Identities concerning Bernoulli and Euler polynomials
| dc.creator | Sun, Zhi-Wei | |
| dc.creator | Pan, Hao | |
| dc.date | 2004-09-02 | |
| dc.date | 2006-11-09 | |
| dc.date.accessioned | 2026-07-07T06:38:46Z | |
| dc.date.available | 2026-07-07T06:38:46Z | |
| dc.description | We establish two general identities for Bernoulli and Euler polynomials, which are of a new type and have many consequences. The most striking result in this paper is as follows: If $n$ is a positive integer, $r+s+t=n$ and $x+y+z=1$, then we have $$rF(s,t;x,y)+sF(t,r;y,z)+tF(r,s;z,x)=0$$ where $$F(s,t;x,y):=\sum_{k=0}^n(-1)^k\binom{s}{k}\binom{t}{n-k}B_{n-k}(x)B_k(y).$$ This symmetric relation implies the curious identities of Miki and Matiyasevich as well as some new identities for Bernoulli polynomials such as $$\sum_{k=0}^n\binom{n}{k}^2B_k(x)B_{n-k}(x)=2\sum^n\Sb k=0 k\not=n-1\endSb\binom{n}{k}\binom{n+k-1}{k}B_k(x)B_{n-k}.$$ | |
| dc.description | 21 pages | |
| dc.identifier | https://arxiv.org/abs/math/0409035 | |
| dc.identifier | http://arxiv.org/abs/math/0409035 | |
| dc.identifier | Acta Arith. 125(2006), 21--39 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/100856 | |
| dc.subject | Number Theory | |
| dc.subject | Combinatorics | |
| dc.subject | 11B68; 05A19 | |
| dc.title | Identities concerning Bernoulli and Euler polynomials | |
| dc.type | text |