Generalizations of orthogonal polynomials

Generalizations of orthogonal polynomials

We give a survey of recent generalizations for orthogonal polynomials that were recently obtained. It concerns not only multidimensional (matrix and vector orthogonal polynomials) and multivariate versions, or multipole (orthogonal rational functions) variants of the classical polynomials but also extensions of the orthogonality conditions (multiple orthogonality). Most of these generalizations...

Some generalizations of the combinatorics of orthogonal polynomials blah blah blah

We generalize the combinatorial interpretations of orthogonal polynomials found in [2] in two directions: first, we show that the interpretation of moments as a generating function of Motzkin paths may be generalized to no only different linear functionals, but to arbitrary sequences of polynomials. Second, we develop several new involutions to describe the L norm of polynomials that satisfy a ...

Orthogonal Polynomials

The theory of orthogonal polynomials plays an important role in many branches of mathematics, such as approximation theory (best approximation, interpolation, quadrature), special functions, continued fractions, differential and integral equations. The notion of orthogonality originated from the theory of continued fractions, but later became an independent (and possibly more important) discipl...

Orthogonal Polynomials

Orthogonal Polynomials

In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed.