2,661 research outputs found

    A combinatorial identity with application to Catalan numbers

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    By a very simple argument, we prove that if l,m,nl,m,n 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 d,r=0,1,2,...d,r=0,1,2,... we construct explicit F(d,r)F(d,r) and G(d,r)G(d,r) such that for any prime p>max⁑{d,r}p>\max\{d,r\} 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 CnC_n denotes the Catalan number (n+1)βˆ’1(2nn)(n+1)^{-1}\binom{2n}{n}. For example, when pβ‰₯5p\geq 5 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

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    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: C(m,n)=(2m)!(2n)!m!(m+n)!n!. C(m,n)=\frac{(2m)!(2n)!}{m!(m+n)!n!}. 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

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    Recently, the numbers Yn(Ξ»)Y_{n}(\lambda ) and the polynomials Yn(x,Ξ»)Y_{n}(x,\lambda) 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 Yn(Ξ»)Y_{n}\left( \lambda \right) .Comment: 17 page
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