A two-sided Faulhaber-like formula involving Bernoulli polynomials

J. Fernando Barbero G.; Juan Margalef-Bentabol; Eduardo J.S. Villaseñor

doi:10.5802/crmath.10

Combinatoire, Physique mathématique

A two-sided Faulhaber-like formula involving Bernoulli polynomials
[Une formule bilatérale de type Faulhaber utilisant les polynômes de Bernoulli]

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

Comptes Rendus. Mathématique, Volume 358 (2020) no. 1, pp. 41-44.

Résumé
Abstract

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 $1^{m} {(n - 1)}^{m} + 2^{m} {(n - 2)}^{m} + \dots + {(n - 1)}^{m} 1^{m}$ où $m$ 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 $1^{m} {(n - 1)}^{m} + 2^{m} {(n - 2)}^{m} + \dots + {(n - 1)}^{m} 1^{m}$ for positive integers $m$ and $n$ .

Reçu le : 2019-05-08
Révisé le : 2019-09-26
Accepté le : 2020-01-21
Publié le : 2020-03-18

DOI : 10.5802/crmath.10

Affiliations des auteurs :

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     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},
}

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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/

Bibliographie
Cité par

[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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