Related papers: On Freud-Sobolev type orthogonal polynomials

The main aim of this work is to apply the study of the asymptotic behaviour of generalized eigenvalues between infinite Hermitian definite positive matrices in an important question regarding the location of zeros of Sobolev orthogonal…

For a fixed $p \in \mathbb{N}$, sequences of polynomials $\{P_n\}$, $n \in \mathbb{N}$, defined by a $(p+2)$-term recurrence relation are related to several topics in Approximation Theory. A $(p+2)$-banded matrix $J$ determines the…

The purpose of this paper is to present a systemic study of some families of q-Euler numbers and polynomials of Norlund's type by using multivariate fermionic p-adic integral on Zp. Moreover, the study of these higher-order q-Euler numbers…

The aim of this paper is to give a new approach to modified $q$-Bernstein polynomials for functions of several variables. By using these polynomials, the recurrence formulas and some new interesting identities related to the second Stirling…

We consider random walk polynomial sequences $(P_n(x))_{n\in\mathbb{N}_0}\subseteq\mathbb{R}[x]$ given by recurrence relations of the form $P_0(x)=1$, $P_1(x)=x$ and $x P_n(x)=a_n P_{n+1}(x)+c_n P_{n-1}(x)\;(n\in\mathbb{N})$, where $a_n$…

New sequences of orthogonal polynomials with ultra-exponential weight functions are discovered. In particular, it gives an explicit solution to the Ditkin-Prudnikov problem (1966). The 3-term recurrence relations, explicit representations,…

We show that a confluent case of the big q-Jacobi polynomials P_n(x;a,b,c;q), which corresponds to a=b=-c, leads to a discrete orthogonality relation for imaginary values of the parameter a (outside of its commonly known domain 0<a<…

We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a semi-classical Laguerre weight and classical solutions of the fourth Painlev\'e equation. We show that the coefficients in these…

We associate to a semisimple complex Lie algebra $\mathfrak{g}$ a sequence of polynomials $P_{\ell,\mathfrak{g}}(x)\in\mathbb{Q}[x]$ in $r$ variables, where $r$ is the rank of $\mathfrak{g}$ and $\ell=0,1,2,\ldots $. The polynomials…

We study a sequence of polynomials orthogonal with respect to a one parameter family of weights $$ w(x):=w(x,t)=\rex^{-t/x}\:x^{\al}(1-x)^{\bt},\quad t\geq 0, $$ defined for $x\in[0,1].$ If $t=0,$ this reduces to a shifted Jacobi weight.…

Let $V$ be a set of isolated points in $\RR^d$. Define a linear functional $\CL$ on the space of real polynomials restricted on $V$, $\CL f = \sum_{x \in V} f(x)\rho(x)$, where $\rho$ is a nonzero function on $V$. Polynomial subspaces that…

Let D_v the difference operator and q-difference operators defined by D_\omega p(x) = \frac{p(x+\omega)-p(x)}{\omega} and D_q p(x) = \frac{p(qx)-p(x)}{(q-1)x}, respectively. Let U and V be two moment regular linear functionals and let…

Our main objective in this work is to show how Sobolev orthogonal polynomials emerge as a useful tool within the framework of spectral methods for boundary-value problems. The solution of a boundary-value problem for a stationary…

We consider two families of polynomials $\mathbb{P}=\polP$ and $\mathbb{Q}=\polQ$\footnote{Here and below we consider only monic polynomials.} orthogonal on the real line with respect to probability measures $\mu$ and $\nu$ respectively.…

The paper is about a Painlev\'e III equation and its relation to isomonodromic families of vector bundles on P^1 with meromorphic connections. The purpose of the paper is two-fold: it offers a conceptual language for the geometrical objects…

Laurent polynomials related to the Hahn-Exton $q$-Bessel function, which are $q$-analogues of the Lommel polynomials, have been introduced by Koelink and Swarttouw. The explicit strong moment functional with respect to which the Laurent…

The purpose of this note is to revive in $L^p$ spaces the original A. Markov ideas to study monotonicity of zeros of orthogonal polynomials. This allows us to prove and improve in a simple and unified way our previous result [Electron.…

For every system $\{ p_n(z) \}_{n=0}^\infty$ of OPRL or OPUC, we construct Sobolev orthogonal polynomials $y_n(z)$, with explicit integral representations involving $p_n$. Two concrete families of Sobolev orthogonal polynomials (depending…

In this paper, the authors deal with the $q$-Genocchi numbers and polynomials with weight zero. They discover some interesting relations via the $p$-adic $q$-integral on $\mathbb{Z}_{p}$ and familiar basis Bernstein polynomials. Finally,…

For the weight function $W_\mu(x) = (1-|x|^2)^\mu$, $\mu > -1$, $\lambda > 0$ and $b_\mu$ a normalizing constant, a family of mutually orthogonal polynomials on the unit ball with respect to the inner product $$ \la f,g \ra = {b_\mu…