Related papers: On a class of determinants
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated…
Two well-known $q$-Hermite polynomials are the continuous and discrete $q$-Hermite polynomials. In this paper we consider a new family of $q$-Hermite polynomials and prove several curious properties about these polynomials. One striking…
One of the most popular and studied recursive series is the Fibonacci sequence. It is challenging to see how Fibonacci numbers can be used to generate other recursive sequences. In our article, we describe some families of integer…
By investigating a recurrence relation about functions, we first give alternative proofs of various identities on Fibonacci numbers and Lucas numbers, and then, make certain well known identities visible via certain trivalent graph…
A term calculus for the proofs in multiplicative-additive linear logic is introduced and motivated as a programming language for channel based concurrency. The term calculus is proved complete for a semantics in linearly distributive…
The Fomin-Zelevinsky Laurent phenomenon states that every cluster variable in a cluster algebra can be expressed as a Laurent polynomial in the variables lying in an arbitrary initial cluster. We give representation-theoretic formulas for…
In this paper, a new kind of resultant, called the determinantal resultant, is introduced. This operator computes the projection of a determinantal variety under suitable hypothesis. As a direct generalization of the resultant of a very…
We introduce and characterize a particularly tractable class of unital type 1 C*-algebras with bounded dimension of irreducible representations. Algebras in this class are called recursive subhomogeneous algebras, and they have an inductive…
We prove recursive formulas involving sums of divisors and sums of triangular numbers and give a variety of identities relating arithmetic functions to divisor functions providing inductive identities for such arithmetic functions.
We introduce a class of doubly indexed real Hermite polynomials and we deal with their related properties like the associated recurrence formulae, Runge's addition formula, generating function and Nielsen's identity.
In this paper, we introduce h(x)-Fibonacci polynomials in an arbitrary finite-dimensional unitary algebra over a field K (K = R,C), which generalize both h(x)-Fibonacci quaternion polynomials and h(x)-Fibonacci octonion polynomials. For…
We obtain new recurrence relations, an explicit formula, and convolution identities for higher order geometric polynomials. These relations generalize known results for geometric polynomials, and lead to congruences for higher order…
A family of formal power series, such that its coefficients satisfy a recursion formula, is characterized in terms of the summability, in the sense of J. P. Ramis, of its elements along certain well chosen directions. We describe a set of…
We survey and prove properties a family of recurrences bears in relation to integer representations, compositions, the Pascal triangle, sums of digits, Nim games and Beatty sequences.
We give an introduction to some of the recent ideas that go under the name "geometric complexity theory". We first sketch the proof of the known upper and lower bounds for the determinantal complexity of the permanent. We then introduce the…
A novel type of approximants is introduced, being based on the ideas of self-similar approximation theory. The method is illustrated by the examples possessing the structure typical of many problems in applied mathematics. Good numerical…
Let $f_{1}, \ldots, f_{k}$ be polynomials defining an algebraic set in affine $n$-space over a finite field. Suppose $k>n$. We prove that there exists a system of polynomials $g_{1}, \ldots, g_{n}$, each being a linear combination with…
In this paper, we consider the problem of representing a multivariate polynomial as the determinant of a definite (monic) symmetric/Hermitian linear matrix polynomial (LMP). Such a polynomial is known as determinantal polynomial.…
The discriminant of a multivariate polynomial with indeterminate coefficients is not necessarily a hypersurface, and characterizing its codimension was an open problem for quite a while. We resolve this problem for the discriminants of…
The aim of this paper is to study finite orthogonal polynomials on a cone of revolution and its surface. We define two classes of finite orthogonal polynomials on the solid cone and derive their corresponding differential equations and…