Related papers: Identities involving Narayana polynomials and Cata…

We define a weighted analog for the multidimensional Catalan numbers, obtain matrix-based recurrences for some of them, and give conditions under which they are periodic. Building on this framework, we introduce two new sequences of…

In this paper, we investigate the weighted Catalan, Motzkin and Schr\"oder numbers together with the corresponding weighted paths. The relation between these numbers is illustrated by three equations, which also lead to some known and new…

We present new proofs of two identities arising in the work of Mourad Ismail using partition theoretic generating function interpretations.

The generating functions of some sequences of Catalan numbers and Narayana polynomials have simple expansions as continued fractions of Jacobi type. We give an overview of these facts and prove analogous results for q-Narayana polynomials…

In this paper we prove some new identities for multiple zeta values and multiple zeta star values of arbitrary depth by using the methods of integral computations of logarithm function and iterated integral representations of series. By…

We prove some combinatorial identities using the Polya urn and the closely related Hoppe urn.

We lift to the multivariate Eulerian polynomials the identity implying that univariate Eulerian polynomials are palindromic. As a consequence of this generalization, we obtain nice combinatorial identities that can be directly extracted…

It is well known that the Catalan number C_n counts dissections of a regular (n+2)-gon into triangles. Here we count such dissections by number of triangles that contain two sides of the polygon among their three edges, leading to a…

Based on weighted noncrossing partitions of type B, we obtain type B analogues of Coker's identities on the Narayana polynomials. A parity reversing involution is given for the alternating sum of Narayana numbers of type B. Moreover, we…

We consider a special class of binomial sums involving harmonic numbers and we prove three identities by using the elementary method of the partial fraction decomposition. Some applications to infinite series and congruences are given.

Some applications of a result, which is proved recently, is considered. We first prove three determinantal identities concerning the binomial coefficient and Stirling numbers of the first and the second kind. We also easily obtain the…

In this note we present a combinatorial proof of an identity involving poly-Bernoulli numbers and Genocchi numbers. We introduce the combinatorial objects, $m-$barred Callan sequences and show that the identity holds in a more general…

This note provide bijective proofs of two combinatorial identities involving generalized Catalan number $C_{m,5}(n)={m\over 5n+m}{5n+m\choose n}$ recently proposed by Sun.

In this paper we derive some interesting identities arising from the orhtogonality of gegenbauer polynomials.

We derive a collection of identities for bivariate Fibonacci and Lucas polynomials using essentially a matrix approach as well as properties of such polynomials when the variables $x$ and $y$ are replaced by polynomials. A wealth of…

By a very simple argument, we prove that if $l,m,n$ 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…

A probability method is provided to prove three classes of combinatorial identities. The method is extremely simple, only one step after the proper probability setup.

We prove a master identity for a class of sequences defined by full-history linear homogeneous recurrences with (non-negative) constant coefficients. The identity is derived in a combinatorial way, providing thus combinatorial proofs for…

Using an elementary approach involving the Euler Beta function and the binomial theorem, we derive two polynomial identities; one of which is a generalization of a known polynomial identity. Two well-known combinatorial identities, namely…

We consider Tuenter polynomials as linear combinations of descending factorials and show that coefficients of these linear combinations are expressed via a Catalan triangle of numbers. We also describe a triangle of coefficients in terms of…