Related papers: A Universal Closed Form For Square Matrix Powers

This paper proposes an elementary solution to a special case of finding all perfect squares that can be written as sum of consecutive integer cubes. It is shown that there are no non-trivial solutions if the perfect square is a prime power,…

Recently the author used certain quaternion orders to demonstrate the universality of some quaternary quadratic forms. Here a further study is done on one of these orders analogous to Hurwitz's proof of the formula for the number of…

A generalized definition of the determinant of matrices is given, which is compatible with the usual determinant for square matrices and keeps many important properties, such as being an alternating multilinear function, keeping…

Our goal in this work is to found a closed form for rational generat- ing functions, these generate a various families of polynomials and generalized polynomials, in order to get the general recursive formula satisfied by these polynomials.

Powers of Fibonacci polynomials are expressed as single sums, improving on a double sum recently seen in the literature.

This paper gives new explicit formulas for sums of powers of integers and their reciprocals.

Finite trigonometric sums occur in various branches of physics, mathematics, and their applications. These sums may contain various powers of one or more trigonometric functions. Sums with one trigonometric function are known, however sums…

The study of solutions to polynomial equations over finite fields has a long history in mathematics and is an interesting area of contemporary research. In recent years the subject has found important applications in the modelling of…

We give a simple recursive formula to obtain the general sum of the first $N$ natural numbers to the $r$th power. Our method allows one to obtain the general formula for the $(r+1)$th power once one knows the general formula for the $r$th…

We obtain explicit factored closed-form expressions for Fibonacci and Lucas sums of the form \mbox{$\sum_{k = 1}^n {F_{2rk}^3 }$} and \mbox{$\sum_{k = 1}^n {L_{2rk}^3 }$}, where $r$~and~$n$ are integers.

In this paper, the exact values of the structured singular values of some generalized stochastic complex matrices is explicit in term of the constant row (column) sum.

This paper introduces a symbolic calculus-based approach for deriving closed-form expressions for the sums of arithmetic sequences. The method extends beyond constant-difference sequences to those with polynomially increasing steps,…

In this paper, closed formulas for the eigenvectors of a particular class of matrices generated by generalized permutation matrices, named generalized circulant matrices, are presented.

This paper presents a method to analyze the powers of a given trilinear form (a special kind of algebraic constructions also called a tensor) and obtain upper bounds on the asymptotic complexity of matrix multiplication. Compared with…

We give a canonical form for a complex matrix, whose square is normal, under transformations of unitary similarity as well as a canonical form for a real matrix, whose square is normal, under transformations of orthogonal similarity.

In this paper we study quadratic forms which are universal when restricted to almost prime inputs, establishing finiteness theorems akin to the Conway--Schneeberger 15 theorem.

In this paper, we obtain a general expression for the entries of the r. (r is integer) power of a certain n-square complex tridiagonal matrix. In addition, we get the complex factorizations of Fibonacci polynomials, Fibonacci and Pell…

In this work, the authors provide closed forms and recurrence expressions for computing the $k$th power of the formal power series, some of them in terms of a determinant of some matrices. As a consequence, we obtain the reciprocal of the…

The cyclotomic matrix is commonly used to arrange cyclotomic numbers in a convenient format. A natural question is whether the structure of the matrix can reflect properties of these numbers. In this article, we examine cyclotomic numbers…

Positive definite forms $f$ which are sums of squares are constructed to have the additional property that the members of any collection of forms whose squares sum to $f$ must share a nontrivial complex root.