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Keywords:
Rota-Baxter operator; Bernoulli number; Bernoulli polynomial
Summary:
We develop the connection between Rota-Baxter operators arisen from algebra and mathematical physics and Bernoulli polynomials. We state that a trivial property of Rota-Baxter operators implies the symmetry of the power sum polynomials and Bernoulli polynomials. We show how Rota-Baxter operators equalities rewritten in terms of Bernoulli polynomials generate identities for the latter.
References:
[1] Agoh, T.: On Bernoulli numbers, I. C. R. Math. Rep. Acad. Sci. Canada, 10, 1988, 7-12, MR 0925293
[2] Agoh, T.: Convolution identities for Bernoulli and Genocchi polynomials. Electron. J. Comb., 21, 1, 2014, 1-14, MR 3192396
[3] Agoh, T., Dilcher, K.: Integrals of products of Bernoulli polynomials. J. Math. Anal. Appl., 381, 1, 2011, 10-16, Elsevier, DOI 10.1016/j.jmaa.2011.03.061 | MR 2796188
[4] Aguiar, M.: Pre-Poisson algebras. Lett. Math. Phys., 54, 4, 2000, 263-277, Springer, DOI 10.1023/A:1010818119040 | MR 1846958
[5] Atkinson, F.V.: Some aspects of Baxter's functional equation. J. Math. Anal. Appl., 7, 1, 1963, 1-30, Elsevier, DOI 10.1016/0022-247X(63)90075-1 | MR 0155196
[6] Baxter, G.: An analytic problem whose solution follows from a simple algebraic identity. Pacific J. Math, 10, 3, 1960, 731-742, DOI 10.2140/pjm.1960.10.731 | MR 0119224
[7] Belavin, A.A., Drinfel'd, V.G.: Solutions of the classical Yang-Baxter equation for simple Lie algebras. Funct. Anal. Appl., 16, 3, 1982, 159-180, DOI 10.1007/BF01081585 | MR 0674005
[8] Carlitz, L.: Note on the integral of the product of several Bernoulli polynomials. J. London Math. Soc., s1-34, 3, 1959, 361-363, Narnia, DOI 10.1112/jlms/s1-34.3.361 | MR 0107022
[9] Cartier, P.: On the structure of free Baxter algebras. Adv. Math., 9, 2, 1972, 253-265, Academic Press, DOI 10.1016/0001-8708(72)90018-7 | MR 0338040
[10] Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann--Hilbert problem I: The Hopf algebra structure of graphs and the main theorem. Commun. Math. Phys., 210, 1, 2000, 249-273, Springer, DOI 10.1007/s002200050779 | MR 1748177
[11] Ebrahimi-Fard, K.: Loday-type algebras and the Rota-Baxter relation. Lett. Math. Phys., 61, 2, 2002, 139-147, Springer, DOI 10.1023/A:1020712215075 | MR 1936573
[12] Ebrahimi-Fard, K.: Rota-Baxter algebras and the Hopf algebra of renormalization. 2006, Ph.D. Thesis, University of Bonn.
[13] Ebrahimi-Fard, K., Guo, L.: Multiple zeta values and Rota-Baxter algebras. Integers, 8, 2--A4, 2008, 1-18, MR 2438289
[14] Gessel, I.M.: On Miki's identity for Bernoulli numbers. J. Number Theory, 110, 1, 2005, 75-82, Elsevier, DOI 10.1016/j.jnt.2003.08.010 | MR 2114674
[15] Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics: a foundation for computer science. 1994, Addison-Wesley Professional, Reading (MA, USA), Second ed. MR 1397498
[16] Gubarev, V.: Rota-Baxter operators on unital algebras. Mosc. Math. J., (Accepted) Preprint arXiv:1805.00723v3.
[17] Gubarev, V., Kolesnikov, P.: Embedding of dendriform algebras into Rota-Baxter algebras. Cent. Eur. J. Math. -- Open Mathematics, 11, 2, 2013, 226-245, Versita, MR 3000640
[18] Guo, L.: An introduction to Rota-Baxter algebra. 2012, International Press Somerville, Higher Education Press, Beijing, Surveys of Modern Mathematics, vol. 4.. MR 3025028
[19] Guo, L., Keigher, W.: Baxter algebras and shuffle products. Adv. Math., 150, 1, 2000, 117-149, DOI 10.1006/aima.1999.1858 | MR 1744484
[20] Kim, D.S., Kim, T.: Bernoulli basis and the product of several Bernoulli polynomials. Int. J. Math. Math. Sci., 2012, 2012, 12 pp, Hindawi, MR 2969368
[21] Kim, D.S., Kim, T., Lee, S.-H., Kim, Y.-H.: Some identities for the product of two Bernoulli and Euler polynomials. Adv. Differ. Equ., 2012, 95, 2012, 14 pp, Springer, MR 2948735
[22] Lehmer, D.H.: A new approach to Bernoulli polynomials. Am. Math. Mon., 95, 10, 1988, 905-911, Taylor & Francis, DOI 10.1080/00029890.1988.11972114 | MR 0979133
[23] Matiyasevich, Yu.: Identities with Bernoulli numbers. 1997, http://logic.pdmi.ras.ru/~yumat/Journal/Bernoulli/bernulli.htm
[24] Miki, H.: A relation between Bernoulli numbers. J. Number Theory, 10, 3, 1978, 297-302, Elsevier, DOI 10.1016/0022-314X(78)90026-4 | MR 0506640
[25] Miller, J.B.: Some properties of Baxter operators. Acta Math. Hung., 17, 3-4, 1966, 387-400, Akadémiai Kiadó, co-published with Springer Science+ Business Media BV MR 0205074
[26] Newsome, N.J., Nogin, M.S., Sabuwala, A.H.: A proof of symmetry of the power sum polynomials using a novel Bernoulli number identity. J. Integer Seq., 20, 2, 2017, 10 pp, MR 3680201
[27] Nielsen, N.: Traité élémentaire des nombres de Bernoulli. 1923, Gauthier-Villars,
[28] Ogievetsky, O., Popov, T.: $R$-matrices in rime. Adv. Theor. Math. Phys., 14, 2, 2010, 439-505, DOI 10.4310/ATMP.2010.v14.n2.a3 | MR 2721653
[29] Ogievetskii, O.V., Schechtman, V.V.: Nombres de Bernoulli et une formule de Schlömilch-Ramanujan. Mosc. Math. J., 10, 4, 2010, 765-788, DOI 10.17323/1609-4514-2010-10-4-765-788 | MR 2791057
[30] Rota, G.-C.: Baxter algebras and combinatorial identities. I. Bull. Am. Math. Soc., 75, 2, 1969, 325-329, DOI 10.1090/S0002-9904-1969-12156-7 | MR 0244070
[31] Semenov-Tyan-Shanskii, M.A.: What is a classical $r$-matrix?. Funct. Anal. its Appl., 17, 1983, 259-272, DOI 10.1007/BF01076717 | MR 0725413
[32] Sury, B., Wang, T., Zhao, F.-Z.: Identities involving reciprocals of binomial coefficients. J. Integer Seq., 7, 2, 2004, 12 pp, MR 2084860
[33] Tuenter, H.J.H.: A symmetry of power sum polynomials and Bernoulli numbers. Am. Math. Mon., 108, 3, 2001, 258-261, Taylor & Francis, DOI 10.1080/00029890.2001.11919750 | MR 1834708
[34] Zagier, D.: Curious and exotic identities for Bernoulli numbers (Appendix). Bernoulli numbers and zeta functions, 2014, 239-262, Springer, MR 3307736
[35] Zhao, J.: Multiple zeta functions, multiple polylogarithms and their special values. 2016, World Scientific, Series on Number Theory and Its Applications, vol. 12.. MR 3469645
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