§18.2 General Orthogonal Polynomials

Referenced by:: §32.15
Permalink:: http://dlmf.nist.gov/18.2

§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

§18.2(i) Definition

Notes:: See Andrews et al. (1999, Definition 5.2.1).
Referenced by:: §18.18(viii), ¶ ‣ §18.19, §18.25(i), §18.3, §3.5(v)
Permalink:: http://dlmf.nist.gov/18.2.i

¶ Orthogonality on Intervals

Keywords:: general orthogonal polynomials, weight functions

Let $(a,b)$ be a finite or infinite open interval in $\Real$ . A system (or set) of polynomials $\{ p_{n}(x)\}$ , $n=0,1,2,\ldots,$ is said to be orthogonal on $(a,b)$ with respect to the weight function $w(x)$ ( $\geq 0$ ) if

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

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $w(x)$ : weight function, $m$ : nonnegative integer, $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ , $a$ : interval endpoint, $b$ : interval endpoint and $x$ : real variable
A&S Ref:: 22.1.1
Referenced by:: ¶ ‣ §18.2(i), §18.2(iii), ¶ ‣ §18.5(iii)
Permalink:: http://dlmf.nist.gov/18.2.E1
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Here $w(x)$ is continuous or piecewise continuous or integrable, and such that $0<\int _{a}^{b}x^{{2n}}w(x)dx<\infty$ 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.)

¶ Orthogonality on Finite Point Sets

Keywords:: general orthogonal polynomials, measure

Let $X$ be a finite set of distinct points on $\Real$ , or a countable infinite set of distinct points on $\Real$ , and $w_{x}$ , $x\in X$ , be a set of positive constants. Then a system of polynomials $\{ p_{n}(x)\}$ , $n=0,1,2,\ldots,$ is said to be orthogonal on $X$ with respect to the weights $w_{x}$ if

18.2.2 $\sum _{{x\in X}}p_{n}(x)p_{m}(x)w_{x}=0,$ $n\neq m$ ,

Symbols:: $\in$ : element of, $w_{x}$ : weights, $X$ : subset of $\Real$ , $m$ : nonnegative integer, $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ and $x$ : real variable
A&S Ref:: 22.1.1
Permalink:: http://dlmf.nist.gov/18.2.E2
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when $X$ is infinite, or

18.2.3 $\sum _{{x\in X}}p_{n}(x)p_{m}(x)w_{x}=0,$ $n,m=0,1,\ldots,N;n\neq m$ ,

Symbols:: $\in$ : element of, $w_{x}$ : weights, $X$ : subset of $\Real$ , $m$ : nonnegative integer, $n$ : nonnegative integer, $N$ : positive integer, $p_{n}(x)$ : polynomial of degree $n$ and $x$ : real variable
A&S Ref:: 22.1.1
Referenced by:: ¶ ‣ §18.2(i), §18.2(iii)
Permalink:: http://dlmf.nist.gov/18.2.E3
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when $X$ is a finite set of $N+1$ distinct points. In the former case we also require

18.2.4 $\sum _{{x\in X}}x^{{2n}}w_{x}<\infty,$ $n=0,1,\dots$ ,

Symbols:: $\in$ : element of, $w_{x}$ : weights, $X$ : subset of $\Real$ , $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.2.E4
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whereas in the latter case the system $\{ p_{n}(x)\}$ is finite: $n=0,1,\ldots,N$ .

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 $0<\int _{a}^{b}x^{{2n}}d\alpha(x)<\infty$ for all $n$ . See McDonald and Weiss (1999, Chapters 3, 4) and Szegö (1975, §1.4).

§18.2(ii) $x$ -Difference Operators

Keywords:: difference operators, general orthogonal polynomials
Referenced by:: §18.20(i), §18.26(iii)
Permalink:: http://dlmf.nist.gov/18.2.ii

If the orthogonality discrete set $X$ is $\{ 0,1,\dots,N\}$ or $\{ 0,1,2,\dots\}$ , then the role of the differentiation operator $\ifrac{d}{dx}$ in the case of classical OP’s (§18.3) is played by $\Delta _{{x}}$ , 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 $(-\infty,\infty)$ or $(0,\infty)$ , then the role of $\ifrac{d}{dx}$ 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)).

§18.2(iii) Normalization

Notes:: See Szegö (1975, §2.2(i)), Chihara (1978, Chapter 1, Theorem 3.3).
Keywords:: general orthogonal polynomials
Referenced by:: ¶ ‣ §18.19, §18.25(iv), §18.3
Permalink:: http://dlmf.nist.gov/18.2.iii

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 $h_{n}=\int _{a}^{b}\left(p_{n}(x)\right)^{2}w(x)dx\text{ or }\sum _{{x\in X}}\left(p_{n}(x)\right)^{2}w_{x},$

Symbols:: $dx$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $w(x)$ : weight function, $w_{x}$ : weights, $X$ : subset of $\Real$ , $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ , $x$ : real variable and $h_{n}$
A&S Ref:: 22.1.2
Referenced by:: §3.5(v)
Permalink:: http://dlmf.nist.gov/18.2.E5
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18.2.6 $\tilde{h}_{n}=\int _{a}^{b}x\left(p_{n}(x)\right)^{2}w(x)dx\text{ or }\sum _{{x\in X}}x\left(p_{n}(x)\right)^{2}w_{x},$

Symbols:: $dx$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $w(x)$ : weight function, $w_{x}$ : weights, $X$ : subset of $\Real$ , $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.2.E6
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and

18.2.7 $p_{n}(x)=k_{n}x^{n}+\tilde{k}_{n}x^{{n-1}}+\tilde{\tilde{k}}_{n}x^{{n-2}}+\cdots,$

Symbols:: $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ , $x$ : real variable and $k_{n}$
A&S Ref:: 22.1.2
Referenced by:: §18.15(vi), §3.5(v)
Permalink:: http://dlmf.nist.gov/18.2.E7
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then two special normalizations are: (i) orthonormal OP’s: $h_{n}=1$ , $k_{n}>0$ ; (ii) monic OP’s: $k_{n}=1$ .

§18.2(iv) Recurrence Relations

Notes:: See Andrews et al. (1999, Theorem 5.2.2 and Remark 5.2.1). For Favard’s theorem see Szegö (1975, §3.2) or Chihara (1978, p. 21).
Keywords:: general orthogonal polynomials
Referenced by:: §18.9(i), §3.5(vi)
Permalink:: http://dlmf.nist.gov/18.2.iv

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

¶ First Form

18.2.8 $p_{{n+1}}(x)=(A_{n}x+B_{n})p_{n}(x)-C_{n}p_{{n-1}}(x),$ $n\geq 0$ .

Symbols:: $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ , $B_{n}$ : real constant, $C_{n}$ : real constant, $x$ : real variable and $A_{n}$ : real constant
A&S Ref:: 22.1.4
Referenced by:: §18.30
Permalink:: http://dlmf.nist.gov/18.2.E8
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Here $A_{n}$ , $B_{n}$ ( $n\geq 0$ ), and $C_{n}$ ( $n\geq 1$ ) are real constants, and $A_{{n-1}}A_{n}C_{n}>0$ for $n\geq 1$ . Then

18.2.9

$A_{n}=\frac{k_{{n+1}}}{k_{n}},$

$B_{n}=\left(\frac{\tilde{k}_{{n+1}}}{k_{{n+1}}}-\frac{\tilde{k}_{n}}{k_{n}}\right)A_{n}=-\frac{\tilde{h}_{n}}{h_{n}}A_{n},$

$C_{n}=\frac{A_{n}\tilde{\tilde{k}}_{n}+B_{n}\tilde{k}_{n}-\tilde{\tilde{k}}_{{n+1}}}{k_{{n-1}}}=\frac{A_{n}}{A_{{n-1}}}\frac{h_{n}}{h_{{n-1}}}.$

Symbols:: $n$ : nonnegative integer, $B_{n}$ : real constant, $C_{n}$ : real constant, $h_{n}$ , $k_{n}$ and $A_{n}$ : real constant
A&S Ref:: 22.1.5
Permalink:: http://dlmf.nist.gov/18.2.E9
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¶ Second Form

18.2.10 $xp_{n}(x)=a_{n}p_{{n+1}}(x)+b_{n}p_{n}(x)+c_{n}p_{{n-1}}(x),$ $n\geq 0$ .

Symbols:: $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ , $a_{n}$ : real constant, $b_{n}$ : real constant, $c_{n}$ : real constant and $x$ : real variable
Referenced by:: ¶ ‣ §18.2(iv)
Permalink:: http://dlmf.nist.gov/18.2.E10
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Here $a_{n}$ , $b_{n}$ ( $n\geq 0$ ), $c_{n}$ ( $n\geq 1$ ) are real constants, and $a_{{n-1}}c_{n}>0$ ( $n\geq 1$ ). Then

18.2.11

$a_{n}=\frac{k_{n}}{k_{{n+1}}},$

$b_{n}=\frac{\tilde{k}_{n}}{k_{n}}-\frac{\tilde{k}_{{n+1}}}{k_{{n+1}}}=\frac{\tilde{h}_{n}}{h_{n}},$

$c_{n}=\frac{\tilde{\tilde{k}}_{n}-a_{n}\tilde{\tilde{k}}_{{n+1}}-b_{n}\tilde{k}_{n}}{k_{{n-1}}}=a_{{n-1}}\frac{h_{n}}{h_{{n-1}}}.$

Symbols:: $n$ : nonnegative integer, $a_{n}$ : real constant, $b_{n}$ : real constant, $c_{n}$ : real constant, $h_{n}$ and $k_{n}$
Permalink:: http://dlmf.nist.gov/18.2.E11
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If the OP’s are orthonormal, then $c_{n}=a_{{n-1}}$ ( $n\geq 1$ ). If the OP’s are monic, then $a_{n}=1$ ( $n\geq 0$ ).

Conversely, if a system of polynomials $\{ p_{n}(x)\}$ satisfies (18.2.10) with $a_{{n-1}}c_{n}>0$ ( $n\geq 1$ ), then $\{ p_{n}(x)\}$ is orthogonal with respect to some positive measure on $\Real$ (Favard’s theorem). The measure is not necessarily of the form $w(x)dx$ nor is it necessarily unique.

§18.2(v) Christoffel–Darboux Formula

Notes:: See Szegö (1975, Theorem 3.2.2 and Eq. (3.2.4)).
Keywords:: Christoffel-Darboux formula, classical orthogonal polynomials, general orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.2.v

18.2.12 $\sum _{{\ell=0}}^{n}\frac{p_{\ell}(x)p_{\ell}(y)}{h_{\ell}}=\frac{k_{n}}{h_{n}k_{{n+1}}}\frac{p_{{n+1}}(x)p_{n}(y)-p_{n}(x)p_{{n+1}}(y)}{x-y},$ $x\neq y$ .

Symbols:: $y$ : real variable, $\ell$ : nonnegative integer, $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ , $x$ : real variable, $h_{n}$ and $k_{n}$
A&S Ref:: 22.12.1
Permalink:: http://dlmf.nist.gov/18.2.E12
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¶ Confluent Form

Keywords:: Christoffel-Darboux formula

18.2.13 $\sum _{{\ell=0}}^{n}\frac{(p_{\ell}(x))^{2}}{h_{\ell}}=\frac{k_{n}}{h_{n}k_{{n+1}}}{\left(p_{{n+1}}^{{\prime}}(x)p_{n}(x)-p_{n}^{{\prime}}(x)p_{{n+1}}(x)\right)}.$

Symbols:: $\ell$ : nonnegative integer, $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ , $x$ : real variable, $h_{n}$ and $k_{n}$
Permalink:: http://dlmf.nist.gov/18.2.E13
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§18.2(vi) Zeros

Notes:: See Andrews et al. (1999, Theorems 5.4.2, 5.4.3).
Keywords:: Chebyshev polynomials, Hermite polynomials, Jacobi polynomials, Laguerre polynomials, Legendre polynomials, classical orthogonal polynomials, general orthogonal polynomials, ultraspherical polynomials
Referenced by:: §18.16(i)
Permalink:: http://dlmf.nist.gov/18.2.vi

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 $m<n$ then between any two zeros of $p_{m}(x)$ there is at least one zero of $p_{n}(x)$ .

For illustrations of these properties see Figures 18.4.1–18.4.7.