Related papers: Determinants containing powers of polynomial seque…
We derive generalizations of a couple of inverse tangent summation identities involving Fibonacci and Lucas numbers. As byproducts we establish many new inverse tangent identities involving the Fibonacci and Lucas numbers.
Let $(p_n)_n$ be a sequence of orthogonal polynomials with respect to the measure $\mu$. Let $T$ be a linear operator acting in the linear space of polynomials $\PP$ and satisfying that $\dgr(T(p))=\dgr(p)-1$, for all polynomial $p$. We…
A linear map between two vector spaces has a very important characteristic: a determinant. In modern theory two generalizations of linear maps are intensively used: to linear complexes (the nilpotent chains of linear maps) and to non-linear…
In this paper we obtained several properties that the characteristic polynomials of the unit-primitive matrix satisfy. In addition, using these properties we have shown that the recurrence relation given as in the formula (1) is true. In…
We study multiple orthogonal polynomials exploiting their explicit determinantal representation in terms of moments. Our reasoning follows that applied to solve the Hermite-Pad\'{e} approximation and interpolation problems. We study also…
Infinite order linear recurrences are studied via kneading matrices and kneading determinants. The concepts of kneading matrix and kneading determinant of an infinite order linear recurrence, introduced in this work, are defined in a purely…
In this paper we consider pentadiagonal $(n+1)\times(n+1)$ matrices with two subdiagonals and two superdiagonals at distances $k$ and $2k$ from the main diagonal where $1\le k<2k\le n$. We give an explicit formula for their determinants and…
An identity is proven that evaluates the determinant of a block tridiagonal matrix with (or without) corners as the determinant of the associated transfer matrix (or a submatrix of it).
We show that several families of polynomials defined via fillings of diagrams satisfy linear recurrences under a natural operation on the shape of the diagram. We focus on key polynomials, (also known as Demazure characters), and Demazure…
Number theorists have studied extensively the connections between the distribution of zeros of the Riemann $\zeta$-function, and of some generalizations, with the statistics of the eigenvalues of large random matrices. It is interesting to…
We study some properties of restricted and associated Fubini numbers. In particular, they have the natural extensions of the original Fubini numbers in the sense of determinants. We also introduce modified Bernoulli and Cauchy numbers and…
In this article we obtain a general polynomial identity in $k$ variables, where $k\geq 2$ is an arbitrary positive integer. We use this identity to give a closed-form expression for the entries of the powers of a $k \times k$ matrix.…
In this expository paper we compute Hankel determinants of some sequences whose generating functions are given by C-fractions and derive orthogonality properties for associated polynomials.
A hierarchy of matrix-valued polynomials which generalize the Jacobi polynomials is found. Defined by a Rodrigues formula, they are also products of a sequence of differential operators. Each class of polynomials is complete, satisfies a…
A composition of birational maps given by Laurent polynomials need not be given by Laurent polynomials; however, sometimes---quite unexpectedly---it does. We suggest a unified treatment of this phenomenon, which covers a large class of…
Some identities that involve the elliptic version of the Cauchy matrices are presented and proved. They include the determinant formula, the formula for the inverse matrix, the matrix product identity and the factorization formula.
This note collects some results and conjectures for the generating functions of the Hankel determinants of certain polynomials which are related to Motzkin paths.
In the past decades, determinants and Pfaffians were found for eigenvalue correlations of various random matrix ensembles. These structures simplify the average over a large number of ratios of characteristic polynomials to integrations…
The purpose of this paper is to compute the asymptotics of determinants of finite sections of operators that are trace class perturbations of Toeplitz operators. For example, we consider the asymptotics in the case where the matrices are of…
The classical Selberg integral contains a power of the Vandermonde determinant. When that power is a square, it is easy to prove Selberg's identity by interpreting it as a determinant of one-variable integrals. We give similar proofs of…