Related papers: Contiguous relations, continued fractions and orth…
The great success of the theory of hypergeometric series in one variable has stimulated the development of a corresponding theory in two and more variables. Horn has investigated the convergence of 34 (14 complete and 20 confluent)…
An analogue of Taylor's formula, which arises by substituting the classical derivative by a divided difference operator of Askey-Wilson type, is developed here. We study the convergence of the associated Taylor series. Our results…
Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. It consists of a four-parameter polynomial with continuous spectrum on the whole real line…
We develop the geometry of Hurwitz continued fractions, a major tool in understanding the approximation properties of complex numbers by ratios of Gaussian integers. Based on a thorough study of the geometric properties of Hurwitz continued…
In the lecture notes we start off with an introduction to the $q$-hypergeometric series, or basic hypergeometric series, and we derive some elementary summation and transformation results. Then the $q$-hypergeometric difference equation is…
Let $F$ be a field of characteristic zero and $W$ be an associative affine $F$-algebra satisfying a polynomial identity (PI). The codimension sequence associated to $W$, $c_n(W)$, is known to be of the form $\Theta (c n^t d^n)$, where $d$…
Let $K$ denote a field, and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy the following two conditions: There exists a basis for…
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 work the interplay between matrix biorthogonal polynomials with respect to a matrix of linear functionals, the $k$-th associated matrix polynomials and the second kind matrix functions, is studied in terms of quasideterminants. A…
For the multiple Fourier series of the periodization of some radial functions on $\mathbb{R}^d$, we investigate the behavior of the spherical partial sum. We show the Gibbs-Wilbraham phenomenon, the Pinsky phenomenon and the third…
It is well-known that orthogonal polynomials on the real line satisfy a three-term recurrence relation and conversely every system of polynomials satisfying a three-term recurrence relation is orthogonal with respect to some positive Borel…
We consider the exactly solvable quantum mechanical systems whose eigenfunctions are described by the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson and Askey-Wilson types. Corresponding to the recurrence relations with…
The purpose of this article is to bring structure to (basic) hypergeometric biorthogonal systems, in particular to the q-Askey scheme of basic hypergeometric orthogonal polynomials. We aim to achieve this by looking at the limits as p->0 of…
We exhibit two distinct renormalization scenarios in many-parameter families of piecewise isometries (PWI) of a rhombus. The rotational component, defined over the quadratic field $\mathbb{K}=\mathbb{Q}(\sqrt{5})$, is fixed. The…
We analyze the asymptotically free massless scalar $\phi^3$ quantum field theory in 6 dimensions, using resurgent asymptotic analysis to find the trans-series solutions which yield the non-perturbative completion of the divergent…
Two sets of infinitely many exceptional orthogonal polynomials related to the Wilson and Askey-Wilson polynomials are presented. They are derived as the eigenfunctions of shape invariant and thus exactly solvable quantum mechanical…
Given a non-negative, decreasing sequence $a$ with sum $1$, we consider all the closed subsets of $[0,1]$ such that the lengths of their complementary open intervals are given by the terms of $a$, the so-called complementary sets. In this…
We prove a suite of dynamical results, including exactness of the transformation and piecewise-analyticity of the invariant measure, for a family of continued fraction systems, including specific examples over reals, complex numbers,…
In recent years, there has been significant progress in the theory of orthogonal polynomials on algebraic curves, particularly on genus 1 surfaces. In this paper, we focus on elliptic orthogonal polynomials and establish several of their…
We use finite connectivity equilibrium replica theory to solve models of finitely connected unit-length vectorial spins, with random pair-interactions which are of the orthogonal matrix type. Since the spins are continuous and the…