Geodesic
Combinatoire, Physique mathématique
A two-sided Faulhaber-like formula involving Bernoulli polynomials
[Une formule bilatérale de type Faulhaber utilisant les polynômes de Bernoulli]
Barbero G., J. Fernando1, 2 ; Margalef-Bentabol, Juan2, 3, 4  ; Villaseñor, Eduardo J.S.2, 5 
1 Instituto de Estructura de la Materia, CSIC. Serrano 123, 28006 Madrid, Spain
2 Grupo de Teorías de Campos y Física Estadística. Instituto Gregorio Millán (UC3M). Unidad Asociada al Instituto de Estructura de la Materia, CSIC, Madrid, Spain
3 Laboratory of Geometry and Dynamical Systems, Department of Mathematics, EPSEB, Universitat Politècnica de Catalunya, BGSMath, Barcelona, Spain
4 Institute for Gravitation and the Cosmos & Physics Department, Penn State, University Park, PA 16802, USA
5 Departamento de Matemáticas, Universidad Carlos III de Madrid. Avda. de la Universidad 30, 28911 Leganés, Spain
Comptes Rendus. Mathématique, Tome 358 (2020) no. 1, pp. 41-44.

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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.

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DOI : 10.5802/crmath.10

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

1 Instituto de Estructura de la Materia, CSIC. Serrano 123, 28006 Madrid, Spain
2 Grupo de Teorías de Campos y Física Estadística. Instituto Gregorio Millán (UC3M). Unidad Asociada al Instituto de Estructura de la Materia, CSIC, Madrid, Spain
3 Laboratory of Geometry and Dynamical Systems, Department of Mathematics, EPSEB, Universitat Politècnica de Catalunya, BGSMath, Barcelona, Spain
4 Institute for Gravitation and the Cosmos & Physics Department, Penn State, University Park, PA 16802, USA
5 Departamento de Matemáticas, Universidad Carlos III de Madrid. Avda. de la Universidad 30, 28911 Leganés, Spain
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {A two-sided {Faulhaber-like} formula involving {Bernoulli} polynomials},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {41--44},
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Barbero G., J. Fernando; Margalef-Bentabol, Juan; Villaseñor, Eduardo J.S. 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. http://geodesic.mathdoc.fr/articles/10.5802/crmath.10/

[1] Barbero G., J. Fernando; Margalef-Bentabol, Juan; Villaseñor, Eduardo J. S. 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] Kolosov, Petro On the relation between binomial theorem and discrete convolution of piecewise defined power function (2016) (https://arxiv.org/abs/1603.02468)

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

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

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