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Identities concerning Bernoulli and Euler polynomials | Zhi-Wei Sun
; Hao Pan
; | Date: |
2 Sep 2004 | Subject: | Number Theory; Combinatorics MSC-class: 11B68; 05A19 | math.NT math.CO | Abstract: | We establish two general identities for Bernoulli and Euler polynomials, these identities of a new type 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 $$F(s,t;x,y):=sum_{k=0}^n(-1)^kinom{s}{k}inom{t}{n-k}B_{n-k}(x)B_k(y).$$ This symmetric relation implies the curious identities of Miki and Matiyasevich as well as some new identities for Bernoulli polynomials such as $$sum_{k=0}^ninom{n}{k}^2B_k(x)B_{n-k}(x)=2sum^nSb k=0 k
ot=n-1endSbinom{n}{k}inom{n+k-1}{k}B_k(x)B_{n-k}.$$ | Source: | arXiv, math.NT/0409035 | Services: | Forum | Review | PDF | Favorites |
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