Related papers: On the third-order Jacobsthal and third-order Jaco…
We compute the matrix elements of $SO(3)$ in any finite-dimensional irreducible representation of $sl_3$. They are expressed in terms of a double sum of products of Krawtchouk and Racah polynomials which generalize the Griffiths-Krawtchouk…
In this article we give the meaning of the determinant for 3D matrices with elements from a field F, and the meaning of 3D inverse matrix. Based on my previous work titled '3D Matrix Rings', we want to constructed the 'general linear group…
The reciprocal Pascal matrix has entries $\binom{i+j}{j}^{-1}$. Explicit formullae for its LU-decomposition, the LU-decomposition of its inverse, and some related matrices are obtained. For all results, $q$-analogues are also presented.
Given an operator L acting on a function space, the J-matrix method consists of finding a sequence y_n of functions such that the operator L acts tridiagonally on y_n with respect to n. Once such a tridiagonalization is obtained, a number…
Let $\F$ be the finite field of odd prime power order $q$, We find explicit expressions for the number of triples $\{\al-1,\al,\al+1 \}$ of consecutive non-zero squares in $\F$ and similarly for the number of triples of consecutive…
In this paper, we study the three-term nested recurrence relation $B(n)=B(n-B(n-1))+B(n-B(n-2))+B(n-B(n-3))$ subject to initial conditions where the first $N$ terms are the integers $1$ through $N$. This recurrence is the three-term analog…
Let $\left(P_{n}\right)_{n\geq0}$ be the sequence of bi-periodic Padovan numbers and let $\left(M_{p_{n}}\right)_{n\geq0}$ be the sequence of bi-periodic Padovan matrices. In this article we study when these matrices are diagonalizable and…
The paper contains the derivation of a general set of recurrence formulas for the calculus of the SU(3) Clebsch-Gordan coefficients. The first six sections are introductory, presenting the notations and placing SU(3) in the framework of the…
We present numerous interesting, mostly new, results involving the $n$-step Fibonacci numbers and $n$-step Lucas numbers and a generalization. Properties considered include recurrence relations, summation identities, including binomial and…
In this paper, new families of generalized Fibonacci and Lucas numbers are introduced. In addition, we present the recurrence relations and the generating functions of the new families for $k=2$.
For an integer \( k \geq 2 \), the sequence of \( k \)-generalized Lucas numbers is defined by the recurrence relation \( L_n^{(k)} = L_{n-1}^{(k)} + \cdots + L_{n-k}^{(k)} \) for all \( n \geq 2 \), with initial conditions \( L_0^{(k)} = 2…
The theme of the first two sections, is to prepare the framework of how from a ``complicated'' family of so called index models $I \in K_1$ we build many and/or complicated structures in a class $K_2$. The index models are…
In this paper we find a third order unimodular matrix, none of whose entries is $1$ or $-1$, such that when each entry of the matrix is replaced by its cube, the resulting matrix is also unimodular. Further, we find third order square…
The Johnson--Lindenstrauss (JL) lemma is a powerful tool for dimensionality reduction in modern algorithm design. The lemma states that any set of high-dimensional points in a Euclidean space can be flattened to lower dimensions while…
In this paper we will study some properties of the matrix representations of symbol algebras of degree three, we study some equations with coefficients in these algebras, we find an octonion algebra in a symbol algebra of degree three, we…
Jacobi-Nijenhuis algebroids are defined as a natural generalization of Poisson-Nijenhuis algebroids, in the case where there exists a Nijenhuis operator on a Jacobi algebroid which is compatible with it. We study modular classes of Jacobi…
Using our recent results on eigenvalues of invariants associated to the Lie superalgebra gl(m|n), we use characteristic identities to derive explicit matrix element formulae for all gl(m|n) generators, particularly non-elementary…
Connections between longest increasing subsequences in random permutations and eigenvalues of random matrices with complex entries have been intensely studied. This note applies properties of random elements of the finite general linear…
We define the generalized Golomb triangular recursion by g_{j,s,lambda}(n) = g_{j,s,lambda}(n - s - g_{j,s,lambda}(n-j)) + \lambda j. For particular choices of the initial conditions, we show that the solution of the recursion is a non-slow…
Let $V$ denote a vector space over C with finite positive dimension. By a {\em Leonard triple} on $V$ we mean an ordered triple of linear operators on $V$ such that for each of these operators there exists a basis of $V$ with respect to…