Remark. This identity is easily verified using the WZ method, in a generalized form [Z] that applies when the summand is a hypergeometric term times a WZ potential function. It holds for all positive n, since it holds for n=1,2,3 (check!), and since the sequence defined by the sum satisfies a certain (homog.) third order linear recurrence equation. To find the recurrence, and its proof, downloa...

Let s and t be variables. Define polynomials {n} in s, t by {0} = 0, {1} = 1, and {n} = s {n− 1}+ t {n− 2} for n ≥ 2. If s, t are integers then the corresponding sequence of integers is called a Lucas sequence. Define an analogue of the binomial coefficients by {n k } = {n}! {k}! {n− k}! where {n}! = {1} {2} · · · {n}. It is easy to see that { n k } is a polynomial in s and t. The purpose of th...

In the present paper combinatorial identities involving q-dual sequences or polynomials with coefficients q-dual sequences are derived. Further, combinatorial identities for q-binomial coefficients(Gaussian coefficients), q-Stirling numbers and q-Bernoulli numbers and polynomials are deduced.

In 2001 Luca proved that no Fermat number can be a nontrivial binomial coefficient. We extend this result to multinomial coefficients.

Abstract. We investigate a (q, t)-generalization of the usual binomial and q-binomial coefficients. It can be viewed as a polynomial in t that depends upon an integer q, with combinatorial interpretations when q is a positive integer, and algebraic interpretations when q is the order of a finite field. These polynomials and their interpretations generalize further in two directions, one relatin...

The triangular array of binomial coefficients, or Pascal’s triangle, is formed by starting with an apex of 1. Every row of Pascal’s triangle can be seen as a line-graph, to each node of which the corresponding binomial coefficient is assigned. We show that the binomial coefficient of a node is equal to the number of ways the line-graph can be constructed when starting with this node and adding ...

The construction of multi-scale image pyramids is used in state-of-the-art methods that perform robust object recognition, such as SIFT and SURF. However, building such image pyramids is computationally expensive, especially when implementations in embedded systems with limited computing resources are considered. Therefore, the use of alternative less expensive approaches are necessary if near ...