Related papers: A conjecture on Exceptional Orthogonal Polynomials
Orthogonal polynomials appear naturally in the study of compositions of M\"obius transformations. In this paper, we consider several classes of orthogonal polynomials associated to non-autonomous perturbations of a parabolic M\"obius map.…
Our work studies sequences of orthogonal polynomials $ \{P_{n}(x)\}_{n=0}^{\infty} $ of the Laguerre-Hahn class, whose Stieltjes functions satisfy a Riccati type differential equation with polynomial coefficients, are subject to a…
We consider orthogonal polynomials on the surface of a double cone or a hyperboloid of revolution, either finite or infinite in axis direction, and on the solid domain bounded by such a surface and, when the surface is finite, by…
Ordinary orthogonal polynomials are uniquely characterized by the three term recurrence relations up to an overall multiplicative constant. We show that the newly discovered M-indexed orthogonal polynomials satisfy 3+2M term recurrence…
In this paper, we give a new proof of a celebrated theorem of J\"orgens which states that every classical convex solution of \[ \det\nabla^2 u (x)=1\quad {in} \mathbb{R}^2 \] has to be a second order polynomial. Our arguments do not use…
The dual discrete $q$-ultraspherical polynomials $D_n^{(s)}(\mu (x;s)|q)$ correspond to indeterminate moment problem and, therefore, have one-parameter family of extremal orthogonality relations. It is shown that special cases of dual…
We construct a new family of quasi-solvable and N-fold supersymmetric quantum systems where each Hamiltonian preserves an exceptional polynomial subspace of codimension 2. We show that the family includes as a particular case the recently…
We investigate generalized Laurent multiple orthogonal polynomials on the unit circle satisfying simultaneous orthogonality conditions with respect to $r$ probability measures or linear functionals on the unit circle. We show that these…
We construct new examples of bispectral dual Hahn polynomials, i.e., orthogonal polynomials with respect to certain superposition of Christoffel and Geronimus transforms of the dual Hahn measure and which are also eigenfunctions of a higher…
Multiple orthogonal polynomials are a generalization of orthogonal polynomials in which the orthogonality is distributed among a number of orthogonality weights. They appear in random matrix theory in the form of special determinantal point…
The Darboux transformations of Krawtchouk polynomials are investigated and all possible exceptional Krawtchouk polynomials obtainable from a single-step Darboux transformation are considered. The properties of these exceptional Krawtchouk…
Classical (maximal) superintegrable systems in $n$ dimensions are Hamiltonian systems with $2n-1$ independent constants of the motion, globally defined, the maximum number possible. They are very special because they can be solved…
The NIST Handbook of Mathematical Functions (2010) and the NIST Digital Library of Mathematical Functions (2025) classify classical orthogonal polynomials through Bochner's 1929 algebraic-differential characterisation and its…
Based on the work of Chen and Its [{\em J. Approx. Theory} {\bf 162} ({2010}) {270--297}], we further study orthogonal polynomials with respect to the singularly perturbed Laguerre weight $w(x;t,\alpha) = {x^\alpha}{\mathrm e^{-…
We present various results on the properties of the four infinite sets of the exceptional $X_{\ell}$ polynomials discovered recently by Odake and Sasaki [{\it Phys. Lett. B} {\bf 679} (2009), 414-417; {\it Phys. Lett. B} {\bf 684} (2010),…
Infinitely many explicit solutions of certain second-order differential equations with an apparent singularity of characteristic exponent -2 are constructed by adjusting the parameter of the multi-indexed Laguerre polynomials.
This note shows that the orbifold Jacobian algebra associated to each invertible polynomial defining an exceptional unimodal singularity is isomorphic to the (usual) Jacobian algebra of the Berglund-H\"{u}bsch transform of an invertible…
Multiple orthogonal polynomials with respect to two weights on the step-line are considered. A connection between different dual spectral matrices, one banded (recursion matrix) and one Hessenberg, respectively, and the Gauss-Borel…
Two families (type $A$ and type $B$) of confluent hypergeometric polynomials in several variables are studied. We describe the orthogonality properties, differential equations, and Pieri type recurrence formulas for these families. In the…
Let $p$ be an odd prime. In the paper, by using the properties of Legendre polynomials we prove some congruences for $\sum_{k=0}^{\frac{p-1}2}\binom{2k}k^2m^{-k}\mod {p^2}$. In particular, we confirm several conjectures of Z.W. Sun. We also…