Related papers: A particular matrix and its relationships with Fib…
We derive a general recurrence relation for squares of Fibonacci-like numbers. Various properties are developed, including double binomial summation identites.
In this research, the Bernoulli polynomials are introduced. The properties of these polynomials are employed to construct the operational matrices of integration together with the derivative and product. These properties are then utilized…
We define and study symmetrized and antisymmetrized multivariate exponential functions. They are defined as determinants and antideterminants of matrices whose entries are exponential functions of one variable. These functions are…
We study matrices which transform the sequence of Fibonacci or Lucas polynomials with even index to those with odd index and vice versa. They turn out to be intimately related to generalized Stirling numbers and to Bernoulli, Genocchi and…
In this paper, we introduce a particular class of matrices. We study the concept of a matrix to be \emph{balanced}. We study some properties of this concept in the context of matrix operations. We examine the behaviour of various matrix…
Motivated by a problem in learning theory, we are led to study the dominant eigenvalue of a class of random matrices. This turns out to be related to the roots of the derivative of random polynomials (generated by picking their roots…
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…
We evaluate determinants of "spiral" matrices, which are matrices in which entries are spiralling from the centre of the matrices towards the outside, with prescribed increments from one entry to the next depending on whether one moves…
In this paper we express the eigenvalues of a sort of real heptadiagonal symmetric matrices as the zeros of explicit rational functions establishing upper and lower bounds for each of them. From these prescribed eigenvalues we compute also…
We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees).…
Matrix determinants play an important role in data analysis, in particular when Gaussian processes are involved. Due to currently exploding data volumes, linear operations - matrices - acting on the data are often not accessible directly…
A technique to compute arbitrary products of a class of Fibonacci $2\times2$ square matrices is proved in this work. General explicit solutions for non autonomous Fibonacci difference equations are obtained from these products. In the…
Using the language of Riordan arrays, we look at two related iterative processes on matrices and determine which matrices are invariant under these processes. In a special case, the invariant sequences that arise are conjectured to have…
We consider the set of $n\times n$ matrices with rational entries having numerator and denominator of size at most $H$ and obtain upper and lower bounds on the number of such matrices of a given rank and then apply them to count such…
We present a prescription for forming matrices with specified eigenvalues and known eigenvectors. With this method, we can form Hermitian, anti-Hermitian, symmetric and general matrices with arbitrary eigenvalues. In addition we propose an…
Determinantal point processes have arisen in diverse settings in recent years and have been investigated intensively. We study basic combinatorial and probabilistic aspects in the discrete case. Our main results concern relationships with…
In this paper, we compute the spectral norms of the matrices related with integer squences and we give two examples related with Fibonacci and Lucas numbers.
We give a combinatorial interpretation of the determinant of a matrix as a generating function over Brauer diagrams in two different but related ways. The sign of a permutation associated to its number of inversions in the Leibniz formula…
The Fibonacci number is the residue of a rational function, from which follows that Fibonacci number summation identities can be derived with the integral representation method, a method also used to derive combinatorial identities. A…
Fibonomial coefficients count the number of specific finite birth self-similar subposets of an infinite non-tree poset naturally related to the Fibonacci tree of rabbits growth process.