708 research outputs found
On sums of binomial coefficients and their applications
In this paper we study recurrences concerning the combinatorial sum
and the alternate sum
, where m>0, and r
are integers. For example, we show that if then
We also apply such results to investigate Bernoulli
and Euler polynomials. Our approach depends heavily on an identity established
by the author [Integers 2(2002)]
On some Binomial Coefficient Identities with Applications
We present a different proof of the following identity due to Munarini, which
generalizes a curious binomial identity of Simons. \begin{align*}
\sum_{k=0}^{n}\binom{\alpha}{n-k}\binom{\beta+k}{k}x^k
&=\sum_{k=0}^{n}(-1)^{n+k}\binom{\beta-\alpha+n}{n-k}\binom{\beta+k}{k}(x+1)^k,
\end{align*} where is a non-negative integer and and are
complex numbers, which are not negative integers. Our approach is based on a
particularly interesting combination of the Taylor theorem and the
Wilf-Zeilberger algorithm. We also generalize a combinatorial identity due to
Alzer and Kouba, and offer a new binomial sum identity. Furthermore, as
applications, we give many harmonic number sum identities. As examples, we
prove that \begin{equation*}
H_n=\frac{1}{2}\sum_{k=1}^{n}(-1)^{n+k}\binom{n}{k}\binom{n+k}{k}H_k
\end{equation*} and \begin{align*}
\sum_{k=0}^{n}\binom{n}{k}^2H_kH_{n-k}=\binom{2n}{n}
\left((H_{2n}-2H_n)^2+H_{n}^{(2)}-H_{2n}^{(2)}\right). \end{align*}Comment: Submitte
Noncommutative Catalan numbers
The goal of this paper is to introduce and study noncommutative Catalan
numbers which belong to the free Laurent polynomial algebra in
generators. Our noncommutative numbers admit interesting (commutative and
noncommutative) specializations, one of them related to Garsia-Haiman
-versions, another -- to solving noncommutative quadratic equations. We
also establish total positivity of the corresponding (noncommutative) Hankel
matrices and introduce accompanying noncommutative binomial coefficients.Comment: 12 pages AM LaTex, a picture and proof of Lemma 3.6 are added,
misprints correcte
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