Related papers: Ordered Products, $W_{\infty}$-Algebra, and Two-Va…
We show how Viennot's combinatorial theory of orthogonal polynomials may be used to generalize some recent results of Sukumar and Hodges on the matrix entries in powers of certain operators in a representation of su(1,1). Our results link…
We study the structure and representation theory of the principal W-algebra $\mathsf{W}^{\mathsf{k}}_{\mathrm{pr}}$ of $\mathsf{V}^{\mathsf{k}}(\mathfrak{psl}_{2|2})$. The defining operator product expansions are computed, as is the Zhu…
In this note, we prove interior a priori first- and second-order estimates for solutions of fully nonlinear degenerate elliptic inequalities structured over the vector fields of Carnot groups, under the main assumption that $u$ is…
Limiting cases are studied of the Koornwinder-Macdonald multivariable generalization of the Askey-Wilson polynomials. We recover recently and not so recently introduced families of hypergeometric orthogonal polynomials in several variables…
A classical (or quantum) superintegrable system on an n-dimensional Riemannian manifold is an integrable Hamiltonian system with potential that admits 2n-1 functionally independent constants of the motion that are polynomial in the momenta,…
We establish a weak Harnack inequality for nonlocal $W^{s,1}$-subminimizers in a complete, connected, doubling metric measure space where $0<s<1$. As a corollary, we prove that $W^{s,1}$-subminimizers are semicontinuous, up to a suitable…
A new class of 2-orthogonal polynomials satisfying orthogonality conditions with respect to a pair of linear functionals $(u_0,u_1)$ was presented in Douak K & Maroni P [On a new class of 2-orthogonal polynomials, I: the recurrence…
Determinantal polynomials play a crucial role in semidefinite programming problems. Helton-Vinnikov proved that real zero (RZ) bivariate polynomials are determinantal. However, it leads to a challenging problem to compute such a…
Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra and $r,m\ge 2$. The universal central extension of the superelliptic current algebra $\mathfrak{g}\otimes A$ is $\widehat{\mathfrak{g}\otimes A}\cong\mathfrak{g}\otimes A…
Let $d\ge1$ be an integer, $W_d$ and $\mathcal{K}_d$ be the Witt algebra and the weyl algebra over the Laurent polynomial algebra $A_d=\mathbb{C} [x_1^{\pm1}, x_2^{\pm1}, ..., x_d^{\pm1}]$, respectively. For any $\mathfrak{gl}_d$-module $M$…
Explicit expressions for restricted partition function $W(s,{\bf d}^m)$ and its quasiperiodic components $W_j(s,{\bf d}^m)$ (called {\em Sylvester waves}) for a set of positive integers ${\bf d}^m = \{d_1, d_2, ..., d_m\}$ are derived. The…
With the aim of completing the previous study by A. Or{\l}owski and the author concerning intertwining maps between induced representations and conjugation representation, termed here weighted class operators, we compute the latter…
New sequences of orthogonal polynomials with ultra-exponential weight functions are discovered. In particular, it gives an explicit solution to the Ditkin-Prudnikov problem (1966). The 3-term recurrence relations, explicit representations,…
A natural family of quantized matrix algebras is introduced. It includes the two best studied such. Located inside ${\s U}_q(A_{2n-1})$, it consists of quadratic algebras with the same Hilbert series as polynomials in $n^2$ variables. We…
Matrix-valued spherical functions related to the quantum symmetric pair for the quantum analogue of $(SU(2) \times SU(2), \text{diag})$ are introduced and studied in detail. The quantum symmetric pair is given in terms of a quantised…
Orthogonal polynomials for a family of weight functions on $[-1,1]^2$, $$ \CW_{\a,\b,\g}(x,y) = |x+y|^{2\a+1} |x-y|^{2\b+1} (1-x^2)^\g(1-y^2)^{\g}, $$ are studied and shown to be related to the Koornwinder polynomials defined on the region…
The explicit solution of the initial-values problem is exhibited of a subclass of the autonomous system of 2 coupled first-order ODE s with second-degree polynomial right-hand sides, hence featuring 12 a prior arbitrary (time-independent)…
We study $2$-representation finite $\mathbb{K}$-algebras obtained from tensor products of tensor algebras of species. In earlier work we computed the higher preprojective algebra of said algebras to be given as Jacobian algebras of certain…
Let $\mathbb K$ be an algebraically closed field of characteristic zero, $A = \mathbb K[x_1,\dots,x_n]$ the polynomial ring, and let $W_n(\mathbb K)$ denote the Lie algebra of all $\mathbb K$-derivations on $A$. The Lie algebra $W_n :=…
We overview classifications of simple infinite-dimensional complex $\mathbb{Z}$-graded Lie (super)algebras of polynomial growth, and their deformations. A subset of such Lie (super)algebras consist of vectorial Lie (super)algebras whose…