José L. Ramírez

Some Combinatorial Properties of the k-Fibonacci and the k-Lucas Quaternions

Abstract In this paper, we define the k-Fibonacci and the k-Lucas quaternions. We investigate the generating functions and Binet formulas for these quaternions. In addition, we derive some sums formulas and identities such as Cassini’s identity.

Some Properties of Convolved -Fibonacci Numbers

We define the convolved k-Fibonacci numbers as an extension of the classical convolved Fibonacci numbers. Then we give some combinatorial formulas involving the k-Fibonacci and k-Lucas numbers. Moreover we obtain the convolved k-Fibonacci numbers from a family of Hessenberg matrices.

On convolved generalized Fibonacci and Lucas polynomials

We define the convolved h(x)-Fibonacci polynomials as an extension of the classical convolved Fibonacci numbers. Then we give some combinatorial formulas involving the h(x)-Fibonacci and h(x)-Lucas polynomials. Moreover we obtain the convolved h(x)-Fibonacci polynomials from a family of Hessenberg matrices.

Incomplete -Fibonacci and -Lucas Numbers

We define the incomplete k-Fibonacci and k-Lucas numbers; we study the recurrence relations and some properties of these numbers.

The linear algebra of the r-Whitney matrices

The linear algebraic theory of the Pascal and Vandermonde matrix is well developed by many authors. In the last two decades many interrelations have been discovered between the mentioned matrices, their generalizations and the Stirling matrices. We follow this direction and discover new matricial relations by using the so-called r-Whitney numbers. Along this way, we develop two natural extensions of the Vandermonde matrix with which we can study and evaluate successive power sums of arithmetic progressions and win new identities for the r-Whitney numbers.

A generalization of the Fibonacci word fractal and the Fibonacci snowflake

In this paper we introduce a family of infinite words that generalize the Fibonacci word and we study their combinatorial properties. We associate with this family of words a family of curves that are like the Fibonacci word fractal and reveal some fractal features. Finally, we describe an infinite family of polyominoes stems from the generalized Fibonacci words and we study some of their geometric properties, such as perimeter and area. These last polyominoes generalize the Fibonacci snowflake and they are double squares polyominoes, i.e., tile the plane by translation in exactly two distinct ways.

GENERATING FRACTAL PATTERNS BY USING p-CIRCLE INVERSION

In this paper, we introduce the p-circle inversion which generalizes the classical inversion with respect to a circle (p = 2) and the taxicab inversion (p = 1). We study some basic properties and we also show the inversive images of some basic curves. We apply this new transformation to well-known fractals such as Sierpinski triangle, Koch curve, dragon curve, Fibonacci fractal, among others. Then we obtain new fractal patterns. Moreover, we generalize the method called circle inversion fractal be means of the p-circle inversion.

Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs

In this paper we studied infinite weighted automata and a general methodology to solve a wide variety of classical lattice path counting problems in an uniform way. This counting problems are related to Dyck paths, Motzkin paths and some generalizations. These methodology uses weighted automata, equations of ordinary generating functions and continued fractions. It is a variation of the one proposed by J. Rutten.

A geometrical construction of inverse points with respect to an ellipse

In this note, we introduce the inverse point respect to an ellipse, which generalizes the classical inversion in a circle. Specifically, we show a geometrical construction of inverse points with respect to an ellipse. For this, we extend the classical method to construct the inverse of a point respect to a circle.

A generalization of the spherical inversion

Abstract In the present article, we introduce a generalization of the spherical inversion. In particular, we define an inversion with respect to an ellipsoid, and prove several properties of this new transformation. The inversion in an ellipsoid is the generalization of the elliptic inversion to the three-dimensional space. We also study the inverse images of planes, spheres, ellipsoids and other curves. We use the software Mathematica to generate these graphics. Finally, we generalize the Pappus chain theorem to ellipsoids.