The q-analogue of Dixon’s identity involves three q-binomial coefficients as summands. We find many variations of it that have beautiful corollories in terms of Fibonomial sums. Proofs involve either several instances of the q-Dixon formula itself or are “mechanical,” i. e., use the q-Zeilberger algorithm

We find a q-analog of the following symmetrical identity involving binomial coefficients ( n m ) and Eulerian numbers An,m, due to Chung, Graham and Knuth [J. Comb., 1 (2010), 29–38]: ∑ k≥0 ( a + b k ) Ak,a−1 = ∑ k≥0 ( a + b k ) Ak,b−1. We give two proofs, using generating function and bijections, respectively.

We give three proofs for the following symmetrical identity involving binomial coefficients ( n m ) and Eulerian numbers 〈 n m 〉 : ∑ k ( a + b k ) 〈 k a − 1 〉 = ∑ k ( a + b k ) 〈 k b − 1 〉 for any positive integers a and b (where we take 〈 0 0 〉 = 0). We also show how this fits into a family of similar (but more complicated) identities for Eulerian numbers.

In this paper, we investigate spectral norms for circulant-type matrices, including circulant, skewcirculant and g-circulant matrices. The entries are product of binomial coefficients with Fibonacci numbers and Lucas numbers, respectively. We obtain identity estimations for these spectral norms. Employing these approaches, we list some numerical tests to verify our results.

This note describes the geometrical pattern of zeroes and ones obtained by reducing modulo two each element of Pascal's triangle formed from binomial coefficients. When an infinite number of rows of Pascal's triangle are included, the limiting pattern is found to be "self-similar," and is characterized by a "fractal dimension" log2 3. Analysis of the pattern provides a simple derivation of the ...

The notion of binomial coefficients (T S ) of finite planar, reduced rooted trees T, S is defined and a recursive formula for its computation is shown. The nonassociative binomial formula (1 + x) = ∑

In this paper, we are concerned with sums involving inverses of binomial coefficients. We study certain sums involving reciprocals of binomial coefficients by using some integrals. Some recurrence relations related to inverses of binomial coefficients are obtained. In addition, we give the approximate values of certain sums involving the inverses of binomial coefficients.

The focus of this paper is the study of generalized Fibonacci polynomials and Fibonomial coefficients. The former are polynomials {n} in variables s, t given by {0} = 0, {1} = 1, and {n} = s{n−1}+t{n−2} for n ≥ 2. The latter are defined by { n k } = {n}!/({k}!{n−k}!) where {n}! = {1}{2} . . . {n}. These quotients are also polynomials in s, t and specializations give the ordinary binomial coeffi...