Related papers: Bender-Dunne Orthogonal Polynomials and Quasi-Exac…
Let R and S be two irreducible root systems spanning the same vector space and having the same Weyl group W, such that S (but not necessarily R) is reduced. For each such pair (R,S) we construct a family of W-invariant orthogonal…
We consider multiple orthogonal polynomials associated with the exponential cubic weight e^{-x^3} over two contours in the complex plane. We study the basic properties of these polynomials, including the Rodrigues formula and…
We construct the orthogonal eigenbasis for a discrete elliptic Ruijsenaars type quantum particle Hamiltonian with hyperoctahedral symmetry. In the trigonometric limit the eigenfunctions in question recover a previously studied $q$-Racah…
Capturing solution near the singular point of any nonlinear SBVPs is challenging because coefficients involved in the differential equation blow up near singularities. In this article, we aim to construct a general method based on…
A long standing question in the theory of orthogonal matrix polynomials is the matrix Bochner problem, the classification of $N \times N$ weight matrices $W(x)$ whose associated orthogonal polynomials are eigenfunctions of a second order…
Over the last decade it has become clear that discrete Painlev\'e equations appear in a wide range of important mathematical and physical problems. Thus, the question of recognizing a given non-autonomous recurrence as a discrete Painlev\'e…
The matrix-valued spherical functions for the pair (K x K, K), K=SU(2), are studied. By restriction to the subgroup A the matrix-valued spherical functions are diagonal. For suitable set of representations we take these diagonals into a…
In this paper we obtain some properties of orthogonal polynomials given by a weight function which is a limit density of a rescaled discrete-time quantum walk on the line.
By using a generalization of Sturm-Liouville problems in discrete spaces, a basic class of symmetric orthogonal polynomials of a discrete variable with four free parameters, which generalizes all classical discrete symmetric orthogonal…
We consider orthogonal polynomials on the unit circle associated with certain semi-classical weight functions. This means that the Pearson-type differential equations satisfied by these weight functions involve two polynomials of degree at…
We consider random polynomials of the form $G_n(z):= \sum_{|\alpha|\leq n} \xi^{(n)}_{\alpha}p_{n,\alpha}(z)$ where $\{\xi^{(n)}_{\alpha}\}_{|\alpha|\leq n}$ are i.i.d. (complex) random variables and $\{p_{n,\alpha}\}_{|\alpha|\leq n}$ form…
We consider two one dimensional nonlinear oscillators, namely (i) Higgs oscillator and (ii) a $k$-dependent nonpolynomial rational potential, where $k$ is the constant curvature of a Riemannian manifold. Both the systems are of position…
We present polynomial-augmented neural networks (PANNs), a novel machine learning architecture that combines deep neural networks (DNNs) with a polynomial approximant. PANNs combine the strengths of DNNs (flexibility and efficiency in…
The analytic solutions of the one-dimensional Schroedinger equation for the trigonometric Rosen-Morse potential reported in the literature rely upon the Jacobi polynomials with complex indices and complex arguments. We first draw attention…
In the present paper, new classes of wavelet functions are presented in the framework of Clifford analysis. Firstly, some classes of orthogonal polynomials are provided based on 2-parameters weight functions. Such classes englobe the well…
In this work we present an explicit representation of the orthonormal Bernstein polynomials and demonstrate that they can be generated from a linear combination of non-orthonormal Bernstein polynomials. In addition, we report a set of $n$…
We introduce the notion of "hypergeometric" polynomials with respect to Newtonian bases. These polynomials are eigenfunctions ($L P_n(x) = \lambda_n P_n(x)$) of some abstract operator $L$ which is 2-diagonal in the Newtonian basis…
We study second-order modular differential equations whose solutions transform equivariantly under the modular group. In the reducible case, we construct all such solutions using an explicit ansatz involving Eisenstein series and the…
We obtain the strong asymptotics of polynomials $p_n(\lambda)$, $\lambda\in\mathbb{C}$, orthogonal with respect to measures in the complex plane of the form $$ e^{-N(|\lambda|^{2s}-t\lambda^s-\overline{t\lambda}^s)}dA(\lambda), $$ where $s$…
We examine various generalizations, e.g. exactly solvable, quasi-exactly solvable and non-Hermitian variants, of a quantum nonlinear oscillator. For all these cases, the same mass function has been used and it has also been shown that the…