Related papers: Calculation of some determinants using the s-shift…
Polynomials are common algebraic structures, which are often used to approximate functions including probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of…
We exhibit explicit expressions, in terms of components, of discriminants, determinants, characteristic polynomials and polynomial identities for matrices of higher rank. We define permutation tensors and in term of them we construct…
Explicit expressions are proven for derivatives of the ratio of a determinant or Pfaffian determinant and a Vandermonde determinant. Such ratios appear for example in general group integrals of Harish-Chandra--Itzykson--Zuber type and in…
We present a variation and generalization of a determinant evaluation of Wilf (math.CO/9809120). His result concerns a matrix whose entries are the coefficients of powers of a given power series; we replace the powers by repeated…
We introduce the notion of the descent set polynomial as an alternative way of encoding the sizes of descent classes of permutations. Descent set polynomials exhibit interesting factorization patterns. We explore the question of when…
The applicability of the factorization method is extended to the case of quantum fractional-differential Hamiltonians. In contrast with the conventional factorization, it is shown that the `factorization energy' is now a…
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…
We investigate determinants of random unitary pencils (with scalar or matrix coefficients), which generalize the characteristic polynomial of a single unitary matrix. In particular we examine moments of such determinants, obtained by…
This paper describes an algorithm which computes the characteristic polynomial of a matrix over a field within the same asymptotic complexity, up to constant factors, as the multiplication of two square matrices. Previously, this was only…
Nonunique factorization in cancellative commutative semigroups is often studied using combinatorial factorization invariants, which assign to each semigroup element a quantity determined by the factorization structure. For numerical…
A natural consequence of the fractional calculus is its extension to a matrix order of differentiation and integration. A matrix-order derivative definition and a matrix-order integration arise from the generalization of the gamma function…
Factorization of numbers with the help of Gauss sums relies on an intimate relationship between the maxima of these functions and the factors. Indeed, when we restrict ourselves to integer arguments of the Gauss sum we profit from a…
A formula is presented for the determinant of the second additive compound of a square matrix in terms of coefficients of its characteristic polynomial. This formula can be used to make claims about the eigenvalues of polynomial matrices,…
For non-negative integers $r$ and $m$, let $S_m^{(r)}(n)$ denote the $r$-fold summation (or hyper-sum) over the first $n$ positive integers to the $m$th powers, with the initial condition $S_m^{(0)}(n) =n^m$. In this paper, we derive a new…
Using a property of the q-shifted factorial, an identity for q-binomial coefficients is proved, which is used to derive the formulas for the q-binomial coefficient for negative arguments. The result is in agreement with an earlier paper…
Factorial designs have broad applications in agricultural, engineering and scientific studies. In constructing and studying properties of factorial designs, traditional design theory treats all factors as nominal. However, this is not…
A new family of polynomials, called cumulant polynomial sequence, and its extensions to the multivariate case is introduced relied on a purely symbolic combinatorial method. The coefficients of these polynomials are cumulants, but depending…
The notion of a descent polynomial, a function in enumerative combinatorics that counts permutations with specific properties, enjoys a revived recent research interest due to its connection with other important notions in combinatorics,…
A new class of distributional transformations is introduced, characterized by equations relating function weighted expectations of test functions on a given distribution to expectations of the transformed distribution on the test function's…
The hyperdeterminant of a polynomial (interpreted as a symmetric tensor) factors into several irreducible factors with multiplicities. Using geometric techniques these factors are identified along with their degrees and their…