Hahn polynomials

Abstract: Riemann-Hilbert problems Jacobi operators Orthogonal polynomials Continued fractions Random matrix theory Equilibrium measures Asymptotics for orthogonal polynomials Universality Bibliography.

Abstract: 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.

Abstract: BASIC THEORY 11 Orthogonal polynomials 12 Properties of orthogonal polynomials 13 Three-term recurrence relation 14 Quadrature rules 15 Classical orthogonal polynomials 16 Kernal polynomials 17 Sobolev orthogonal polynomials 18 Orthogonal polynomials on the semicircle 19 Notes to chapter 1 COMPUTATIONAL METHODS 21 Moment-based methods 22 Discretization methods 23 Computing Cauchy integrals of orthogonal polynomials 24 Modification algorithms 25 Computing Sobolev orthogonal polynomials 26 Notes to chapter 2 APPLICATIONS 31 Quadrature 32 Least squares approximation 33 Moment-preserving spline approximation 34 Slowly convergent series 35 Notes to chapter 3