Geodesic
On a new identity for double sum related to Bernoulli numbers
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 17 (2024) no. 5, pp. 609-612 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Let $m$, $n$ and $l$ be integers with $0\leqslant l\leqslant m+n$. It is the main purpose of this paper to give an identity for the sum: $$\mathop{\sum_{a=0}^{m} \sum_{b=0}^{n}}_{a+b\geqslant m+n-l}B_{m-a}B_{n-b}\frac{\binom{m}{a}\binom{n}{b}}{a+b+1}\binom{a+b+1}{m+n-l},$$ where $B_m$ $(m=0,1,2,\dots)$ is the Bernoulli number. As corollary we prove that the above sum equal to $\dfrac{1}{2}$ when $l=0$.
Keywords: Bernoulli number, generating function.
Mots-clés : Bernoulli polynomial 
@article{JSFU_2024_17_5_a5,
     author = {Brahim Mittou},
     title = {On a new identity for double sum related to {Bernoulli} numbers},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {609--612},
     year = {2024},
     volume = {17},
     number = {5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2024_17_5_a5/}
}
TY  - JOUR
AU  - Brahim Mittou
TI  - On a new identity for double sum related to Bernoulli numbers
JO  - Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
PY  - 2024
SP  - 609
EP  - 612
VL  - 17
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/JSFU_2024_17_5_a5/
LA  - en
ID  - JSFU_2024_17_5_a5
ER  - 
%0 Journal Article
%A Brahim Mittou
%T On a new identity for double sum related to Bernoulli numbers
%J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
%D 2024
%P 609-612
%V 17
%N 5
%U http://geodesic.mathdoc.fr/item/JSFU_2024_17_5_a5/
%G en
%F JSFU_2024_17_5_a5
Brahim Mittou. On a new identity for double sum related to Bernoulli numbers. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 17 (2024) no. 5, pp. 609-612. http://geodesic.mathdoc.fr/item/JSFU_2024_17_5_a5/

[1] T.Agoh, K.Dilcher, “Integrals of products of Bernoulli polynomials”, J. Math. Anal. Appl., 381 (2011), 10–16 | DOI | MR | Zbl

[2] T.M.Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976 | MR | Zbl

[3] T.Arakawa, T.Ibukiyama, M.Kaneko, Bernoulli Numbers and Zeta Functions, Springer, Japan, 2014 | MR | Zbl

[4] H.Cohen, Number Theory, v. II, Analytic and Modern Tools, Springer-Verlag, Berlin, 2007 | DOI | MR

[5] P.Vassilev, M.V.Missana, “On one remarkable identity involving Bernoulli numbers”, Notes on Number Theory and Discrete Mathematics, 11 (2005), 22–24