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.

Abstract (VO)
Résumé

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 $n$ .

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.

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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