We list the so-called Askey-scheme of hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation and generating functions of all classes of orthogonal polynomials in this scheme. In chapeter 2 we give all limit relation between different classes of orthogonal polynomials listed in the Askey-scheme. 
In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme. We give their definition, orthogonality relation, three term recurrence relation and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal polynomials. Finally in chapter 5 we point out how the `classical` hypergeometric orthogonal polynomials of the Askey-scheme can be obtained from their q-analogues.

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The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue

We present a historical survey of Gauss-Christoffel quadrature formulae, beginning with Gauss’ discovery of his well-known method of approximate integration and the early contributions of Jacobi and Christoffel, but emphasizing the more recent advances made after the emergence of powerful digital computing machinery. One group of inquiry concerns the development of the quadrature formula itself, e.g. the inclusion of preassigned nodes and the admission of multiple nodes, as well as other generalizations of the quadrature sum. Another is directed towards the widening of the class of integrals made accessible to Gauss-Christoffel quadrature. These include integrals with nonpositive measures of integration and singular principal value integrals. An account of the error and convergence theory will also be given, as well as a discussion of modern methods for generating Gauss-Christoffel formulae, and a survey of numerical tables.

A Survey of Gauss-Christoffel Quadrature Formulae

This book presents the first in-depth, complete, and unified theoretical discussion of the two most important classes of algorithms for solving matrix eigenvalue problems: QR-like algorithms for dense problems and Krylov subspace methods for sparse problems. The author discusses the theory of the generic GR algorithm, including special cases (for example, QR, SR, HR), and the development of Krylov subspace methods. Also addressed are a generic Krylov process and the Arnoldi and various Lanczos algorithms, which are obtained as special cases. The chapter on product eigenvalue problems provides further unification, showing that the generalized eigenvalue problem, the singular value decomposition problem, and other product eigenvalue problems can all be viewed as standard eigenvalue problems. The author provides theoretical and computational exercises in which the student is guided, step by step, to the results. Some of the exercises refer to a collection of MATLAB programs compiled by the author that are available on a Web site that supplements the book. Audience: Readers of this book are expected to be familiar with the basic ideas of linear algebra and to have had some experience with matrix computations. This book is intended for graduate students in numerical linear algebra. It will also be useful as a reference for researchers in the area and for users of eigenvalue codes who seek a better understanding of the methods they are using. Contents: Preface; Chapter 1: Preliminary Material; Chapter 2: Basic Theory of Eigensystems; Chapter 3: Elimination; Chapter 4: Iteration; Chapter 5: Convergence; Chapter 6: The Generalized Eigenvalue Problem; Chapter 7: Inside the Bulge; Chapter 8: Product Eigenvalue Problems; Chapter 9: Krylov Subspace Methods; Bibliography; Index.

The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods

An efficient algorithm for the accurate computation of Gauss--Legendre and Gauss--Jacobi quadrature nodes and weights is presented. The algorithm is based on Newton's root-finding method with initial guesses and function evaluations computed via asymptotic formulae. The $n$-point quadrature rule is computed in $\mathcal{O}(n)$ operations to an accuracy of essentially double precision for any $n\geq 100$.

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Fast and accurate computation of gauss-legendre and gauss-jacobi quadrature nodes and weights ∗

(1 < p ≤ ∞) [LS87f]. (2) [HR88a]. (2m− 2) [KL88]. (A0, A1)θ1 [Xu87a]. (α, β) [Pie88a, Fin88a]. (d ≥ 1) [Wsc85a]. (λ) [DM85b]. (Z/2) [Car86b]. (nα) [Sch85h]. (φ)2 [BM89c]. (τ − λ)u = f [Wei87r]. (x, t) [Lum87, Lum89]. (X1 −X3, X2 −X3) [SW87]. 0 [Caz88, Kas86, Pro87]. 0 < p < 1 [Cle87]. 1 [Bak85a, DD85, Drm87, Eli88, FT88a, Gek86d, HN88, Kos86a, LT89, Pet89a, Pro87, Tan87, vdG89]. 1/4 [KS86e]. 1 ≤ q < 2 [Gue86]. 2 [BPPS87, Cam88, Cat85a, ES87b, Gan85e, Gol86a, HRL89g, Hei85, Hua86, Kan89, KB86, Li86, LT89, Mil87b, Mur85a, Qui85b, SP89, Shi85, Spe86, Wal85b, Wan86]. 2m− 2 [Kos88b]. 2m− 3 [Kos88b]. 2m− 4 [Kos88b]. 2× 2 [Vog88]. 3 [Aso89, BPPS87, BW85c, BG88b, Che86d, Fis86, Gab85, Gu87a, HLM85b, Kam89b, Kir89c, Lev85c, Mil85c, Néd86, Pet86, Ron86, Sch85b, ST88, Tur88b, Wan86, Wen85, tDP89, vdW86]. 4 [Bau88a, Don85a, FKV88, Kha88, Kir89l, SS86, Seg85b, Wal85b]. 5 [Ito89, Kir89e, SV85]. 5(4) [Cas86]. 5819539783680 [KSX87]. 6 [PH89, Žub88].

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A Complete Bibliography of Lecture Notes in Mathematics (1985{1989)

In this paper we obtain two asymptotic formulas for the zeros $ \lambda _{n,k}^{(\alpha )},k = 1,2, \ldots ,n, $ of the Laguerre polynomials $ L_n^{(\alpha )}(x) $, as n → ∞ and α is fixed. These formulas are in terms of the zeros of the Bessel function J (x) and in terms of the zeros of the Airy function Ai(χ). They hold for k — 1, 2, ..., [qn] and for k — [pn], [pn] + 1, ..., n respectively, where p and q are fixed numbers in the interval (0, 1).

Uniform Approximations for the Zeros of Laguerre Polynomials

Uniform asymptotic representations for the zeros of the Whittaker functions M
κ,μ(z) and W
κ,μ(z) are derived from well-known uniform asymptotic expansions. The approximation formulas involve the zeros of Bessel or Airy functions.

New Asymptotics for the Zeros of Whittaker's Functions

In questo lavoro si studiano le formule di quadratura della forma

$$\int\limits_{ - 1}^1 {(1 - x)^a (1 + x)^\beta f(x)dx = \sum\limits_{i = 0}^{r - 1} {[A_i f^{(i)} ( - 1) + B_i f^{(i)} (1)] + K_n (\alpha ,\beta ;r)\sum\limits_{i = 1}^n f } (x_{n,i} )} ,$$

(α>−1, β>−1), conAi,Bi,Kn reali e nodi realixn,i compresi nell'intervallo(−1,1) valide per polinomi di grado ≤2n+2r−1. Nella prima parte si prova che c'e validita per polinomi di grado esattamente2n+2r−1 se e solo se$\alpha = \beta = - \frac{1}{2}$r=0 oppurer=1. Nella seconda parte si considera il problema dell'esistenza della formula

$$\int\limits_{ - 1}^1 {(1 - x^2 )^{\lambda - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} f(x)} dx = A_n f( - 1) + B_n f(1) + C_n \sum\limits_{i = 1}^n {f(x_{n,i} )}$$

per polinomi di grado ≤n+2. Alcuni risultati numerici sono dati per,$\lambda = \frac{1}{2}$.

Alcuni problemi relativi alle formule di quadratura di tipo Tchebycheff

In questo lavoro si studiano le due formule di quadratura

$$\int\limits_{ - 1}^1 {(1 - x^2 )^{\lambda - 1/2} f(x)dx = C_n^{ (\lambda )} \sum\limits_{i = 1}^n f (x_{n,i} ) + R_n \left[ f \right]} ,$$

(1)



$$\int\limits_{ - 1}^1 {(1 - x^2 )^{\lambda - 1/2} f(x)dx = A_n^{ (\lambda )} \left[ {f\left( { - 1} \right) + f\left( 1 \right)} \right] + K_n^{ (\lambda )} \sum\limits_{i = 1}^n f (\bar x_{n,i} ) + \bar R_n \left[ f \right]} ,$$

(2)

con 0<λ<1, e si stabiliscono disuguaglianze per il loro gradoN di precisione polinomiale.

Sul grado di precisione di formule di quadratura del tipo di tchebycheff

In this paper a two-term approximation for the weights n,k (α, β) of the Gauss-Jacobi numerical quadrature rule is given. This approximation, which is obtained by using some recent results on the uniform asymptotic expansion of Jacobi polynomials and their zeros, holds as n → ∞ for k = 1, 2,..., [qn], where q is a fixed number in the interval (0,1).

Luigi Gatteschi

Papers

Uniform Approximations for the Zeros of Laguerre Polynomials

New Asymptotics for the Zeros of Whittaker's Functions

Alcuni problemi relativi alle formule di quadratura di tipo Tchebycheff

Sul grado di precisione di formule di quadratura del tipo di tchebycheff

Uniform Approximation of Christoffel Numbers for Jacobi Weight