2,661 research outputs found
A combinatorial identity with application to Catalan numbers
By a very simple argument, we prove that if are nonnegative integers
then \sum_{k=0}^l(-1)^{m-k}\binom{l}{k}\binom{m-k}{n}\binom{2k}{k-2l+m}
=\sum_{k=0}^l\binom{l}{k}\binom{2k}{n}\binom{n-l}{m+n-3k-l}.
On the basis of this identity, for we construct explicit
and such that for any prime we have
\sum_{k=1}^{p-1}k^r C_{k+d}\equiv \cases F(d,r)(mod p)& if 3|p-1, \\G(d,r)\
(mod p)& if 3|p-2,
where denotes the Catalan number . For
example, when is a prime, we have
\sum_{k=1}^{p-1}k^2C_k\equiv\cases-2/3 (mod p)& if 3|p-1, \1/3 (mod p)& if
3|p-2;
and
\sum_{0<k<p-4}\frac{C_{k+4}}k \equiv\cases 503/30 (mod p)& if 3|p-1, -100/3
(mod p)& if 3|p-2.
This paper also contains some new recurrence relations for Catalan numbers.Comment: 22 page
Canonical characters on quasi-symmetric functions and bivariate Catalan numbers
Every character on a graded connected Hopf algebra decomposes uniquely as a
product of an even character and an odd character (Aguiar, Bergeron, and
Sottile, math.CO/0310016).
We obtain explicit formulas for the even and odd parts of the universal
character on the Hopf algebra of quasi-symmetric functions. They can be
described in terms of Legendre's beta function evaluated at half-integers, or
in terms of bivariate Catalan numbers:
Properties of characters and of quasi-symmetric functions are then used to
derive several interesting identities among bivariate Catalan numbers and in
particular among Catalan numbers and central binomial coefficients
Combinatorial identities associated with new families of the numbers and polynomials and their approximation values
Recently, the numbers and the polynomials
have been introduced by the second author [22]. The purpose
of this paper is to construct higher-order of these numbers and polynomials
with their generating functions. By using these generating functions with their
functional equations and derivative equations, we derive various identities and
relations including two recurrence relations, Vandermonde type convolution
formula, combinatorial sums, the Bernstein basis functions, and also some well
known families of special numbers and their interpolation functions such as the
Apostol--Bernoulli numbers, the Apostol--Euler numbers, the Stirling numbers of
the first kind, and the zeta type function. Finally, by using Stirling's
approximation for factorials, we investigate some approximation values of the
special case of the numbers .Comment: 17 page
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