18 Orthogonal Polynomials General Orthogonal Polynomials 18.1 Notation 18.3 Definitions

§18.2 General Orthogonal Polynomials

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Referenced by:: §32.15, Erratum (V1.2.0) §18.2
Permalink:: http://dlmf.nist.gov/18.2
See also:: Annotations for Ch.18

§18.2(i) Definition
§18.2(ii) $x$ -Difference Operators
§18.2(iii) Standardization and Related Constants
§18.2(iv) Recurrence Relations
§18.2(v) Christoffel–Darboux Formula
§18.2(vi) Zeros
§18.2(vii) Quadratic Transformations
§18.2(viii) Uniqueness of Orthogonality Measure and Completeness
§18.2(ix) Moments
§18.2(x) Orthogonal Polynomials and Continued Fractions
§18.2(xi) Some Special Classes of General Orthogonal Polynomials
§18.2(xii) Other Special Constructions Involving General OP’s

§18.2(i) Definition

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Notes:: See Andrews et al. (1999, Definition 5.2.1).
Referenced by:: §18.18(viii), §18.19, §18.25(i), §18.3, §18.38(iii), §3.5(v), Erratum (V1.2.0) §18.2
Permalink:: http://dlmf.nist.gov/18.2.i
Addition (effective with 1.2.0):: Equations (18.2.1_5), (18.2.4_5) were added.
See also:: Annotations for §18.2 and Ch.18

Orthogonality on Intervals

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Keywords:: definition, general orthogonal polynomials, on intervals, weight functions
See also:: Annotations for §18.2(i), §18.2 and Ch.18

Let $(a,b)$ be a finite or infinite open interval in $\mathbb{R}$ . A system (or set) of polynomials $\{p_{n}(x)\}$ , $n=0,1,2,\ldots$ , where $p_{n}(x)$ has degree $n$ as in §18.1(i), is said to be orthogonal on $(a,b)$ with respect to the weight function $w(x)$ ( $\geq 0$ ) if

18.2.1		$\int_{a}^{b}p_{n}(x)p_{m}(x)w(x)\,\mathrm{d}x=0,$
$n\neq m$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $p_{n}(x)$ : polynomial of degree $n$ , $w(x)$ : weight function, $m$ : nonnegative integer, $n$ : nonnegative integer, $a$ : interval endpoint, $b$ : interval endpoint and $x$ : real variable A&S Ref: 22.1.1 Referenced by: §18.2(i), §18.2(iii), §18.2(viii), §18.5(iii) Permalink: http://dlmf.nist.gov/18.2.E1 Encodings: TeX, pMML, png See also: Annotations for §18.2(i), §18.2(i), §18.2 and Ch.18

Here $w(x)$ is continuous or piecewise continuous or integrable such that

18.2.1_5		$0<\int_{a}^{b}w(x)\,\mathrm{d}x<\infty,$
		$\int_{a}^{b}\|x\|^{n}w(x)\,\mathrm{d}x<\infty,$
	$n\in\mathbb{N}$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $\mathbb{N}$ : set of all positive integers, $w(x)$ : weight function, $n$ : nonnegative integer, $a$ : interval endpoint, $b$ : interval endpoint and $x$ : real variable Source: Szegő (1975, §2.2(1)) Referenced by: §18.2(xi), §18.2(i), Erratum (V1.2.0) §18.2 Permalink: http://dlmf.nist.gov/18.2.E1_5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.2(i), §18.2(i), §18.2 and Ch.18

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 Countable Sets

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Keywords:: general orthogonal polynomials, on countable point sets
Modification (effective with 1.2.0):: This paragraph was previously called “Orthogonality on Finite Point Sets”.
See also:: Annotations for §18.2(i), §18.2 and Ch.18

Let $X$ be a finite set of distinct points on $\mathbb{R}$ , or a countable infinite set of distinct points on $\mathbb{R}$ , and $w_{x}$ , $x\in X$ , be a set of positive constants. Then a system of polynomials $\{p_{n}(x)\}$ , $n=0,1,2,\ldots$ , 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, $p_{n}(x)$ : polynomial of degree $n$ , $w_{x}$ : weights, $X$ : subset of $\mathbb{R}$ , $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.1.1 Referenced by: §18.2(iii) Permalink: http://dlmf.nist.gov/18.2.E2 Encodings: TeX, pMML, png See also: Annotations for §18.2(i), §18.2(i), §18.2 and Ch.18

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, $N$ : positive integer, $p_{n}(x)$ : polynomial of degree $n$ , $w_{x}$ : weights, $X$ : subset of $\mathbb{R}$ , $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.1.1 Referenced by: §18.2(i), §18.2(iv), §18.2(iii), §18.30(iii) Permalink: http://dlmf.nist.gov/18.2.E3 Encodings: TeX, pMML, png See also: Annotations for §18.2(i), §18.2(i), §18.2 and Ch.18

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

18.2.4		$\sum_{x\in X}\|x\|^{n}w_{x}<\infty,$
$n=0,1,\dots$ ,
ⓘ Symbols: $\in$ : element of, $w_{x}$ : weights, $X$ : subset of $\mathbb{R}$ , $n$ : nonnegative integer and $x$ : real variable Permalink: http://dlmf.nist.gov/18.2.E4 Encodings: TeX, pMML, png See also: Annotations for §18.2(i), §18.2(i), §18.2 and Ch.18

whereas in the latter case the system $\{p_{n}(x)\}$ is finite: $n=0,1,\ldots,N$ .

Orthogonality on General Sets

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Keywords:: Lebesgue–Stieltjes measure, general orthogonal polynomials, measure, on general sets, theory
Addition (effective with 1.2.0):: This paragraph was added.
See also:: Annotations for §18.2(i), §18.2 and Ch.18

More generally than (18.2.1)–(18.2.3), $w(x)\,\mathrm{d}x$ may be replaced in (18.2.1) by $\,\mathrm{d}\mu(x)$ , where the measure $\mu$ is the Lebesgue–Stieltjes measure $\mu_{\alpha}$ corresponding to a bounded nondecreasing function $\alpha$ on the closure of $(a,b)$ with an infinite number of points of increase, and such that $\int_{a}^{b}|x|^{n}\,\mathrm{d}\mu(x)<\infty$ for all $n$ . See §1.4(v), McDonald and Weiss (1999, Chapters 3, 4) and Szegő (1975, §1.4). Then

18.2.4_5		$\int_{a}^{b}p_{n}(x)p_{m}(x)\,\mathrm{d}\mu(x)=0,$
$n\neq m$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $p_{n}(x)$ : polynomial of degree $n$ , $m$ : nonnegative integer, $n$ : nonnegative integer, $a$ : interval endpoint, $b$ : interval endpoint and $x$ : real variable Source: Szegő (1975, (2.2.4)); For $n\neq m$ . Referenced by: §18.2(iii), §18.2(iv), §18.2(i), §18.2(v), §18.2(x), Erratum (V1.2.0) §18.2 Permalink: http://dlmf.nist.gov/18.2.E4_5 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.2(i), §18.2(i), §18.2 and Ch.18

§18.2(ii) $x$ -Difference Operators

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Keywords:: difference operators, general orthogonal polynomials, $x$ -difference operators
Referenced by:: §18.20(i), §18.26(iii)
Permalink:: http://dlmf.nist.gov/18.2.ii
See also:: Annotations for §18.2 and Ch.18

If the orthogonality discrete set $X$ is $\{0,1,\dots,N\}$ or $\{0,1,2,\dots\}$ , then the role of the differentiation operator $\ifrac{\mathrm{d}}{\mathrm{d}x}$ 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 $\ifrac{\mathrm{d}}{\mathrm{d}x}$ 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) Standardization and Related Constants

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Notes:: See Szegő (1975, §2.2(1)), Chihara (1978, Chapter 1, Theorem 3.3).
Referenced by:: §18.19, §18.25(iv), §18.3, §18.30(vii), §18.38(iii), Erratum (V1.2.0) §18.2
Permalink:: http://dlmf.nist.gov/18.2.iii
Addition (effective with 1.2.0):: Previously the name of this subsection was “Normalization”. Equation (18.2.5_5) was added.
See also:: Annotations for §18.2 and Ch.18

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 standardizations.

Constants

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See also:: Annotations for §18.2(iii), §18.2 and Ch.18

Throughout this chapter we will use constants $h_{n}$ and $k_{n}$ , and variants of these, related to OP’s $p_{n}(x)$ .

The $h_{n}$ are defined as:

18.2.5	$\displaystyle h_{n}$	$\displaystyle=\int_{a}^{b}\left(p_{n}(x)\right)^{2}w(x)\,\mathrm{d}x$
	$\displaystyle\text{ or }h_{n}$	$\displaystyle=\sum_{x\in X}\left(p_{n}(x)\right)^{2}w_{x}$
	$\displaystyle\text{ or }h_{n}$	$\displaystyle=\int_{a}^{b}\left(p_{n}(x)\right)^{2}\,\mathrm{d}\mu(x).$
ⓘ Defines: $h_{n}$ (locally) Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $p_{n}(x)$ : polynomial of degree $n$ , $w(x)$ : weight function, $w_{x}$ : weights, $X$ : subset of $\mathbb{R}$ , $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.1.2 Referenced by: §18.2(xii), §18.2(v), §18.2(viii), §3.5(v), Erratum (V1.2.0) for Equations (18.2.5), (18.2.6) Permalink: http://dlmf.nist.gov/18.2.E5 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §18.2(iii), §18.2(iii), §18.2 and Ch.18

Thus

18.2.5_5		$\int_{a}^{b}p_{n}(x)p_{m}(x)w(x)\,\mathrm{d}x=h_{n}\delta_{n,m},$
ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $p_{n}(x)$ : polynomial of degree $n$ , $w(x)$ : weight function, $m$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $h_{n}$ Referenced by: §18.2(xii), §18.2(xii), §18.2(iii), Erratum (V1.2.0) §18.2 Permalink: http://dlmf.nist.gov/18.2.E5_5 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.2(iii), §18.2(iii), §18.2 and Ch.18

and similar extensions for (18.2.4_5) and (18.2.2).

The constants $\tilde{h}_{n}$ , $k_{n}$ , $\tilde{k}_{n}$ and $\tilde{\tilde{k}}_{n}$ are defined as:

18.2.6	$\displaystyle\tilde{h}_{n}$	$\displaystyle=\int_{a}^{b}x\left(p_{n}(x)\right)^{2}w(x)\,\mathrm{d}x$
	$\displaystyle\text{ or }\tilde{h}_{n}$	$\displaystyle=\sum_{x\in X}x\left(p_{n}(x)\right)^{2}w_{x}$
	$\displaystyle\text{ or }\tilde{h}_{n}$	$\displaystyle=\int_{a}^{b}x\left(p_{n}(x)\right)^{2}\,\mathrm{d}\mu(x),$
ⓘ Defines: $\tilde{h}_{n}$ (locally) Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $p_{n}(x)$ : polynomial of degree $n$ , $w(x)$ : weight function, $w_{x}$ : weights, $X$ : subset of $\mathbb{R}$ , $n$ : nonnegative integer and $x$ : real variable Referenced by: Erratum (V1.2.0) for Equations (18.2.5), (18.2.6) Permalink: http://dlmf.nist.gov/18.2.E6 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §18.2(iii), §18.2(iii), §18.2 and Ch.18

and

18.2.7		$p_{n}(x)=k_{n}x^{n}+\tilde{k}_{n}x^{n-1}+\tilde{\tilde{k}}_{n}x^{n-2}+\cdots,$
ⓘ Defines: $k_{n}$ (locally), $\tilde{k}_{n}$ (locally) and $\tilde{\tilde{k}}_{n}$ (locally) Symbols: $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.1.2 Referenced by: §18.15(vi), §18.2(v), §3.5(v) Permalink: http://dlmf.nist.gov/18.2.E7 Encodings: TeX, pMML, png See also: Annotations for §18.2(iii), §18.2(iii), §18.2 and Ch.18

where $\tilde{k}_{0}=0$ and $\tilde{\tilde{k}}_{n}=0$ for $n=0,1$ .

Standardizations

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Keywords:: general orthogonal polynomials, monic, orthonormal
See also:: Annotations for §18.2(iii), §18.2 and Ch.18

The classical orthogonal polynomials are defined with:

(i) the traditional OP standardizations of Table 18.3.1, where each is defined in terms of the above constants.

Two, more specialized, standardizations are:

(ii) monic OP’s: $k_{n}=1$ .

(iii) orthonormal OP’s: $h_{n}=1$ (and usually, but not always, $k_{n}>0$ );

The constant function $p_{0}(x)$ will often, but not always, be identically $1$ (see, for example, (18.2.11_8)), $p_{-1}(x)=0$ in all cases, by convention, as indicated in §18.1(i).

§18.2(iv) Recurrence Relations

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Keywords:: general orthogonal polynomials, recurrence relations
Notes:: See Andrews et al. (1999, Theorem 5.2.2 and Remark 5.2.1). For Favard’s theorem see Szegő (1975, §3.2) or Chihara (1978, p. 21).
Referenced by:: §18.9(i), §18.9(i), §3.5(vi), Erratum (V1.0.19) for Notation, Erratum (V1.2.0) §18.2
Permalink:: http://dlmf.nist.gov/18.2.iv
Addition (effective with 1.2.0):: Equations (18.2.9_5), (18.2.11_1)–(18.2.11_9) were added.
See also:: Annotations for §18.2 and Ch.18

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

First Form

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See also:: Annotations for §18.2(iv), §18.2 and Ch.18

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: $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer, $B_{n}$ : real constant, $C_{n}$ : real constant, $x$ : real variable and $A_{n}$ : real constant A&S Ref: 22.1.4 Referenced by: §18.2(iv), §18.2(iv), §18.2(iv), §18.2(viii), §18.30, §18.30(vi), §18.35(i) Permalink: http://dlmf.nist.gov/18.2.E8 Encodings: TeX, pMML, png See also: Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

Here $A_{n}$ , $B_{n}$ ( $n\geq 0$ ), and $C_{n}$ ( $n\geq 1$ ) are real constants. Then

18.2.9	$\displaystyle A_{n}$	$\displaystyle=\frac{k_{n+1}}{k_{n}},$
	$\displaystyle B_{n}$	$\displaystyle=\left(\frac{\tilde{k}_{n+1}}{k_{n+1}}-\frac{\tilde{k}_{n}}{k_{n}% }\right)A_{n}=-\frac{\tilde{h}_{n}}{h_{n}}A_{n},$
	$\displaystyle C_{n}$	$\displaystyle=\frac{A_{n}\tilde{\tilde{k}}_{n}+B_{n}\tilde{k}_{n}-\tilde{% \tilde{k}}_{n+1}}{k_{n-1}}=\frac{A_{n}}{A_{n-1}}\frac{h_{n}}{h_{n-1}}.$
ⓘ Defines: $B_{n}$ : real constant (locally), $C_{n}$ : real constant (locally) and $A_{n}$ : real constant (locally) Symbols: $n$ : nonnegative integer, $h_{n}$ , $\tilde{h}_{n}$ , $k_{n}$ , $\tilde{k}_{n}$ and $\tilde{\tilde{k}}_{n}$ A&S Ref: 22.1.5 Permalink: http://dlmf.nist.gov/18.2.E9 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

Hence $A_{n-1}A_{n}C_{n}=A_{n}^{2}\ifrac{h_{n}}{h_{n-1}}$ ( $n\geq 1$ ), so

18.2.9_5		$A_{n-1}A_{n}C_{n}>0,$
$n\geq 1$ .
ⓘ Symbols: $n$ : nonnegative integer, $C_{n}$ : real constant and $A_{n}$ : real constant Referenced by: §18.2(iv), §18.2(iv), §18.2(viii), §18.30(vi), §18.36(v), Erratum (V1.2.0) §18.2 Permalink: http://dlmf.nist.gov/18.2.E9_5 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

The OP’s are orthonormal iff $C_{n}=A_{n}/A_{n-1}$ ( $n\geq 1$ ) and $h_{0}=1$ . The OP’s are monic iff $A_{n}=1$ ( $n\geq 0$ ) and $k_{0}=1$ .

Second Form

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See also:: Annotations for §18.2(iv), §18.2 and Ch.18

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: $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer, $a_{n}$ : real constant, $b_{n}$ : real constant, $c_{n}$ : real constant and $x$ : real variable Referenced by: §18.2(iv), §18.2(iv), §18.2(iv), §18.2(viii), §18.35(i) Permalink: http://dlmf.nist.gov/18.2.E10 Encodings: TeX, pMML, png See also: Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

Here $a_{n}$ , $b_{n}$ ( $n\geq 0$ ), $c_{n}$ ( $n\geq 1$ ) are real constants. Then

18.2.11	$\displaystyle a_{n}$	$\displaystyle=\frac{k_{n}}{k_{n+1}},$
	$\displaystyle b_{n}$	$\displaystyle=\frac{\tilde{k}_{n}}{k_{n}}-\frac{\tilde{k}_{n+1}}{k_{n+1}}=% \frac{\tilde{h}_{n}}{h_{n}},$
	$\displaystyle c_{n}$	$\displaystyle=\frac{\tilde{\tilde{k}}_{n}-a_{n}\tilde{\tilde{k}}_{n+1}-b_{n}% \tilde{k}_{n}}{k_{n-1}}=a_{n-1}\frac{h_{n}}{h_{n-1}}.$
ⓘ Defines: $a_{n}$ : real constant (locally), $b_{n}$ : real constant (locally) and $c_{n}$ : real constant (locally) Symbols: $n$ : nonnegative integer, $h_{n}$ , $\tilde{h}_{n}$ , $k_{n}$ , $\tilde{k}_{n}$ and $\tilde{\tilde{k}}_{n}$ Permalink: http://dlmf.nist.gov/18.2.E11 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

Hence

18.2.11_1		$\sum_{j=0}^{n}b_{j}=\frac{\tilde{k}_{n+1}}{k_{n+1}}.$
ⓘ Symbols: $n$ : nonnegative integer, $b_{n}$ : real constant, $k_{n}$ and $\tilde{k}_{n}$ Referenced by: §18.2(iv), Erratum (V1.2.0) §18.2 Permalink: http://dlmf.nist.gov/18.2.E11_1 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

Furthermore, $a_{n-1}c_{n}=a_{n-1}^{2}\ifrac{h_{n}}{h_{n-1}}$ ( $n\geq 1$ ), so

18.2.11_2		$a_{n-1}c_{n}>0,$
$n\geq 1$ .
ⓘ Symbols: $n$ : nonnegative integer, $a_{n}$ : real constant and $c_{n}$ : real constant Referenced by: §18.2(viii) Permalink: http://dlmf.nist.gov/18.2.E11_2 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

The OP’s are orthonormal iff $c_{n}=a_{n-1}$ ( $n\geq 1$ ) and $h_{0}=1$ . The OP’s are monic iff $a_{n}=1$ ( $n\geq 0$ ) and $k_{0}=1$ .

The coefficients $A_{n},B_{n},C_{n}$ in the first form and $a_{n},b_{n},c_{n}$ in the second form are related by

18.2.11_3	$\displaystyle a_{n}$	$\displaystyle=A_{n}^{-1},\quad b_{n}=-A_{n}^{-1}B_{n},\quad c_{n}=A_{n}^{-1}C_% {n};$
18.2.11_3	$\displaystyle A_{n}$	$\displaystyle=a_{n}^{-1},\quad B_{n}=-a_{n}^{-1}b_{n},\quad C_{n}=a_{n}^{-1}c_% {n}.$
ⓘ Symbols: $n$ : nonnegative integer, $a_{n}$ : real constant, $b_{n}$ : real constant and $c_{n}$ : real constant Permalink: http://dlmf.nist.gov/18.2.E11_3 Encodings: TeX, TeX, pMML, pMML, png, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

Monic and Orthonormal Forms

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See also:: Annotations for §18.2(iv), §18.2 and Ch.18

Assume that the $p_{n}(x)$ are monic, so $A_{n}=1=a_{n}$ . Then, with

18.2.11_4	$\displaystyle\alpha_{n}$	$\displaystyle\equiv b_{n}=-B_{n},$
18.2.11_4	$\displaystyle\beta_{n}$	$\displaystyle\equiv c_{n}=C_{n}=\ifrac{h_{n}}{h_{n-1}},$
ⓘ Symbols: $\equiv$ : equals by definition, $n$ : nonnegative integer and $h_{n}$ Referenced by: §18.2(iv) Permalink: http://dlmf.nist.gov/18.2.E11_4 Encodings: TeX, TeX, pMML, pMML, png, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

the monic recurrence relations (18.2.8) and (18.2.10) take the form

18.2.11_5		$\displaystyle xp_{n}(x)$	$\displaystyle=p_{n+1}(x)+\alpha_{n}p_{n}(x)+\beta_{n}p_{n-1}(x),$
	$n\geq 1$ ,
		$\displaystyle p_{1}(x)$	$\displaystyle=x-\alpha_{0},$
		$\displaystyle p_{0}(x)$	$\displaystyle=1.$
ⓘ Symbols: $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.2(ix), §18.2(viii), §18.2(x), §18.2(x), §18.2(x), §18.35(i), §18.40(ii) Permalink: http://dlmf.nist.gov/18.2.E11_5 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

Remarks

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Keywords:: Jacobi matrix
See also:: Annotations for §18.2(iv), §18.2 and Ch.18

If polynomials $p_{n}(x)$ are generated by recurrence relation (18.2.8) under assumption of inequality (18.2.9_5) (or similarly for the other three forms) then the $p_{n}(x)$ are orthogonal by Favard’s theorem, see §18.2(viii), in that the existence of a bounded non-decreasing function $\alpha(x)$ on $(a,b)$ yielding the orthogonality realtion (18.2.4_5) is guaranteed.

If the polynomials $p_{n}(x)$ ( $n=0,1,\ldots,N$ ) are orthogonal on a finite set $X$ of $N+1$ distinct points as in (18.2.3), then the polynomial $p_{N+1}(x)$ of degree $N+1$ , up to a constant factor defined by (18.2.8) or (18.2.10), vanishes on $X$ .

The recurrence relations (18.2.10) can be equivalently written as

18.2.11_9		$\begin{pmatrix}b_{0}&a_{0}&&0\\[6.0pt] c_{1}&b_{1}&a_{1}&\\ &c_{2}&\ddots&\ddots\\ 0&&\ddots&\ddots\end{pmatrix}\begin{pmatrix}p_{0}(x)\\[6.0pt] p_{1}(x)\\ \vdots\\ \vdots\end{pmatrix}=x\begin{pmatrix}p_{0}(x)\\[6.0pt] p_{1}(x)\\ \vdots\\ \vdots\end{pmatrix}.$
ⓘ Symbols: $p_{n}(x)$ : polynomial of degree $n$ and $x$ : real variable Referenced by: §18.2(iv), §18.2(vi), Erratum (V1.2.0) §18.2 Permalink: http://dlmf.nist.gov/18.2.E11_9 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

The matrix on the left-hand side is an (infinite tridiagonal) Jacobi matrix. This matrix is symmetric iff $c_{n}=a_{n-1}$ ( $n\geq 1$ ).

§18.2(v) Christoffel–Darboux Formula

ⓘ

Keywords:: Christoffel–Darboux formula, general orthogonal polynomials, sums of products
Notes:: See Szegő (1975, Theorem 3.2.2 and Eq. (3.2.4)).
Referenced by:: Erratum (V1.2.0) §18.2
Permalink:: http://dlmf.nist.gov/18.2.v
Addition (effective with 1.2.0):: Equations (18.2.12_5), (18.2.14)–(18.2.20) were added.
See also:: Annotations for §18.2 and Ch.18

With notation (18.2.4_5), (18.2.5), (18.2.7)

18.2.12		$K_{n}(x,y)\equiv\sum_{\ell=0}^{n}\frac{p_{\ell}(x)p_{\ell}(y)}{h_{\ell}}=\frac% {k_{n}}{h_{n}k_{n+1}}\frac{p_{n+1}(x)p_{n}(y)-p_{n}(x)p_{n+1}(y)}{x-y},$
$x\neq y$ ,
ⓘ Symbols: $\equiv$ : equals by definition, $(\NVar{a},\NVar{b})$ : open interval, $y$ : real variable, $p_{n}(x)$ : polynomial of degree $n$ , $\ell$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable, $h_{n}$ and $k_{n}$ A&S Ref: 22.12.1 Referenced by: §18.2(v), Erratum (V1.2.0) for Equations (18.2.12), (18.2.13) Permalink: http://dlmf.nist.gov/18.2.E12 Encodings: TeX, pMML, png Addition (effective with 1.2.0): The left-hand side was updated to include the definition of the Christoffel–Darboux kernel $K_{n}(x,y)$ . See also: Annotations for §18.2(v), §18.2 and Ch.18

Kernel property

ⓘ

See also:: Annotations for §18.2(v), §18.2 and Ch.18

18.2.12_5		$\int_{a}^{b}f(y)K_{n}(x,y)\,\mathrm{d}\mu(y)=\begin{cases}f(x),&\text{$f\in% \operatorname{Span}(p_{0},p_{1},\ldots,p_{n})$,}\\ 0,&\text{$\int_{a}^{b}f(x)p_{\ell}(x)\,\mathrm{d}\mu(x)=0$\quad($\ell=0,1,% \ldots,n$).}\end{cases}$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $(\NVar{a},\NVar{b})$ : open interval, $y$ : real variable, $p_{n}(x)$ : polynomial of degree $n$ , $\ell$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.2(v), Erratum (V1.2.0) §18.2 Permalink: http://dlmf.nist.gov/18.2.E12_5 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.2(v), §18.2(v), §18.2 and Ch.18

Confluent Form

ⓘ

Keywords:: Christoffel–Darboux formula, confluent form
See also:: Annotations for §18.2(v), §18.2 and Ch.18

18.2.13		$K_{n}(x,x)=\sum_{\ell=0}^{n}\frac{(p_{\ell}(x))^{2}}{h_{\ell}}=\frac{k_{n}}{h_% {n}k_{n+1}}{\left(p_{n+1}^{\prime}(x)p_{n}(x)-p_{n}^{\prime}(x)p_{n+1}(x)% \right)}.$
ⓘ Symbols: $(\NVar{a},\NVar{b})$ : open interval, $p_{n}(x)$ : polynomial of degree $n$ , $\ell$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable, $h_{n}$ and $k_{n}$ Referenced by: Erratum (V1.2.0) for Equations (18.2.12), (18.2.13) Permalink: http://dlmf.nist.gov/18.2.E13 Encodings: TeX, pMML, png Addition (effective with 1.2.0): The left-hand side was updated to include the definition of the confluent form of the Christoffel–Darboux kernel $K_{n}(x,x)$ . See also: Annotations for §18.2(v), §18.2(v), §18.2 and Ch.18

Kernel Polynomials

ⓘ

Keywords:: contiguous relations, kernel polynomials
Notes:: See Chihara (1978, Ch. I, §7).
See also:: Annotations for §18.2(v), §18.2 and Ch.18

Assume $y\notin(a,b)$ in (18.2.12). Then the kernel polynomials

18.2.14		$q_{n}(x)=q_{n}(x;y)\equiv K_{n}(x,y)=\sum_{\ell=0}^{n}\frac{p_{\ell}(x)p_{\ell% }(y)}{h_{\ell}}$
ⓘ Symbols: $\equiv$ : equals by definition, $(\NVar{a},\NVar{b})$ : open interval, $y$ : real variable, $p_{n}(x)$ : polynomial of degree $n$ , $q$ : real variable, $\ell$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $h_{n}$ Referenced by: §18.2(v), Erratum (V1.2.0) §18.2 Permalink: http://dlmf.nist.gov/18.2.E14 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.2(v), §18.2(v), §18.2 and Ch.18

are OP’s with orthogonality relation

18.2.15		$\int_{a}^{b}q_{n}(x)q_{m}(x)\|y-x\|\,\mathrm{d}\mu(x)=0,$
$n\neq m$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $y$ : real variable, $q$ : real variable, $m$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable Permalink: http://dlmf.nist.gov/18.2.E15 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.2(v), §18.2(v), §18.2 and Ch.18

Between the systems $\{p_{n}(x)\}$ and $\{q_{n}(x)\}$ there are the contiguous relations

18.2.16	$\displaystyle q_{n}(x)-q_{n-1}(x)$	$\displaystyle=\frac{p_{n}(y)}{h_{n}}p_{n}(x),$
ⓘ Symbols: $y$ : real variable, $p_{n}(x)$ : polynomial of degree $n$ , $q$ : real variable, $n$ : nonnegative integer, $x$ : real variable and $h_{n}$ Referenced by: §18.9(ii) Permalink: http://dlmf.nist.gov/18.2.E16 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.2(v), §18.2(v), §18.2 and Ch.18
18.2.17	$\displaystyle p_{n}(y)p_{n+1}(x)-p_{n+1}(y)p_{n}(x)$	$\displaystyle=\frac{h_{n}k_{n+1}}{k_{n}}(x-y)q_{n}(x).$
ⓘ Symbols: $y$ : real variable, $p_{n}(x)$ : polynomial of degree $n$ , $q$ : real variable, $n$ : nonnegative integer, $x$ : real variable, $h_{n}$ and $k_{n}$ Referenced by: §18.9(ii), Erratum (V1.2.0) §18.2 Permalink: http://dlmf.nist.gov/18.2.E17 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.2(v), §18.2(v), §18.2 and Ch.18

§18.2(vi) Zeros

ⓘ

Keywords:: Chebyshev polynomials, Hermite polynomials, Jacobi matrix, Jacobi polynomials, Laguerre polynomials, Legendre polynomials, classical orthogonal polynomials, distribution, general orthogonal polynomials, of orthogonal polynomials, truncated, ultraspherical polynomials, zeros
Notes:: See Andrews et al. (1999, Theorems 5.4.2, 5.4.3).
Referenced by:: §18.16(i), Erratum (V1.2.0) §18.2
Permalink:: http://dlmf.nist.gov/18.2.vi
Addition (effective with 1.2.0):: The equation (18.2.18) and discussion immediately surrounding it were added.
See also:: Annotations for §18.2 and Ch.18

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.

For usage of the zeros of an OP in Gauss quadrature see §3.5(v). When the Jacobi matrix in (18.2.11_9) is truncated to an $n\times n$ matrix

18.2.18		$\mathbf{J}_{n}=\begin{pmatrix}b_{0}&a_{0}&&&0\\[6.0pt] c_{1}&b_{1}&a_{1}&&\\ &c_{2}&\ddots&\ddots&\\ &&\ddots&\ddots&a_{n-2}\\[6.0pt] 0&&&c_{n-1}&b_{n-1}\end{pmatrix}$
ⓘ Symbols: $n$ : nonnegative integer Referenced by: §18.2(vi) Permalink: http://dlmf.nist.gov/18.2.E18 Encodings: TeX, pMML, png See also: Annotations for §18.2(vi), §18.2 and Ch.18

then the zeros of $p_{n}(x)$ are the eigenvalues of $\mathbf{J}_{n}$ (see also §3.5(vi)).

Discriminants

ⓘ

Keywords:: discriminant, general orthogonal polynomials
Addition (effective with 1.2.0):: This paragraph was added.
See also:: Annotations for §18.2(vi), §18.2 and Ch.18

Let $x_{1},\ldots,x_{n}$ be the zeros of the OP $p_{n}$ , so

18.2.19		$p_{n}(x)=k_{n}\prod_{j=1}^{n}(x-x_{j}).$
ⓘ Symbols: $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer, $x$ : real variable and $k_{n}$ Permalink: http://dlmf.nist.gov/18.2.E19 Encodings: TeX, pMML, png See also: Annotations for §18.2(vi), §18.2(vi), §18.2 and Ch.18

The discriminant of $p_{n}$ is defined by

18.2.20		$\operatorname{Disc}\left(p_{n}\right)=k_{n}^{2n-2}\prod_{1\leq i<j\leq n}(x_{i% }-x_{j})^{2}.$
ⓘ Defines: $\operatorname{Disc}\left(\NVar{x}\right)$ : discriminant function Symbols: $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer, $x$ : real variable and $k_{n}$ Source: Ismail (2009, (3.1.8)) Referenced by: §18.16(vii), §18.2(v), Erratum (V1.2.0) §18.2 Permalink: http://dlmf.nist.gov/18.2.E20 Encodings: TeX, pMML, png See also: Annotations for §18.2(vi), §18.2(vi), §18.2 and Ch.18

See Ismail (2009, §3.4) for another expression of the discriminant in the case of a general OP.

§18.2(vii) Quadratic Transformations

ⓘ

Keywords:: general orthogonal polynomials, quadratic transformations
Referenced by:: Erratum (V1.2.0) §18.2, Version 1.2.0 (March 27, 2024)
Permalink:: http://dlmf.nist.gov/18.2.vii
Addition (effective with 1.2.0):: This subsection was added.
See also:: Annotations for §18.2 and Ch.18

For OP’s $\{p_{n}(x)\}$ on $\mathbb{R}$ with respect to an even weight function $w(x)$ we have

18.2.21		$p_{n}(-x)=(-1)^{n}p_{n}(x),$
ⓘ Symbols: $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.2(vii), Erratum (V1.2.0) §18.2 Permalink: http://dlmf.nist.gov/18.2.E21 Encodings: TeX, pMML, png See also: Annotations for §18.2(vii), §18.2 and Ch.18

so we can put

18.2.22	$\displaystyle r_{n}(x^{2})$	$\displaystyle\equiv p_{2n}(x),$
	$\displaystyle s_{n}(x^{2})$	$\displaystyle\equiv x^{-1}p_{2n+1}(x),$
	$\displaystyle v(x^{2})$	$\displaystyle\equiv w(x).$
ⓘ Symbols: $\equiv$ : equals by definition, $p_{n}(x)$ : polynomial of degree $n$ , $w(x)$ : weight function, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.7(ii) Permalink: http://dlmf.nist.gov/18.2.E22 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §18.2(vii), §18.2 and Ch.18

Then $\{r_{n}(x)\}$ are OP’s on $(0,\infty)$ with respect to weight function $x^{-\frac{1}{2}}v(x)$ and $\{s_{n}(x)\}$ are OP’s on $(0,\infty)$ with respect to weight function $x^{\frac{1}{2}}v(x)$ .

As a slight variant let $\{p_{n}(x)\}$ be OP’s with respect to an even weight function $w(x)$ on $(-1,1)$ . Then (18.2.21) still holds and we can put

18.2.23	$\displaystyle r_{n}(2x^{2}-1)$	$\displaystyle\equiv p_{2n}(x),$
	$\displaystyle s_{n}(2x^{2}-1)$	$\displaystyle\equiv x^{-1}p_{2n+1}(x),$
	$\displaystyle v(2x^{2}-1)$	$\displaystyle\equiv w(x).$
ⓘ Symbols: $\equiv$ : equals by definition, $p_{n}(x)$ : polynomial of degree $n$ , $w(x)$ : weight function, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.33(iii), §18.7(ii) Permalink: http://dlmf.nist.gov/18.2.E23 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §18.2(vii), §18.2 and Ch.18

Then $\{r_{n}(x)\}$ are OP’s on $(-1,1)$ with respect to weight function $(1+x)^{-\frac{1}{2}}v(x)$ and $\{s_{n}(x)\}$ are OP’s on $(-1,1)$ with respect to weight function $(1+x)^{\frac{1}{2}}v(x)$ .

See Chihara (1978, Ch. I, §8).

§18.2(viii) Uniqueness of Orthogonality Measure and Completeness

ⓘ

Keywords:: Favard’s theorem, complete, general orthogonal polynomials
Referenced by:: §18.2(iv), (18.35.6_2)
Permalink:: http://dlmf.nist.gov/18.2.viii
Addition (effective with 1.2.0):: This subsection was added.
See also:: Annotations for §18.2 and Ch.18

If a system of polynomials $\{p_{n}(x)\}$ satisfies any of the formula pairs (recurrence relation and coefficient inequality) (18.2.8), (18.2.9_5) or (18.2.10), (18.2.11_2) or (18.2.11_5), (18.2.11_6) or (18.2.11_8), (18.2.11_6) then $\{p_{n}(x)\}$ is orthogonal with respect to some positive measure on $\mathbb{R}$ (Favard’s theorem). The measure is not necessarily absolutely continuous (i.e., of the form $w(x)\,\mathrm{d}x$ ) nor is it necessarily unique, up to a positive constant factor. However, if OP’s have an orthogonality relation on a bounded interval, then their orthogonality measure is unique, up to a positive constant factor.

A system $\{p_{n}(x)\}$ of OP’s satisfying (18.2.1) and (18.2.5) is complete if each $f(x)$ in the Hilbert space $L_{w}^{2}((a,b))$ can be approximated in Hilbert norm by finite sums $\sum_{n}\lambda_{n}p_{n}(x)$ . For such a system, functions $f\in L_{w}^{2}((a,b))$ and sequences $\{\lambda_{n}\}$ ( $n=0,1,2,\ldots$ ) satisfying $\sum_{n=0}^{\infty}h_{n}|\lambda_{n}|^{2}<\infty$ can be related to each other in a similar way as was done for Fourier series in (1.8.1) and (1.8.2):

18.2.24		$\lambda_{n}={h_{n}^{-1}}\int_{a}^{b}f(x)p_{n}(x)w(x)\,\mathrm{d}x$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $p_{n}(x)$ : polynomial of degree $n$ , $w(x)$ : weight function, $n$ : nonnegative integer, $x$ : real variable and $h_{n}$ Referenced by: §18.18(i) Permalink: http://dlmf.nist.gov/18.2.E24 Encodings: TeX, pMML, png See also: Annotations for §18.2(viii), §18.2 and Ch.18

if and only if

18.2.25		$f(x)=\sum_{n=0}^{\infty}{\lambda_{n}}p_{n}(x)$
ⓘ Symbols: $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.18(i), §18.2(xi) Permalink: http://dlmf.nist.gov/18.2.E25 Encodings: TeX, pMML, png See also: Annotations for §18.2(viii), §18.2 and Ch.18

(convergence in $L_{w}^{2}((a,b))$ ). A system of OP’s with unique orthogonality measure is always complete, see Shohat and Tamarkin (1970, Theorem 2.14). In particular, a system of OP’s on a bounded interval is always complete.

§18.2(ix) Moments

ⓘ

Keywords:: Hankel determinant, general orthogonal polynomials, moments
Permalink:: http://dlmf.nist.gov/18.2.ix
Addition (effective with 1.2.0):: This subsection was added.
See also:: Annotations for §18.2 and Ch.18

The moments for an orthogonality measure $\,\mathrm{d}\mu(x)$ are the numbers

18.2.26		$\mu_{n}=\int_{a}^{b}x^{n}\,\mathrm{d}\mu(x),$
$n=0,1,2,\ldots$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.2(x), §18.34(ii) Permalink: http://dlmf.nist.gov/18.2.E26 Encodings: TeX, pMML, png See also: Annotations for §18.2(ix), §18.2 and Ch.18

The Hankel determinant $\Delta_{n}$ of order $n$ is defined by $\Delta_{0}=1$ and

18.2.27		$\Delta_{n}=\begin{vmatrix}\mu_{0}&\mu_{1}&\ldots&\mu_{n-1}\\ \mu_{1}&\mu_{2}&\ldots&\mu_{n}\\ \vdots&\vdots&&\vdots\\ \mu_{n-1}&\mu_{n}&\ldots&\mu_{2n-2}\end{vmatrix},$
$n=1,2,\ldots$ .
ⓘ Symbols: $\det$ : determinant and $n$ : nonnegative integer Source: Gautschi (2004, (2.1.1)) Referenced by: §18.2(ix) Permalink: http://dlmf.nist.gov/18.2.E27 Encodings: TeX, pMML, png See also: Annotations for §18.2(ix), §18.2 and Ch.18

Also define determinants $\Delta_{n}^{\prime}$ by $\Delta_{0}^{\prime}=0$ , $\Delta_{1}^{\prime}=\mu_{1}$ and

18.2.28		$\Delta_{n}^{\prime}=\begin{vmatrix}\mu_{0}&\mu_{1}&\ldots&\mu_{n-2}&\mu_{n}\\ \mu_{1}&\mu_{2}&\ldots&\mu_{n-1}&\mu_{n+1}\\ \vdots&\vdots&&\vdots&\vdots\\ \mu_{n-1}&\mu_{n}&\ldots&\mu_{2n-3}&\mu_{2n-1}\end{vmatrix},$
$n=2,3,\ldots$ .
ⓘ Symbols: $\det$ : determinant and $n$ : nonnegative integer Source: Gautschi (2004, (2.1.2)) Referenced by: §18.2(ix) Permalink: http://dlmf.nist.gov/18.2.E28 Encodings: TeX, pMML, png See also: Annotations for §18.2(ix), §18.2 and Ch.18

The monic OP’s $p_{n}(x)$ with respect to the measure $\,\mathrm{d}\mu(x)$ can be expressed in terms of the moments by

18.2.29		$p_{n}(x)={\frac{1}{\Delta_{n}}}\begin{vmatrix}\mu_{0}&\mu_{1}&\ldots&\mu_{n}\\ \mu_{1}&\mu_{2}&\ldots&\mu_{n+1}\\ \vdots&\vdots&&\vdots\\ \mu_{n-1}&\mu_{n}&\ldots&\mu_{2n-1}\\ 1&x&\ldots&x^{n}\end{vmatrix},$
$n=1,2,\ldots$ .
ⓘ Symbols: $\det$ : determinant, $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer and $x$ : real variable Proved: Gautschi (2004, Theorem 2.1)(proved) Permalink: http://dlmf.nist.gov/18.2.E29 Encodings: TeX, pMML, png See also: Annotations for §18.2(ix), §18.2 and Ch.18

The recurrence coefficients $\alpha_{n}$ and $\beta_{n}$ in (18.2.11_5) can be expressed in terms of the determinants (18.2.27) and (18.2.28) by

18.2.30		$\displaystyle\alpha_{n}$	$\displaystyle=\frac{\Delta_{n+1}^{\prime}}{\Delta_{n+1}}-\frac{\Delta_{n}^{% \prime}}{\Delta_{n}},$
	$n=0,1,2,\ldots$ .
		$\displaystyle\beta_{n}$	$\displaystyle=\frac{\Delta_{n+1}\Delta_{n-1}}{\Delta_{n}^{2}},$
	$n=1,2,\ldots$ .
ⓘ Symbols: $n$ : nonnegative integer Proved: Gautschi (2004, Theorem 2.2)(proved) Referenced by: §18.2(ix), §18.40(ii) Permalink: http://dlmf.nist.gov/18.2.E30 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.2(ix), §18.2 and Ch.18

It is to be noted that, although formally correct, the results of (18.2.30) are of little utility for numerical work, as Hankel determinants are notoriously ill-conditioned. See Gautschi (2004, p. 54), and Golub and Meurant (2010, pp. 56, 57). Alternatives for numerical calculation of the recursion coefficients in terms of the moments are discussed in these references, and in §18.40(ii).

§18.2(x) Orthogonal Polynomials and Continued Fractions

ⓘ

Keywords:: Markov’s theorem, and corecursive OP’s, associated orthogonal polynomials, continued fractions, corecursive orthogonal polynomials, denominator polynomials, general orthogonal polynomials, monic, numerator polynomials
Referenced by:: §18.30(vii), §18.30
Permalink:: http://dlmf.nist.gov/18.2.x
Addition (effective with 1.2.0):: This subsection was added.
See also:: Annotations for §18.2 and Ch.18

In this subsection fix the recurrence coefficients $\alpha_{n}$ ( $n=0,1,2,\ldots$ ) and $\beta_{n}$ ( $n=1,2,\ldots$ ) as in (18.2.11_5), with $p_{n}(x)$ the corresponding monic OP’s and with $\,\mathrm{d}\mu(x)$ , $a$ and $b$ as in the orthogonality relation (18.2.4_5). Define the first associated monic orthogonal polynomials $p_{n}^{(1)}(x)$ as monic OP’s satisfying

18.2.31		$\displaystyle p_{0}^{(1)}(x)$	$\displaystyle=1,$
		$\displaystyle p_{1}^{(1)}(x)$	$\displaystyle=x-\alpha_{1},$
		$\displaystyle xp_{n}^{(1)}(x)$	$\displaystyle=p_{n+1}^{(1)}(x)+\alpha_{n+1}p_{n}^{(1)}(x)+\beta_{n+1}p_{n-1}^{% (1)}(x),$
	$n=1,2,\ldots$ ,
ⓘ Symbols: $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.2(x), §18.30 Permalink: http://dlmf.nist.gov/18.2.E31 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §18.2(x), §18.2 and Ch.18

where the first indicates that the indices of the recursion coefficients $\alpha_{n}$ , $\beta_{n}$ of (18.2.31) have been incremented by $1$ , when compared to those of (18.2.11_5). More generally, §18.30 defines the recurrence relation of the $c$ th associated monic OP by means of a similar shift by $c$ in (18.2.11_5).

The OP’s $p_{n}^{(1)}(x)$ may also be calculated from the original recursion (18.2.11_5), but with independent initial conditions for $p_{0},p_{1}$ :

18.2.32		$\displaystyle p_{0}^{(0)}(x)$	$\displaystyle=0,$
		$\displaystyle p_{1}^{(0)}(x)$	$\displaystyle=1,$
		$\displaystyle xp_{n}^{(0)}(x)$	$\displaystyle=p_{n+1}^{(0)}(x)+\alpha_{n}p_{n}^{(0)}(x)+\beta_{n}p_{n-1}^{(0)}% (x),$
	$n=1,2,\ldots$ ,
ⓘ Symbols: $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.30(vii), §18.30 Permalink: http://dlmf.nist.gov/18.2.E32 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §18.2(x), §18.2 and Ch.18

resulting in $p_{n}^{(0)}(x)=p_{n-1}^{(1)}(x)$ , by simple comparison of the two recursions. The $p_{n}^{(0)}(x)$ are the monic corecursive orthogonal polynomials. These relationships are further explored in §§18.30(vi) and 18.30(vii).

The polynomials $p_{n}^{(1)}(x)$ may be also be directly expressed in terms of the $p_{n}{(x)}$ of (18.2.11_5):

18.2.33		$p_{n-1}^{(1)}(z)=\frac{1}{\mu_{0}}\int_{a}^{b}\frac{p_{n}(z)-p_{n}(x)}{z-x}\,% \mathrm{d}\mu(x),$
$z\in\mathbb{C}\backslash[a,b]$ , $n=1,2,\ldots$ ,
ⓘ Symbols: $[\NVar{a},\NVar{b}]$ : closed interval, $\mathbb{C}$ : complex plane, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $p_{n}(x)$ : polynomial of degree $n$ , $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Proved: Chihara (1978, Chapter III, §4)(proved) Permalink: http://dlmf.nist.gov/18.2.E33 Encodings: TeX, pMML, png See also: Annotations for §18.2(x), §18.2 and Ch.18

with moment $\mu_{0}$ defined in (18.2.26).

Using the terminology of §1.12(ii), the $n$ -th approximant of the continued fraction

18.2.34		$\cfrac{1}{x-\alpha_{0}-\cfrac{\beta_{1}}{x-\alpha_{1}-\cfrac{\beta_{2}}{x-% \alpha_{2}-\cdots}}}$
ⓘ Symbols: $x$ : real variable Referenced by: §18.13, §18.30(vi) Permalink: http://dlmf.nist.gov/18.2.E34 Encodings: TeX, pMML, png See also: Annotations for §18.2(x), §18.2 and Ch.18

is given by

18.2.35		$F_{n}(x)=\cfrac{1}{x-\alpha_{0}-\cfrac{\beta_{1}}{x-\alpha_{1}-\cfrac{\beta_{2% }}{x-\alpha_{2}-\cdots}}}\frac{\beta_{n-1}}{x-\alpha_{n-1}}.$
ⓘ Symbols: $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.30(vi), §18.30(vii) Permalink: http://dlmf.nist.gov/18.2.E35 Encodings: TeX, pMML, png See also: Annotations for §18.2(x), §18.2 and Ch.18

Then

18.2.36		$F_{n}(x)=\frac{p_{n-1}^{(1)}(x)}{p_{n}(x)}=\frac{p_{n}^{(0)}(x)}{p_{n}(x)}=% \frac{1}{\mu_{0}}\sum_{k=1}^{n}\frac{w_{k}}{x-x_{k}},$
ⓘ Symbols: $p_{n}(x)$ : polynomial of degree $n$ , $w_{x}$ : weights, $k$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable Proved: Chihara (1978, Chapter III, Theorem 4.3)(proved) Referenced by: §18.13, §18.2(x) Permalink: http://dlmf.nist.gov/18.2.E36 Encodings: TeX, pMML, png See also: Annotations for §18.2(x), §18.2 and Ch.18

where $x_{1},x_{2},\ldots,x_{n}$ are the zeros of $p_{n}(x)$ and

18.2.37		$w_{k}=\int_{a}^{b}\frac{p_{n}(x)}{(x-x_{k})p_{n}^{\prime}(x_{k})}\,\mathrm{d}% \mu(x),$
$k=1,2,\ldots,n$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $p_{n}(x)$ : polynomial of degree $n$ , $w_{x}$ : weights, $k$ : nonnegative integer, $n$ : nonnegative integer and $x$ : real variable Permalink: http://dlmf.nist.gov/18.2.E37 Encodings: TeX, pMML, png See also: Annotations for §18.2(x), §18.2 and Ch.18

are the Christoffel numbers, see also (3.5.18). Because of (18.2.36) the OP’s $p_{n}(x)$ are also called monic denominator polynomials and the OP’s $p_{n-1}^{(1)}(x)$ , or, equivalently, the $p_{n}^{(0)}(x)$ , are called the monic numerator polynomials.

Assume that the interval $[a,b]$ is bounded. Markov’s theorem states that

18.2.38		$\lim_{n\to\infty}F_{n}(z)=\frac{1}{\mu_{0}}\int_{a}^{b}\frac{\,\mathrm{d}\mu(x% )}{z-x},$
$z\in\mathbb{C}\backslash[a,b]$ .
ⓘ Symbols: $[\NVar{a},\NVar{b}]$ : closed interval, $\mathbb{C}$ : complex plane, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $z$ : complex variable, $n$ : nonnegative integer and $x$ : real variable Proved: Chihara (1978, Chapter III, Theorem 4.3)(proved) Referenced by: §18.40(ii) Permalink: http://dlmf.nist.gov/18.2.E38 Encodings: TeX, pMML, png See also: Annotations for §18.2(x), §18.2 and Ch.18

See Chihara (1978, pp. 86–89), and, in slightly different notation, Ismail (2009, §§2.3, 2.6, 2.10), where it is assumed that $\mu_{0}=1$ . See also the extended development of these ideas in §§18.30(vi), 18.30(vii), and in §18.40(ii) where they form the basis for one method of solving the classical moment problem.

§18.2(xi) Some Special Classes of General Orthogonal Polynomials

ⓘ

Referenced by:: §18.39(iv)
Permalink:: http://dlmf.nist.gov/18.2.xi
Addition (effective with 1.2.0):: This subsection was added.
See also:: Annotations for §18.2 and Ch.18

The Szegő Class $\mathcal{G}$

ⓘ

Keywords:: Szegő class, equiconvergence theorem, general orthogonal polynomials
See also:: Annotations for §18.2(xi), §18.2 and Ch.18

This is the class of weight functions $w$ on $(-1,1)$ such that, in addition to (18.2.1_5),

18.2.39		$\int_{-1}^{1}\frac{\left\|\ln\left(w(x)\right)\right\|}{\sqrt{1-x^{2}}}\,\mathrm% {d}x<\infty.$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $w(x)$ : weight function and $x$ : real variable Source: Szegő (1975, §12.1) Referenced by: §18.35(ii) Permalink: http://dlmf.nist.gov/18.2.E39 Encodings: TeX, pMML, png See also: Annotations for §18.2(xi), §18.2(xi), §18.2 and Ch.18

For OP’s $p_{n}(x)$ with weight function in the class $\mathcal{G}$ there are asymptotic formulas as $n\to\infty$ , respectively for $x\in\mathbb{C}$ outside $[-1,1]$ and for $x\in[-1,1]$ , see Szegő (1975, Theorems 12.1.2, 12.1.4). Under further conditions on the weight function there is an equiconvergence theorem, see Szegő (1975, Theorem 13.1.2). This says roughly that the series (18.2.25) has the same pointwise convergence behavior as the same series with $p_{n}(x)=T_{n}\left(x\right)$ , a Chebyshev polynomial of the first kind, see Table 18.3.1.

Generalizations of the Szegő Class

ⓘ

See also:: Annotations for §18.2(xi), §18.2 and Ch.18

Nevai (1979, p.39) defined the class $\mathcal{S}$ of orthogonality measures with support inside $[-1,1]$ such that the absolutely continuous part $w(x)\,\mathrm{d}x$ has $w$ in the Szegő class $\mathcal{G}$ . For OP’s with orthogonality measure in $\mathcal{S}$ Nevai (1979, pp. 148–150) generalized Szegő’s equiconvergence theorem. In further generalizations of the class $\mathcal{S}$ discrete mass points $x_{k}$ outside $[-1,1]$ are allowed. If these $x_{k}$ satisfy $\sum_{k}(|x_{k}|-1)^{\ifrac{1}{2}}<\infty$ then Szegő type asymptotics outside $[-1,1]$ can be given for the corresponding OP’s, see Simon (2011, Corollary 3.7.2 and following).

The Nevai class ${\mathbf{M}}(a,b)$

ⓘ

Keywords:: Nevai–Blumenthal class, general orthogonal polynomials
See also:: Annotations for §18.2(xi), §18.2 and Ch.18

The class $\mathbf{M}(a,b)$ ( $a>0$ , $b\in\mathbb{R}$ ), introduced by Nevai (1979, p.10), consists of all orthogonality measures $\,\mathrm{d}\mu$ such that the coefficients $\sqrt{\beta_{n}}$ and $\alpha_{n}$ in the recurrence relation (18.2.11_8) for the corresponding orthonormal OP’s satisfy

18.2.40	$\displaystyle\lim_{n\to\infty}\sqrt{\beta_{n}}$	$\displaystyle=\tfrac{1}{2}a,$
18.2.40	$\displaystyle\lim_{n\to\infty}\alpha_{n}$	$\displaystyle=b.$
ⓘ Symbols: $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/18.2.E40 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.2(xi), §18.2(xi), §18.2 and Ch.18

If $\,\mathrm{d}\mu\in{\mathbf{M}}(a,b)$ then the interval $[b-a,b+a]$ is included in the support of $\,\mathrm{d}\mu$ , and outside $[b-a,b+a]$ the measure $\,\mathrm{d}\mu$ only has discrete mass points $x_{k}$ such that $b\pm a$ are the only possible limit points of the sequence $\{x_{k}\}$ , see Máté et al. (1991, Theorem 10). Part of this theorem was already proved by Blumenthal (1898). Therefore this class is also called the Nevai–Blumenthal class.

§18.2(xii) Other Special Constructions Involving General OP’s

ⓘ

Referenced by:: (18.14.3_5), (18.14.8), Erratum (V1.2.0) §18.2, Version 1.2.0 (March 27, 2024)
Permalink:: http://dlmf.nist.gov/18.2.xii
Addition (effective with 1.2.0):: This subsection was added.
See also:: Annotations for §18.2 and Ch.18

Poisson kernel

ⓘ

Keywords:: Poisson kernel, general orthogonal polynomials
See also:: Annotations for §18.2(xii), §18.2 and Ch.18

For OP’s $p_{n}$ with $h_{n}$ and orthogonality relation as in (18.2.5) and (18.2.5_5), the Poisson kernel is defined by

18.2.41		$P_{z}(x,y)=\sum_{n=0}^{\infty}\frac{p_{n}(x)p_{n}(y)}{h_{n}}z^{n},$
$\|z\|<1$ ,
ⓘ Symbols: $(\NVar{a},\NVar{b})$ : open interval, $y$ : real variable, $p_{n}(x)$ : polynomial of degree $n$ , $z$ : complex variable, $n$ : nonnegative integer, $x$ : real variable and $h_{n}$ Referenced by: §18.18(vii), §18.2(xii) Permalink: http://dlmf.nist.gov/18.2.E41 Encodings: TeX, pMML, png See also: Annotations for §18.2(xii), §18.2(xii), §18.2 and Ch.18

for $x, y$ in the support of the orthogonality measure and $z$ such that the series in (18.2.41) converges absolutely for all these $x, y$ . Instances where the Poisson kernel is nonnegative are of special interest, see Ismail (2009, Theorem 4.7.12).

Degree lowering and raising differentiation formulas and structure relations

ⓘ

Keywords:: degree lowering and raising, differentiation formulas, orthogonal polynomials, semi-classical, semi-classical orthogonal polynomials, structure relation
See also:: Annotations for §18.2(xii), §18.2 and Ch.18

For a large class of OP’s $p_{n}$ there exist pairs of differentiation formulas

18.2.42	$\displaystyle\pi_{n}(x)p_{n}^{\prime}(x)+A_{n}(x)p_{n}(x)$	$\displaystyle=\lambda_{n}p_{n-1}(x),$
ⓘ Symbols: $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.9(iii) Permalink: http://dlmf.nist.gov/18.2.E42 Encodings: TeX, pMML, png See also: Annotations for §18.2(xii), §18.2(xii), §18.2 and Ch.18
18.2.43	$\displaystyle\pi_{n}(x)p_{n}^{\prime}(x)+B_{n}(x)p_{n}(x)$	$\displaystyle=\mu_{n}p_{n+1}(x),$
ⓘ Symbols: $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer and $x$ : real variable Permalink: http://dlmf.nist.gov/18.2.E43 Encodings: TeX, pMML, png See also: Annotations for §18.2(xii), §18.2(xii), §18.2 and Ch.18

see Ismail (2009, (3.2.3), (3.2.10)). If $A_{n}(x)$ and $B_{n}(x)$ are polynomials of degree independent of $n$ , and moreover $\pi_{n}(x)$ is a polynomial $\pi(x)$ independent of $n$ then

18.2.44		$\pi(x)p_{n}^{\prime}(x)=\sum_{j=n-s}^{n+t}a_{n,j}p_{j}(x)$
ⓘ Symbols: $p_{n}(x)$ : polynomial of degree $n$ , $t$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.2(xii), §18.9(iii) Permalink: http://dlmf.nist.gov/18.2.E44 Encodings: TeX, pMML, png See also: Annotations for §18.2(xii), §18.2(xii), §18.2 and Ch.18

for certain coefficients $a_{n,j}$ with $s, t$ independent of $n$ . Then the OP’s are called semi-classical and (18.2.44) is called a structure relation.

Sheffer Polynomials

ⓘ

Keywords:: Bernoulli polynomials, Euler polynomials, Hahn class orthogonal polynomials, Hermite polynomials, Laguerre polynomials, Sheffer polynomials, as Sheffer polynomials, definition, delta operator, generating function, which are also Sheffer polynomials, which are also orthogonal
See also:: Annotations for §18.2(xii), §18.2 and Ch.18

Polynomials $p_{n}(x)$ of degree $n$ ( $n=0,1,2,\ldots$ ) are called Sheffer polynomials if they are generated by a generating function of the form

18.2.45		$f(t)\exp\left(xu(t)\right)=\sum_{n=0}^{\infty}c_{n}p_{n}(x)\,\frac{t^{n}}{n!}\,,$
ⓘ Symbols: $\exp\NVar{z}$ : exponential function, $!$ : factorial (as in $n!$ ), $p_{n}(x)$ : polynomial of degree $n$ , $t$ : real variable, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.2(xii), §18.2(xii) Permalink: http://dlmf.nist.gov/18.2.E45 Encodings: TeX, pMML, png See also: Annotations for §18.2(xii), §18.2(xii), §18.2 and Ch.18

where $f(t)$ and $u(t)$ are formal power series in $t$ , with $f(0)=1$ , $u(0)=0$ and $u^{\prime}(0)=1$ . Often a standardization $c_{n}=1$ is taken. If $v(s)$ is the formal power series such that $v(u(t))=t$ then a property equivalent to (18.2.45) with $c_{n}=1$ is that

18.2.46		$v\left({D}_{x}\right)p_{n}(x)=np_{n-1}(x).$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer and $x$ : real variable Referenced by: Erratum (V1.2.0) §18.2 Permalink: http://dlmf.nist.gov/18.2.E46 Encodings: TeX, pMML, png See also: Annotations for §18.2(xii), §18.2(xii), §18.2 and Ch.18

The operator ${D}_{x}$ is a delta operator, i.e., ${D}_{x}$ commutes with translation in the variable $x$ and ${D}_{x}x$ is a nonzero constant.

The generating functions (18.12.13), (18.12.15), (18.23.3), (18.23.4), (18.23.5) and (18.23.7) for Laguerre, Hermite, Krawtchouk, Meixner, Charlier and Meixner–Pollaczek polynomials, respectively, can be written in the form (18.2.45). In fact, these are the only OP’s which are Sheffer polynomials (with Krawtchouk polynomials being only a finite system)

The Bernoulli polynomials $B_{n}\left(x\right)$ and Euler polynomials $E_{n}\left(x\right)$ are examples of Sheffer polynomials which are not OP’s, see the generating functions (24.2.3) and (24.2.8). For other examples of Sheffer polynomials, not in DLMF, see Roman (1984).

For further details see Meixner (1934), Sheffer (1939), Rota et al. (1973) and Butzer and Koornwinder (2019).

Monotonic Weight Functions

ⓘ

Keywords:: general orthogonal polynomials, inequalities, inequality, monotonic weight function
See also:: Annotations for §18.2(xii), §18.2 and Ch.18

For OP’s $p_{n}$ on $[a,b]$ with weight function $w(x)$ and orthogonality relation (18.2.5_5) assume that $b<\infty$ and $w(x)$ is non-decreasing in the interval $[a,b]$ . Then the functions $\sqrt{w(x)}p_{n}(x)$ attain their maximum in $[a,b]$ for $x=b$ . See Szegő (1975, Theorem 7.2).

Equations (18.14.3_5) and (18.14.8), both for $\alpha=0$ , can be seen as special cases of this result for Jacobi and Laguerre polynomials, respectively.

18.1 Notation 18.3 Definitions

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18.2.11_7		$q_{n}(x)=\ifrac{p_{n}(x)}{\sqrt{h_{n}}},$
$n\geq 0$ .
ⓘ Symbols: $p_{n}(x)$ : polynomial of degree $n$ , $q$ : real variable, $n$ : nonnegative integer, $x$ : real variable and $h_{n}$ Permalink: http://dlmf.nist.gov/18.2.E11_7 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

18.2.11_8		$\displaystyle xq_{n}(x)$	$\displaystyle=\sqrt{\beta_{n+1}}\,q_{n+1}(x)+\alpha_{n}q_{n}(x)+\sqrt{\beta_{n% }}\,q_{n-1}(x),$
	$n\geq 1$
		$\displaystyle q_{1}(x)$	$\displaystyle=(x-\alpha_{0})/\sqrt{h_{0}\beta_{1}},$
		$\displaystyle q_{0}(x)$	$\displaystyle=1/\sqrt{h_{0}},$
ⓘ Symbols: $q$ : real variable, $n$ : nonnegative integer, $x$ : real variable and $h_{n}$ Referenced by: §18.2(xi), §18.2(iii), §18.2(viii), §18.40(ii) Permalink: http://dlmf.nist.gov/18.2.E11_8 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

18.2.11_6		$\beta_{n}>0,$
$n\geq 1$ .
ⓘ Symbols: $n$ : nonnegative integer Referenced by: §18.2(viii) Permalink: http://dlmf.nist.gov/18.2.E11_6 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18