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A note on sums of products of Bernoulli numbers. (English) Zbl 1222.11031

Let \((B_n)\) be the Bernoulli numbers. The author presents an explicit formula for the sum \[ \sum_{r_1,\dots,r_n\geq 0\atop r_1+\dots+r_n=r}\binom{2r}{2r_1,\dots,2r_n}B_{2r_1}\dots B_{2r_n}, \] where \(\binom{2r}{2r_1,\dots,2r_n}\) are the multinomial coefficients.

MSC:

11B68 Bernoulli and Euler numbers and polynomials
11B65 Binomial coefficients; factorials; \(q\)-identities
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[1] Choi, J., Explicit formulas for Bernoulli polynomials of order \(n\), Indian J. Pure Appl. Math., 27, 667-674 (1996) · Zbl 0860.11009
[2] Kanemitsu, S.; Tanigawa, Y.; Yoshimoto, M., On multiple Hurwitz zeta-function values at rational arguments, Acta Arith., 107, 45-67 (2003) · Zbl 1054.11043
[3] Srivastava, H. M.; Choi, J., Series Associated with the Zeta and Related Functions (2001), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, Boston and London · Zbl 1014.33001
[4] Vardi, I., Determinants of Laplacians and multiple gamma functions, SIAM J. Math. Anal., 19, 493-507 (1988) · Zbl 0641.33003
[5] Mellin, Hj., Die Dirichletschen Reihen die zahlentheoretischen Funktionen und unendlichen Produkte von endlichen Geschlecht, Acta Soc. Sci. Fenn., 31, 1-48 (1902)
[6] Dilcher, K., Sums of products of Bernoulli numbers, J. Number Theory, 60, 23-41 (1996) · Zbl 0863.11011
[7] Kim, T., Sums of products of \(q\)-Bernoulli numbers, Arch. Math., 76, 190-195 (2001) · Zbl 0986.11010
[8] Miki, H., A relation between Bernoulli numbers, J. Number Theory, 10, 297-302 (1978) · Zbl 0379.10007
[9] Petojevi, A., New sums of products of Bernoulli numbers, Integral Transform. Spec. Funct., 19, 105-114 (2008) · Zbl 1173.11308
[10] Petojevi, A., A note about the sums of products of Bernoulli numbers, Novi Sad J. Math., 37, 123-128 (2007) · Zbl 1199.11056
[11] Petojević, A.; Srivastava, H. M., Computation of Euler’s type sums of the products of Bernoulli numbers, Appl. Math. Lett., 22, 796-801 (2009) · Zbl 1228.11025
[12] Simsek, Y.; Kurt, V.; Kim, D., New approach to the complete sum of products of the twisted \((h, q)\)-Bernoulli numbers and polynomials, J. Nonlinear Math. Phys., 14, 44-56 (2007) · Zbl 1163.11015
[13] Sita Rama Chandra Rao, R.; Davis, B., Some identities involving the Riemann zeta function, II, Indian J. Pure Appl. Math., 17, 1175-1186 (1986) · Zbl 0614.10013
[14] Eie, M., A note on Bernoulli numbers and Shintani generalized Bernoulli polynomials, Trans. Amer. Math. Soc., 348, 1117-1136 (1996) · Zbl 0864.11043
[15] Apostol, T. M., Introduction to analytic number theory, (Undergraduate Texts in Mathematics (1976), Springer-Verlag: Springer-Verlag New York, Heidelberg) · Zbl 0043.04403
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