6,386 research outputs found
Identities concerning Bernoulli and Euler polynomials
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 is a positive integer, and
, then we have where
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}.Comment: 21 page
Consecutive primes and Legendre symbols
Let be any positive integer and let . We
show that for some constanst there are infinitely many integers
with such that
for all ,
where denotes the -th prime, and denotes the
Legendre symbol for any odd prime . We also prove that under the Generalized
Riemann Hypothesis there are infinitely many positive integers such that
is a primitive root modulo for any distinct and
among .Comment: 12 pages, final published versio
On 2-adic orders of some binomial sums
We prove that for any nonnegative integers and the binomial sum is divisible by
, where denotes the number of
1's in the binary expansion of . This confirms a recent conjecture of Guo
and Zeng.Comment: 6 page
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