Related papers: Efficient computation of Laguerre polynomials

We propose a simple uniform lower bound on the spacings between the successive zeros of the Laguerre polynomials $L_n^{(\alpha)}$ for all $\alpha>-1$. Our bound is sharp regarding the order of dependency on $n$ and $\alpha$ in various…

Laguerre's rootfinding algorithm is highly recommended although most of its properties are known only by empirical evidence. In view of this, we prove the first sufficient convergence criterion. It is applicable to simple roots of…

The polynomial eigenvalue problem arises in many applications and has received a great deal of attention over the last decade. The use of root-finding methods to solve the polynomial eigenvalue problem dates back to the work of…

We consider a $(q,y)$-analogue of Laguerre polynomials $L^{(\alpha)}_n(x;y;q)$ for integral $\alpha\geq -1$, which turns out to be a rescaled version of Al-Salam--Chihara polynomials. A combinatorial interpretation for the $(q,y)$-Laguerre…

We develop a new type of orthogonal polynomial, the modified discrete Laguerre (MDL) polynomials, designed to accelerate the computation of bosonic Matsubara sums in statistical physics. The MDL polynomials lead to a rapidly convergent…

We study the asymptotic zero distribution of the rescaled Laguerre polynomials, $\displaystyle L_n^{(\alpha_n)}(nz)$, with the parameter $\alpha_n$ varying in such a way that $\displaystyle \lim_{n\rightarrow \infty}\alpha_n/n=-1$. The…

We are presenting a method for computing the Fourier coefficients of a given polynomial regression by using the trapezoidal rule for numerical integration. As function basis we use the orthogonal Legendre polynomials. The results are…

We derive a system of difference equations satisfied by the three-term recurrence coefficients of some families of discrete orthogonal polynomials.

In this paper we consider the strong asymptotic behavior of Laguerre polynomials in the complex plane. The leading behavior is well known from Perron and Mehler-Heine formulas, but higher order coefficients, which are important in the…

We study the asymptotic behavior of Laguerre polynomials $L_n^{(\alpha_n)}(nz)$ as $n \to \infty$, where $\alpha_n$ is a sequence of negative parameters such that $-\alpha_n/n$ tends to a limit $A > 1$ as $n \to \infty$. These polynomials…

We construct fast algorithms for evaluating transforms associated with families of functions which satisfy recurrence relations. These include algorithms both for computing the coefficients in linear combinations of the functions, given the…

We discuss numerical solution of Altarelli-Parisi equations in a Laguerre-polynomial method and in a brute-force method. In the Laguerre method, we get good accuracy by taking about twenty Laguerre polynomials in the flavor-nonsinglet case.…

We present two new classes of orthogonal functions, log orthogonal functions (LOFs) and generalized log orthogonal functions (GLOFs), which are constructed by applying a $\log$ mapping to Laguerre polynomials. We develop basic approximation…

A complete numerical implementation, in both singlet and non-singlet sectors, of a very elegant method to solve the QCD Evolution equations, due to Furmanski and Petronzio, is presented. The algorithm is directly implemented in x-space by a…

We consider the problem of finding a function defined on $(0,\infty)$ from a countable set of values of its Laplace transform. The problem is severely ill-posed. We shall use the expansion of the function in a series of Laguerre polynomials…

We investigate a numerical solution of the flavor-nonsinglet Altarelli-Parisi equation with next-to-leading-order $\alpha_s$ corrections by using Laguerre polynomials. Expanding a structure function (or a quark distribution) and a splitting…

Using the theory of orthogonal polynomials, their associated recursion relations and differential formulas we develop a method for evaluating new integrals. The method is illustrated by obtaining a closed-form expression for the value of an…

This paper makes two main contributions. First, we present a pedagogical review of the derivation of the three-term recurrence relation for Legendre polynomials, without relying on the classical Legendre differential equation, Rodrigues'…

We define the family of truncated Laguerre polynomials $P_n(x;z)$, orthogonal with respect to the linear functional $\ell$ defined by $$\langle{\ell,p\rangle}=\int_{0}^zp(x)x^\alpha e^{-x}dx,\qquad\alpha>-1.$$ The connection between…

Associated to a finite measure on the real line with finite moments are recurrence coefficients in a three-term formula for orthogonal polynomials with respect to this measure. These recurrence coefficients are frequently inputs to modern…