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)

Asymptotics and bounds for the zeros of Laguerre polynomials: a survey

It is shown that certain asymptotic approximations are upper or lower bounds for the zeros $\theta _{n,k} (\alpha ,\beta )$ of Jacobi polynomials $P_n^{(\alpha ,\beta )} (\cos \theta )$. The procedure for deriving these bounds is based on the Sturm comparison theorem. Numerical examples are given to illustrate the sharpness of the new inequalities.

New inequalities for the zeros of Jacobi polynomials

(1983). On the concavity of zeros of bessel functions. Applicable Analysis: Vol. 16, No. 4, pp. 261-278.

On the concavity of zeros of bessel functions

Let P, ' -) (x) be the Jacobi polynomial of degree n. For -2 < a, 8 < 2 and 0 < 0 < x, it is proved that (sn2)a 7 co ).8+ 7 p(a, ) (cos 0) I < rF(q + I ) (n + q Nq2 2 f,( I) \ n where q = max(a, ,B) and N = n + 2(a +,B + 1). When a=f= 0, this reduces to a sharpened form of the well-known Bernstein inequality for the Legendre polynomial.

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A Bernstein-type inequality for the Jacobi polynomial

We consider a new asymptotic approximation of Jacobi polynomials P n (α,β) (cosϑ) and we obtain a realistic and explicit bound for the corresponding error term. The approximation is of Hilb's type and is uniformly valid for 0 0. Bounds for the error term in the asymptotic approximation of the zeros of P n (α,β) (cosϑ) are also given.

Luigi Gatteschi

Papers

Asymptotics and bounds for the zeros of Laguerre polynomials: a survey

New inequalities for the zeros of Jacobi polynomials

On the concavity of zeros of bessel functions

A Bernstein-type inequality for the Jacobi polynomial

The bounds for the error term of an asymptotic approximation of Jacobi polynomials