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We carry out some algebraic and analytic properties of a new class of orthogonal polyanalytic polynomials, including their operational formulas, recurrence relations, generating functions, integral representations and different…
The Painlev\'{e} equations arise from the study of Hankel determinants generated by moment matrices, whose weights are expressed as the product of ``classical" weights multiplied by suitable ``deformation factors", usually dependent on a…
A new, recursive method of calculating matrix elements of polynomial hamiltonians is proposed. It is particularly suitable for the recent algebraic studies of the supersymmetric Yang-Mills quantum mechanics in any dimensions. For the D=2…
In this contribution we consider the sequence $\{Q_{n}^{\lambda}\}_{n\geq 0} $ of monic polynomials orthogonal with respect to the following inner product involving differences \begin{equation*} \langle p,q\rangle…
The eigenvalues of a self-adjoint nxn matrix A can be put into a decreasing sequence $\lambda=(\lambda_1,...,\lambda_n)$, with repetitions according to multiplicity, and the diagonal of A is a point of $R^n$ that bears some relation to…
We introduce a new family of multiple orthogonal polynomials satisfying orthogonality conditions with respect to two weights $(w_1,w_2)$ on the positive real line, with $w_1(x)=x^\alpha e^{-x}$ the gamma density and $w_2(x) = x^\alpha…
In this paper we classify the irreducible quasifinite highest weight modules over the orthogonal and symplectic types Lie subalgebras of the Lie algebra of the matrix quantum pseudo differential operators. We also realize them in terms of…
Considering a differential operator of third order that does not increase the degree of polynomials, we analyse some properties of elements of the dual space of 2-orthogonal polynomial eigenfunctions. In two classes of such generic…
We characterize all the multiple orthogonal threefold symmetric polynomial sequences whose sequence of derivatives is also multiple orthogonal. Such a property is commonly called the Hahn property and it is an extension of the concept of…
There is a commutative algebra of differential-difference operators, with two parameters, associated to any dihedral group with an even number of reflections. The intertwining operator relates this algebra to the algebra of partial…
We give two examples of algebras of differential operators associated to families of matrix valued orthogonal polynomials arising from representations of SU$(N+1)$. The first one gives a commutative algebra and the second one a…
We introduce orthogonal polynomials $M_j^{\mu,\ell}(x)$ as eigenfunctions of a certain self-adjoint fourth order differential operator depending on two parameters $\mu\in\mathbb{C}$ and $\ell\in\mathbb{N}_0$. These polynomials arise as…
In this work we study orthogonal polynomials via polynomial mappings in the framework of the $H_q-$semiclassical class. We consider two monic orthogonal polynomial sequences $\{p_n (x)\}_{n\geq0}$ and $\{q_n(x)\}_{n\geq0}$ such that $$…
This paper constructs weight-shifting integral operators for Maass forms on the full modular group SL(2,Z). Under the weight parity condition t = k (mod 2), the operator utilizes an automorphic kernel constructed via Poincare series from a…
We present an explicit difference operator diagonalized by the Macdonald polynomials associated with an (arbitrary) admissible pair of irreducible reduced crystallographic root systems. By the duality symmetry, this gives rise to an…
In the literature concerning the Laguerre-type weight function $x^\lambda w_0(x), x\in[0,+\infty)$, the Jacobi-type weight function $(1-x)^{\alpha}(1+x)^{\beta}w_0(x),x\in[-1,1]$, and the shifted Jacobi-type weight function…
We study a family of higher-derivative conformal operators $P_{2k}^{(2)}$ acting on transverse-traceless symmetric 2-tensors on generic Einstein spaces. They are a natural generalization of the well-known construction for scalars. We first…
We characterize those sequences of weighted isobaric polynomials as defined in math.CO/0106213 which belong to the kernel of the linear operator $D_{11} - \sum_{j=1}^k a_j t_j D_{2j} - mD_2$, and we characterize those linear operators of…
This paper defines and investigates nonsymmetric Macdonald polynomials with values in an irreducible module of the Hecke algebra of type $A_{N-1}$. These polynomials appear as simultaneous eigenfunctions of Cherednik operators. Several…
We present the first systematic extension of the classical Hermite-Laguerre quadratic correspondence to the matrix-valued setting. Starting from a Hermite-type weight matrix W(x) = exp(-x^2) Z(x) with W(x) = W(-x), the change of variables y…