A two-sided Faulhaber-like formula involving Bernoulli polynomials

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

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]

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

Comptes Rendus. Mathématique, Tome 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 :

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

@article{CRMATH_2020__358_1_41_0,
     author = {Barbero G., J. Fernando and Margalef-Bentabol, Juan and Villase\~nor, Eduardo J.S.},
     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 = {http://www.numdam.org/articles/10.5802/crmath.10/}
}

TY  - JOUR
AU  - Barbero G., J. Fernando
AU  - Margalef-Bentabol, Juan
AU  - Villaseñor, Eduardo J.S.
TI  - A two-sided Faulhaber-like formula involving Bernoulli polynomials
JO  - Comptes Rendus. Mathématique
PY  - 2020
SP  - 41
EP  - 44
VL  - 358
IS  - 1
PB  - Académie des sciences, Paris
UR  - http://www.numdam.org/articles/10.5802/crmath.10/
DO  - 10.5802/crmath.10
LA  - en
ID  - CRMATH_2020__358_1_41_0
ER  -

%0 Journal Article
%A Barbero G., J. Fernando
%A Margalef-Bentabol, Juan
%A Villaseñor, Eduardo J.S.
%T A two-sided Faulhaber-like formula involving Bernoulli polynomials
%J Comptes Rendus. Mathématique
%D 2020
%P 41-44
%V 358
%N 1
%I Académie des sciences, Paris
%U http://www.numdam.org/articles/10.5802/crmath.10/
%R 10.5802/crmath.10
%G en
%F CRMATH_2020__358_1_41_0

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://www.numdam.org/articles/10.5802/crmath.10/

Bibliographie
Cité par

[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

Cité par Sources :