In this chapter we define the special functions used in this volume and state the properties relevant to the treatment of orthogonal polynomials. We also state a few facts from complex analysis used in the later parts.

We begin by outlining some methods for getting information on zeros of orthogonal polynomials. Some of the main ones depend on the orthogonality measure, the recurrence relation and, if available, the differential equation for the polynomials.

In Section 10.2 we give results obtained by these methods for general classes of orthogonal polynomials. Sections 10.3 to 10.8 give specific applications to, and special results for, Jacobi, ultraspherical, Legendre, Laguerre, Hermite, and other polynomials.

Other chapters contain material on zeros. Zeros of Bessel polynomials are discussed in Section 3.13.

The continuous q-ultraspherical and continuous q-Hermite polynomials first appeared in Rogers’ work on the Rogers–Ramanujan identities in 1893–95 (Askey and Ismail, 1983). They belong to the Fejér class of polynomials having a generating function of the form

∑n=0∞ϕn(cosθ)tn=|F(reiθ)|2, (7.0.1)

where F(z) is analytic in a neighborhood of z=0. Feldheim (1941) and Lanzewizky (1941) independently proved that the only orthogonal generalized polynomials in the Fejér class are either the ultraspherical polynomials or the q-ultraspherical polynomials or special cases of them. They proved that F has to be F1 or F2, or some limiting cases of them, where

These polynomials appeared first in Meixner (1934) as orthogonal polynomials of Sheffer A-type zero relative to ddx. This is equivalent to having a generating function of the form

∑n=0∞​pn(x)cntn=A(t)exp(xH(t)),A(t)=∑n=0∞​antn,  H(t)=∑n=0∞​hn+1tn+1, (5.1.1)

for some sequence (cn)n with a0cnh1≠0 for all n. Their recurrence relation is

One way to generalize orthogonal polynomials on subsets of ℝ is to consider orthogonality on curves in the complex plane. Among these generalizations, the most developed theory is the general theory of orthogonal polynomial on the unit circle T. The basic sources for this chapter are Grenander and Szegő (1958), Szegő ([1939] 1975), Geronimus (1961, 1962), Simon (2004a,b), Ismail (2005b, Chapters 8 and 17), and recent papers which will be cited in the appropriate places.

In what follows we shall use Simon’s abbreviation OPUC for orthogonal polynomials on the unit circle.

The Al-Salam–Chihara polynomials appeared in a characterization problem regarding convolutions of orthogonal polynomials. Al-Salam and Chihara (1976) only recorded the three-term recurrence relation and a generating function. The weight function was first found by Askey and Ismail (1983, 1984), who also named the polynomials after the ones who first identified them.

A basic hypergeometric representation is

pn(x;t1,t2|q)=ϕ32(q−n,t1eiθ,t1e−iθt1t2,0|q,q), (8.1.1)

and the orthogonality relation is

Suppose we are given a positive Borel measure μ on ℝ with infinite support whose moments

mn := ∫Rxndμ(x)

exist for n=0,1,…. We normalize μ by m0=1. The distribution function Fμ is right continuous and defined by

Fμ(x)=μ((−∞,x])=∫−∞xdμ(t). (2.1.1)

A polynomial sequence (φn(x))n is a sequence of polynomials such that φn has exact degree n. Such a sequence is monic if φn(x)−xn has degree at most n−1.

A birth and death process is a stationary Markov process whose states are labeled by nonnegative integers and whose transition probabilities

pm,n(t)=Pr{X(t)=n|X(0)=m} (4.1.1)

satisfy certain conditions as t→0+:

{pn,n+1(t)=λnt=o(t),pn,n−1(t)=μnt+o(t),pn,n(t)=1−(λn+μn)t+o(t),pn,m(t)=o(t),|m−n|>1.

It is assumed that

λn>0,n≥0 and μ0≥0,μn>0,n>0. (4.1.2)

Among the classical (scalar-valued) families of orthogonal polynomials with rich and deep connections to several branches of mathematics, the Jacobi polynomials occupy a distinguished role.

In this contribution we describe a way of obtaining some families of matrix-valued orthogonal polynomials of arbitrary dimension and depending on two parameters α, β, which extends the scalar theory in many respects. We will achieve this goal by focusing on a group representation approach. In the scalar case the Jacobi polynomials appeared in several concrete mathematical physics problems in the hands of people like Laplace and Legendre. The group-theoretical interpretation, in the hands of E. Cartan and H. Weyl, is of more recent vintage.

The Jacobi polynomials are defined by

Pn(α,β)(x)=(α+1)nn!F12(−n,α+β+n+1;α+1;(1−x)/2), (3.1.1)

and satisfy the orthogonality relations

∫−11Pm(α,β)(x)Pn(α,β)(x)(1−x)α(1+x)βdx=hn(α,β)δm,n, (3.1.2)

hn(α,β)=2α+β+1Γ(α+n+1)Γ(β+n+1)n!Γ(α+β+n+1)(α+β+2n+1). (3.1.3)