Comptes Rendus
no. 1
Combinatorics, Mathematical Physics
A two-sided Faulhaber-like formula involving Bernoulli polynomials
J. Fernando Barbero G.1, 2; Juan Margalef-Bentabol1, 3, 4 ; Eduardo J.S. Villaseñor5, 1 
1 Grupo de Teorías de Campos y Física Estadística. Instituto Gregorio Millán (UC3M). Unidad Asociada al Instituto de Estructura de la Materia, CSIC, Madrid, Spain
2 Instituto de Estructura de la Materia, CSIC. Serrano 123, 28006 Madrid, Spain
3 Institute for Gravitation and the Cosmos & Physics Department, Penn State, University Park, PA 16802, USA
4 Laboratory of Geometry and Dynamical Systems, Department of Mathematics, EPSEB, Universitat Politècnica de Catalunya, BGSMath, Barcelona, Spain
5 Departamento de Matemáticas, Universidad Carlos III de Madrid. Avda. de la Universidad 30, 28911 Leganés, Spain
Comptes Rendus. Mathématique, Volume 358 (2020) no. 1, pp. 41-44.

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.

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 où m et n sont des entiers positifs.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.10

J. Fernando Barbero G. 1, 2; Juan Margalef-Bentabol 1, 3, 4; Eduardo J.S. Villaseñor 5, 1

1 Grupo de Teorías de Campos y Física Estadística. Instituto Gregorio Millán (UC3M). Unidad Asociada al Instituto de Estructura de la Materia, CSIC, Madrid, Spain
2 Instituto de Estructura de la Materia, CSIC. Serrano 123, 28006 Madrid, Spain
3 Institute for Gravitation and the Cosmos & Physics Department, Penn State, University Park, PA 16802, USA
4 Laboratory of Geometry and Dynamical Systems, Department of Mathematics, EPSEB, Universitat Politècnica de Catalunya, BGSMath, Barcelona, Spain
5 Departamento de Matemáticas, Universidad Carlos III de Madrid. Avda. de la Universidad 30, 28911 Leganés, Spain
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     title = {A two-sided {Faulhaber-like} formula involving {Bernoulli} polynomials},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {41--44},
     publisher = {Acad\'emie des sciences, Paris},
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J. Fernando Barbero G.; Juan Margalef-Bentabol; Eduardo J.S. Villaseñor. A two-sided Faulhaber-like formula involving Bernoulli polynomials. Comptes Rendus. Mathématique, Volume 358 (2020) no. 1, pp. 41-44. doi : 10.5802/crmath.10. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.10/

[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 | Zbl

[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 | Zbl

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