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This book presents an introduction to orthogonal polynomials, with an algebraic flavor, based on linear functionals defining the orthogonality and the Jacobi matrices associated with them. Basic properties of their zeros, as well as quadrature rules, are discussed. A key point is the analysis of those functionals satisfying Pearson equations (semiclassical case) and the hierarchy based on their class. The book's structure reflects the fact that its content is based on a set of lectures delivered by one of the authors at the first Orthonet Summer School in Seville, Spain in 2016. The presentation of the material is self contained and will be valuable to students and researchers interested in a novel approach to the study of orthogonal polynomials, focusing on their analytic properties.
Moment functionals on P and orthogonal polynomials
Existence of orthogonal polynomial sequences
Three-term recurrence relation
Christoffel–Darboux kernel polynomials
Polynomials of the first kind and the Stieltjes function
Continued fractions
Continued fractions and orthogonal polynomials
Zeros of orthogonal polynomials
The interlacing property of zeros
Gauss quadrature rules
Symmetric functionals
LU factorization
Transformations of moment functionals
Canonical Christoffel transformation
Canonical Geronimus transformation
Uvarov transformation
Classical orthogonal polynomials
The linear differential operator and its solutions
Weight function and inner product
Classical functionals
Electrostatic interpretation for the zeros of classical orthogonal polynomials
Equilibrium points on a bounded interval with charged end points
Equilibrium points on the complex plane: The Bessel case
Classical orthogonal polynomials and the inverse problem
Semiclassical functionals
Examples of semiclassical orthogonal polynomials
The Askey scheme
Hahn polynomials
Jacobi polynomials
Meixner polynomials
Krawtchouk polynomials
Laguerre polynomials
Bessel polynomials
Charlier polynomials
Hermite polynomials
Limit relations