Orthogonal Polynomials

In this chapter, we give an overview of the links between Riordan arrays and orthogonal polynomials, and then we study some specialized areas including classical and semi-classical orthogonal polynomials defined by Riordan arrays, orthogonal polynomials that can be described as the moment sequences of Riordan arrays, applications of exponential Riordan arrays to the Toda lattice equations, and combinatorial polynomials that are moments of Riordan arrays. Orthogonal polynomials enjoy a special place both in pure mathematics and in applied mathematics. Stieltjes, in studying the moment sequences associated to families of orthogonal polynomials, defined the integral that now bears his name. Chebyshev and others, by putting the theory of orthogonal polynomials on a firm basis, provided a tool that has proved invaluable to mathematicians working in the area of approximation of functions, in the area of differential equations, and in many branches of mathematical physics. Traditional orthogonal polynomials are studied on the real line and on the circle. More sophisticated approaches study orthogonal polynomials defined on curves. We shall see that the orthogonal polynomials that can be defined by Riordan arrays are defined either over finite intervals on the real line, or intervals and some discrete points, or in the case of exponential Riordan arrays, over intervals of infinite extent, such as [ 0, ∞) or even (- ∞, ∞). Families of orthogonal polynomials over the real line are typically associated with a measure, which is often realized through a density or weight function. In the case of orthogonal polynomials defined by ordinary Riordan arrays, this weight function can be determined. However, in the case of orthogonal polynomials determined by exponential Riordan arrays, such weight functions are only known in special cases. The production matrix plays a vital role in this theory, as it is precisely when the production matrix is tri-diagonal in form that the corresponding Riordan array (either ordinary or exponential) defines a family of orthogonal polynomials. In the theory of orthogonal polynomials, a distinction is made between so-called “classical” orthogonal polynomials and those that are not “classical”. Such a distinction can also be made for those orthogonal polynomials that can be defined by Riordan arrays.