. · Zbl 0907.11024
[7] M. Hirose, K. Imatomi, H. Murahara and S. Saito, Ohno type relations for classical and finite multiple zeta-star values, preprint, arXiv:1806.09299.
[8] M. E. Hoffman, Quasi-symmetric functions and mod \(p\) multiple harmonic sums, Kyushu J. Math., 69 (2015), 345-366. · Zbl 1382.11066 · doi:10.2206/kyushujm.69.345
[9] Y. Ohno, A generalization of the duality and sum formulas on the multiple zeta values, J. Number Theory, 74 (1999), 39-43. · Zbl 0920.11063 · doi:10.1006/jnth.1998.2314
[10] K. Oyama, Ohno-type relation for finite multiple zeta values, Kyushu J. Math., 72 (2018), 277-285. · Zbl 1458.11133 · doi:10.2206/kyushujm.72.277
[11] H. Prodinger, A \(q\)-analogue of a formula of Hernandez obtained by inverting a result of Dilcher, Australas. J. Combin., 21 (2000), 271-274. · Zbl 0951.05006
[12] S. Roman, The harmonic logarithms and the binomial formula, J. Combin. Theory Ser. A, 63 (1993), 143-163. · Zbl 0774.05004 · doi:10.1016/0097-3165(93)90030-C
[13] J. Rosen, Asymptotic relations for truncated multiple zeta values, J. London Math. Soc., 91 (2015), 554-572. · Zbl 1391.11106 · doi:10.1112/jlms/jdu084
[14] S. Saito and N. Wakabayashi, Sum formula for finite multiple zeta values, J. Math. Soc. Japan, 67 (2015), 1069-1076. · Zbl 1329.11093 · doi:10.2969/jmsj/06731069
[15] K. Sakugawa and S. Seki, On functional equations of finite multiple polylogarithms, J. Algebra, 469 (2017), 323-357. · Zbl 1406.11088 · doi:10.1016/j.jalgebra.2016.07.035
[16] S. Seki, The \(\boldsymbol{p} \)-adic duality for the finite star-multiple polylogarithms, Tohoku Math. J. (2), 71 (2019), 111-122. · Zbl 1443.11182 · doi:10.2748/tmj/1552100444
[17] S. Seki and S. Yamamoto, A new proof of the duality of multiple zeta values and its generalizations, Int. J. Number Theory, 15 (2019), 1261-1265. · Zbl 1443.11183 · doi:10.1142/S1793042119500702
[18] Z.-H. Sun, Congruences concerning Bernoulli numbers and Bernoulli polynomials, Discrete Appl. Math., 105 (2000), 193-223. · Zbl 0990.11008 · doi:10.1016/S0166-218X(00)00184-0
[19] L. Van Hamme, Advanced problem: 6407, Amer. Math. Monthly, 89 (1982), 703-704.
[20] J. Zhao, Wolstenholme type theorem for multiple harmonic sums, Int. J. Number Theory, 4 (2008), 73-106. · Zbl 1218.11005 · doi:10.1142/S1793042108001146
[21] J. Zhao, Finite multiple zeta values and finite Euler sums, preprint, arXiv:1507.04917.
[22] X. · Zbl 1115.11006 · doi:10.1090/S0002-9939-06-08777-6
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