Related papers: On orthogonal polynomials spanning a non-standard …
Orthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We…
We study two families of orthogonal polynomials with respect to the weight function $w(t)(t^2-\|x\|^2)^{\mu-\frac12}$, $\mu > -\frac 12$, on the cone $\{(x,t): \|x\| \le t, \, x \in \mathbb{R}^d, t >0\}$ in $\mathbb{R}^{d+1}$. The first…
For orthogonal polynomials defined by compact Jacobi matrix with exponential decay of the coefficients, precise properties of orthogonality measure is determined. This allows showing uniform boundedness of partial sums of orthogonal…
The $(-1)$-Jacobi, Bannai-Ito, and $(-1)$-Meixner-Pollaczek polynomials are studied in [Trans. Amer. Math. Soc. 364 (2012), 5491-5507], [Adv. Math. 229 (2012), 2123-2158], and [Stud. Appl. Math. 153 (2024), e12728], respectively, through…
We introduce a family of weight matrices $W$ of the form $T(t)T^*(t)$, $T(t)=e^{\mathscr{A}t}e^{\mathscr{D}t^2}$, where $\mathscr{A}$ is certain nilpotent matrix and $\mathscr{D}$ is a diagonal matrix with negative real entries. The weight…
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…
It is well known that the classical families of orthogonal polynomials are characterized as eigenfunctions of a second order linear differential/difference operator. In this paper we present a study of classical orthogonal polynomials in a…
We introduce a family of exactly-solvable two-dimensional Hamiltonians whose wave functions are given in terms of Laguerre and exceptional Jacobi polynomials. The Hamiltonians contain purely quantum terms which vanish in the classical limit…
The aim of this paper is to bring into the picture a new phenomenon in the theory of orthogonal matrix polynomials satisfying second order differential equations. The last few years have witnessed some examples of a (fixed) family of…
We present two infinite sequences of polynomial eigenfunctions of a Sturm-Liouville problem. As opposed to the classical orthogonal polynomial systems, these sequences start with a polynomial of degree one. We denote these polynomials as…
We produce a new family of polynomials f(x) over fields K of characteristic 2 which are exceptional, in the sense that f(x)-f(y) has no absolutely irreducible factors in K[x,y] besides the scalar multiples of x-y; when K is finite, this…
We give some structural formulas for the family of matrix-valued orthogonal polynomials of size $2\times 2$ introduced by C. Calder\'on et al. in an earlier work, which are common eigenfunctions of a differential operator of hypergeometric…
We give an analog of exceptional polynomials in the matrix valued setting by considering suitable factorizations of a given second order differential operator and performing Darboux transformations. Orthogonality and density of the…
This paper revisits the notion of classical orthogonal polynomials from a broader functional-analytic point of view. It is intended neither as a survey of known results nor as a review of the literature, but rather as a conceptual…
We study a family of orthogonal polynomials which satisfy (apart from a 3-term recurrence relation) an eigenvalue equation involving a third order differential operator of Dunkl-type. The orthogonality measure of these polynomials consists…
The multi-indexed Laguerre and Jacobi polynomials form a complete set of orthogonal polynomials. They satisfy second-order differential equations but not three term recurrence relations, because of the 'holes' in their degrees. The…
Simple derivation is presented of the four families of infinitely many shape invariant Hamiltonians corresponding to the exceptional Laguerre and Jacobi polynomials. Darboux-Crum transformations are applied to connect the well-known shape…
The so-called exceptional orthogonal X1-polynomials arise as eigen functions of a Sturm-Liouville problem. In this paper, a generic classification of these polynomials is presented based on Pearson distributions family. Then, six special…
Several important families of orthogonal polynomials on the real line are called ``hypergeometric'' since they can be explicitly described in terms of some hypergeometric series $_pF_q$ that uses the degree $n$ of the polynomial as a…
The Krawtchouck polynomials arise naturally in both coding theory and probability theory and have been studied extensively from these points of view. However, very little is known about their irreducibility and Galois properties. Just like…