Related papers: A deformation of Hermite polynomials
We show that a natural class of orthogonal polynomials on large spheres in $N$ dimensions tend to Hermite polynomials in the large-$N$ limit. We determine the behavior of the spherical Laplacian as well as zonal harmonic polynomials in the…
We provide an algebraic interpretation for two classes of continuous $q$-polynomials. Rogers' continuous $q$-Hermite polynomials and continuous $q$-ultraspherical polynomials are shown to realize, respectively, bases for representation…
We study a rational version of the double affine Hecke algebra associated to the nonreduced affine root system of type $(C^\vee_n,C_n)$. A certain representation in terms of difference-reflection operators naturally leads to the definition…
In this note we explicit the notion of Hermite interpolant of a multivariate symmetric polynomial, generalizing the notion of Lagrange interpolant to the case when there are roots coalescence, an extension of the results on the symmetric…
For a non-zero parameter $q$, we define Hermite trace polynomials, which are multivariate polynomials indexed by permutations. We prove several combinatorial properties for them, such as expansions and product formulas. The linear…
A polynomial ensemble is a probability density function for the position of $n$ real particles of the form $\frac{1}{Z_n} \, \prod_{j<k} (x_k-x_j) \, \det \left[ f_k (x_j) \right]_{j,k=1}^n$, for certain functions $f_1, \ldots, f_n$. Such…
We investigate spectral conditions on Hermitian matrices of roots of unity. Our main results are conjecturally sharp upper bounds on the number of residue classes of the characteristic polynomial of such matrices modulo ideals generated by…
We study the irreducibility of Wronskian Hermite polynomials labelled by partitions. It is known that these polynomials factor as a power of x times a remainder polynomial. We show that the remainder polynomial is irreducible for the…
Quadratic descent of hermitian and skew hermitian forms over division algebras with involution of the first kind in arbitrary characteristic is investigated and a criterion, in terms of systems of quadratic forms, is obtained. A refined…
In this paper, we construct a new family of q-Hermite polynomials denoted by Hn(x,s|q). Main properties and relations are established and proved. In addition, is deduced a sequence of novel polynomials, Ln(. ,.|q), which appear to be…
We consider the set $\mathcal{M}_n(\mathbb{Z}; H)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain upper and lower bounds on the number of distinct irreducible characteristic polynomials which correspond to…
Consider the Wronskians of the classical Hermite polynomials $$H_{\lambda, l}(x):=\mathrm{Wr}(H_l(x),H_{k_1}(x),\ldots, H_{k_n}(x)), \quad l \in \mathbb Z_{\geq 0},$$ where $k_i=\lambda_i+n-i, \,\, i=1,\dots, n$ and $\lambda=(\lambda_1,…
In the present paper, we study deformations of polar weighted homogeneous polynomials which are also polar weighted homogeneous polynomials. We describe a round handle decomposition of the Milnor fibration of a deformation of a polar…
We prove that the complex Hermite polynomials H_{m,n} on the complex plane $\mathbb{C}$ can be realized as the Fourier-Wigner transform $\mathcal{V}$ of the well-known real Hermite functions $h_n$ on real line $\mathbb{R}$. This reduces…
In this paper, a polynomial-time algorithm is given to compute the generalized Hermite normal form for a matrix F over Z[x], or equivalently, the reduced Groebner basis of the Z[x]-module generated by the column vectors of F. The algorithm…
Supposing that $A(z)$ is an exponential polynomial of the form $$ A(z)=H_0(z)+H_1(z)e^{\zeta_1z^n}+\cdots +H_m(z)e^{\zeta_mz^n}, $$ where $H_j$'s are entire and of order $<n$, it is demonstrated that the function $H_0(z)$ and the geometric…
We consider Hermitian random matrices of the form $H = W + \lambda V$, where $W$ is a Wigner matrix and $V$ a diagonal random matrix independent of $W$. We assume subexponential decay for the matrix entries of $W$ and we choose $\lambda…
We study Wronskians of Hermite polynomials labelled by partitions and use the combinatorial concepts of cores and quotients to derive explicit expressions for their coefficients. These coefficients can be expressed in terms of the…
We study analytic deformations of holomorphic foliations given by homogeneous integrable one-forms in the complex affine space $\mathbb C^n$. The deformation is supposed to be of first order (order one in the parameter). We also assume that…
In this paper, we study properties of polynomials over division rings. Moreover, we present formulas for finding roots of some polynomials