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Comptes Rendus. Mathématique
Combinatoire, Physique mathématique
A two-sided Faulhaber-like formula involving Bernoulli polynomials
[Une formule bilatérale de type Faulhaber utilisant les polynômes de Bernoulli]
Comptes Rendus. Mathématique, Tome 358 (2020) no. 1, pp. 41-44.

Nous donnons une nouvelle identité utilisant les polynômes de Bernoulli et les coefficient binomiaux. Ceci fournit, en particulier, une formule de type Faulhaber pour des sommes de la forme 1 m (n-1) m +2 m (n-2) m ++(n-1) m 1 m m et n sont des entiers positifs.

We give a new identity involving Bernoulli polynomials and combinatorial numbers. This provides, in particular, a Faulhaber-like formula for sums of the form 1 m (n-1) m +2 m (n-2) m ++(n-1) m 1 m for positive integers m and n.

Reçu le : 2019-05-07
Révisé le : 2019-09-25
Accepté le : 2020-01-20
Publié le : 2020-03-18
DOI : https://doi.org/10.5802/crmath.10
@article{CRMATH_2020__358_1_41_0,
     author = {J. Fernando Barbero G. and Juan Margalef-Bentabol and Eduardo J.S. Villase\~nor},
     title = {A two-sided Faulhaber-like formula involving Bernoulli polynomials},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {41--44},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {1},
     year = {2020},
     doi = {10.5802/crmath.10},
     language = {en},
     url = {comptes-rendus.academie-sciences.fr/mathematique/item/CRMATH_2020__358_1_41_0/}
}
J. Fernando Barbero G.; Juan Margalef-Bentabol; Eduardo J.S. Villaseñor. A two-sided Faulhaber-like formula involving Bernoulli polynomials. Comptes Rendus. Mathématique, Tome 358 (2020) no. 1, pp. 41-44. doi : 10.5802/crmath.10. https://comptes-rendus.academie-sciences.fr/mathematique/item/CRMATH_2020__358_1_41_0/

[1] J. Fernando Barbero G.; Juan Margalef-Bentabol; Eduardo J. S. Villaseñor On the distribution of the eigenvalues of the area operator in loop quantum gravity, Class. Quant. Grav., Volume 35 (2018) no. 6, 065008, 17 pages | MR 3768349 | Zbl 1386.83055

[2] Petro Kolosov On the relation between binomial theorem and discrete convolution of piecewise defined power function (2016) (https://arxiv.org/abs/1603.02468)

[3] N. J. A. Sloane The On-Line Encyclopedia of Integer Sequences, 2010 (http://oeis.org)

[4] Zhi-Wei Sun Combinatorial identities in dual sequences, Eur. J. Comb., Volume 24 (2003) no. 6, pp. 709-718 | MR 1995582 | Zbl 1024.05010