- §18.2(i) Definition
- §18.2(ii) $x$-Difference Operators
- §18.2(iii) Normalization
- §18.2(iv) Recurrence Relations
- §18.2(v) Christoffel–Darboux Formula
- §18.2(vi) Zeros

Let $(a,b)$ be a finite or infinite open interval in $\mathrm{\mathbb{R}}$. A system (or set)
of polynomials $\{{p}_{n}(x)\}$, $n=0,1,2,\mathrm{\dots},$ is said to be
*orthogonal on* $(a,b)$ *with respect to the weight function*
$w(x)$ ($\ge 0$) *if*

18.2.1 | $${\int}_{a}^{b}{p}_{n}(x){p}_{m}(x)w(x)dx=0,$$ | ||

$n\ne m$. | |||

Here $w(x)$ is continuous or piecewise continuous or integrable, and such that $$ for all $n$.

It is assumed throughout this chapter that for each polynomial ${p}_{n}(x)$
that is orthogonal on an open interval $(a,b)$ the variable $x$ is confined to
the closure of $(a,b)$ *unless indicated otherwise.*
(However, under appropriate conditions almost all equations given in the chapter
can be continued analytically to various complex values of the variables.)

Let $X$ be a finite set of distinct points on $\mathrm{\mathbb{R}}$, or a countable infinite
set of distinct points on $\mathrm{\mathbb{R}}$, and ${w}_{x}$, $x\in X$, be a set of positive
constants. Then a system of polynomials $\{{p}_{n}(x)\}$, $n=0,1,2,\mathrm{\dots},$ is
said to be *orthogonal* on $X$ with respect to the *weights* ${w}_{x}$ if

18.2.2 | $$\sum _{x\in X}{p}_{n}(x){p}_{m}(x){w}_{x}=0,$$ | ||

$n\ne m$, | |||

when $X$ is infinite, or

18.2.3 | $$\sum _{x\in X}{p}_{n}(x){p}_{m}(x){w}_{x}=0,$$ | ||

$n,m=0,1,\mathrm{\dots},N;n\ne m$, | |||

when $X$ is a finite set of $N+1$ distinct points. In the former case we also require

18.2.4 | $$ | ||

$n=0,1,\mathrm{\dots}$, | |||

whereas in the latter case the system $\{{p}_{n}(x)\}$ is finite: $n=0,1,\mathrm{\dots},N$.

More generally than (18.2.1)–(18.2.3), $w(x)dx$ may be replaced in (18.2.1) by a positive measure $d\alpha (x)$, where $\alpha (x)$ is a bounded nondecreasing function on the closure of $(a,b)$ with an infinite number of points of increase, and such that $$ for all $n$. See McDonald and Weiss (1999, Chapters 3, 4) and Szegö (1975, §1.4).

If the orthogonality discrete set $X$ is $\{0,1,\mathrm{\dots},N\}$ or $\{0,1,2,\mathrm{\dots}\}$, then the role of the differentiation operator $d/dx$ in the case of classical OP’s (§18.3) is played by ${\mathrm{\Delta}}_{x}$, the forward-difference operator, or by ${\nabla}_{x}$, the backward-difference operator; compare §18.1(i). This happens, for example, with the Hahn class OP’s (§18.20(i)).

If the orthogonality interval is $(-\mathrm{\infty},\mathrm{\infty})$ or $(0,\mathrm{\infty})$, then the role of $d/dx$ can be played by ${\delta}_{x}$, the central-difference operator in the imaginary direction (§18.1(i)). This happens, for example, with the continuous Hahn polynomials and Meixner–Pollaczek polynomials (§18.20(i)).

The orthogonality relations (18.2.1)–(18.2.3) each determine the polynomials ${p}_{n}(x)$ uniquely up to constant factors, which may be fixed by suitable normalization.

If we define

18.2.5 | $${h}_{n}={\int}_{a}^{b}{\left({p}_{n}(x)\right)}^{2}w(x)dx\text{or}\sum _{x\in X}{\left({p}_{n}(x)\right)}^{2}{w}_{x},$$ | ||

18.2.6 | $${\stackrel{~}{h}}_{n}={\int}_{a}^{b}x{\left({p}_{n}(x)\right)}^{2}w(x)dx\text{or}\sum _{x\in X}x{\left({p}_{n}(x)\right)}^{2}{w}_{x},$$ | ||

and

18.2.7 | $${p}_{n}(x)={k}_{n}{x}^{n}+{\stackrel{~}{k}}_{n}{x}^{n-1}+{\stackrel{~}{\stackrel{~}{k}}}_{n}{x}^{n-2}+\mathrm{\cdots},$$ | ||

then two special normalizations are:
(i) *orthonormal OP’s*: ${h}_{n}=1$, ${k}_{n}>0$;
(ii) *monic OP’s*: ${k}_{n}=1$.

As in §18.1(i) we assume that ${p}_{-1}(x)\equiv 0$.

18.2.8 | $${p}_{n+1}(x)=({A}_{n}x+{B}_{n}){p}_{n}(x)-{C}_{n}{p}_{n-1}(x),$$ | ||

$n\ge 0$. | |||

Here ${A}_{n}$, ${B}_{n}$ ($n\ge 0$), and ${C}_{n}$ ($n\ge 1$) are real constants, and ${A}_{n-1}{A}_{n}{C}_{n}>0$ for $n\ge 1$. Then

18.2.9 | ${A}_{n}$ | $={\displaystyle \frac{{k}_{n+1}}{{k}_{n}}},$ | ||

${B}_{n}$ | $=\left({\displaystyle \frac{{\stackrel{~}{k}}_{n+1}}{{k}_{n+1}}}-{\displaystyle \frac{{\stackrel{~}{k}}_{n}}{{k}_{n}}}\right){A}_{n}=-{\displaystyle \frac{{\stackrel{~}{h}}_{n}}{{h}_{n}}}{A}_{n},$ | |||

${C}_{n}$ | $={\displaystyle \frac{{A}_{n}{\stackrel{~}{\stackrel{~}{k}}}_{n}+{B}_{n}{\stackrel{~}{k}}_{n}-{\stackrel{~}{\stackrel{~}{k}}}_{n+1}}{{k}_{n-1}}}={\displaystyle \frac{{A}_{n}}{{A}_{n-1}}}{\displaystyle \frac{{h}_{n}}{{h}_{n-1}}}.$ | |||

18.2.10 | $$x{p}_{n}(x)={a}_{n}{p}_{n+1}(x)+{b}_{n}{p}_{n}(x)+{c}_{n}{p}_{n-1}(x),$$ | ||

$n\ge 0$. | |||

Here ${a}_{n}$, ${b}_{n}$ ($n\ge 0$), ${c}_{n}$ ($n\ge 1$) are real constants, and ${a}_{n-1}{c}_{n}>0$ ($n\ge 1$). Then

18.2.11 | ${a}_{n}$ | $={\displaystyle \frac{{k}_{n}}{{k}_{n+1}}},$ | ||

${b}_{n}$ | $={\displaystyle \frac{{\stackrel{~}{k}}_{n}}{{k}_{n}}}-{\displaystyle \frac{{\stackrel{~}{k}}_{n+1}}{{k}_{n+1}}}={\displaystyle \frac{{\stackrel{~}{h}}_{n}}{{h}_{n}}},$ | |||

${c}_{n}$ | $={\displaystyle \frac{{\stackrel{~}{\stackrel{~}{k}}}_{n}-{a}_{n}{\stackrel{~}{\stackrel{~}{k}}}_{n+1}-{b}_{n}{\stackrel{~}{k}}_{n}}{{k}_{n-1}}}={a}_{n-1}{\displaystyle \frac{{h}_{n}}{{h}_{n-1}}}.$ | |||

If the OP’s are orthonormal, then ${c}_{n}={a}_{n-1}$ ($n\ge 1$). If the OP’s are monic, then ${a}_{n}=1$ ($n\ge 0$).

Conversely, if a system of polynomials $\{{p}_{n}(x)\}$ satisfies (18.2.10) with ${a}_{n-1}{c}_{n}>0$ ($n\ge 1$), then $\{{p}_{n}(x)\}$ is orthogonal with respect to some positive measure on $\mathrm{\mathbb{R}}$ (Favard’s theorem). The measure is not necessarily of the form $w(x)dx$ nor is it necessarily unique.

18.2.12 | $$\sum _{\mathrm{\ell}=0}^{n}\frac{{p}_{\mathrm{\ell}}(x){p}_{\mathrm{\ell}}(y)}{{h}_{\mathrm{\ell}}}=\frac{{k}_{n}}{{h}_{n}{k}_{n+1}}\frac{{p}_{n+1}(x){p}_{n}(y)-{p}_{n}(x){p}_{n+1}(y)}{x-y},$$ | ||

$x\ne y$. | |||

18.2.13 | $$\sum _{\mathrm{\ell}=0}^{n}\frac{{({p}_{\mathrm{\ell}}(x))}^{2}}{{h}_{\mathrm{\ell}}}=\frac{{k}_{n}}{{h}_{n}{k}_{n+1}}\left({p}_{n+1}^{\prime}(x){p}_{n}(x)-{p}_{n}^{\prime}(x){p}_{n+1}(x)\right).$$ | ||

All $n$ zeros of an OP ${p}_{n}(x)$ are simple, and they are located in the interval of orthogonality $(a,b)$. The zeros of ${p}_{n}(x)$ and ${p}_{n+1}(x)$ separate each other, and if $$ then between any two zeros of ${p}_{m}(x)$ there is at least one zero of ${p}_{n}(x)$.