18 Orthogonal Polynomials Classical Orthogonal Polynomials 18.8 Differential Equations 18.10 Integral Representations

§18.9 Recurrence Relations and Derivatives

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Referenced by:: Erratum (V1.2.0) §18.9
Permalink:: http://dlmf.nist.gov/18.9
See also:: Annotations for Ch.18

§18.9(i) Recurrence Relations
§18.9(ii) Contiguous Relations in the Parameters and the Degree
§18.9(iii) Derivatives

§18.9(i) Recurrence Relations

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Keywords:: classical orthogonal polynomials, recurrence relations
Notes:: For (18.9.1), (18.9.2) see Szegő (1975, (4.5.1)). For Table 18.9.1, Rows 2, 9, 10, see Szegő (1975, (4.7.17), (5.1.10), (5.5.8)), respectively; Rows 3 and 4 are rewritings of elementary trigonometric identities in view of (18.5.1), (18.5.2); Row 7 is the special case $\alpha=\beta=0$ of (18.9.2).
Referenced by:: §1.10(xi), §18.5(iv), Erratum (V1.0.7) for Other, Erratum (V1.2.0) §18.9
Permalink:: http://dlmf.nist.gov/18.9.i
Addition (effective with 1.2.0):: Equations (18.9.2_1), (18.9.2_2) and Table 18.9.2 were added.
Addition (effective with 1.0.7):: Initial values $p_{0}(x)$ and $p_{1}(x)$ for the general recurrence relation have been added in terms of the coefficients $A_{0}$ and $B_{0}$ , and these coefficients have been explicitly specified for Jacobi polynomials in this subsection.
Suggested 2013-06-24 by Christopher Gilbreth
See also:: Annotations for §18.9 and Ch.18

The notation of §18.2(iv) will be used.

First Form

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Keywords:: Jacobi polynomials, classical orthogonal polynomials, recurrence relations
Addition (effective with 1.2.0):: This paragraph was added.
See also:: Annotations for §18.9(i), §18.9 and Ch.18

18.9.1		$p_{n+1}(x)=(A_{n}x+B_{n})p_{n}(x)-C_{n}p_{n-1}(x),$
ⓘ Defines: $A_{n}$ : coefficient (locally), $B_{n}$ : coefficient (locally) and $C_{n}$ : coefficient (locally) Symbols: $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.21(ii), Table 18.9.1, §18.9(i) Permalink: http://dlmf.nist.gov/18.9.E1 Encodings: TeX, pMML, png See also: Annotations for §18.9(i), §18.9(i), §18.9 and Ch.18

with initial values $p_{0}(x)=1$ and $p_{1}(x)=A_{0}x+B_{0}$ .

For $p_{n}(x)=P^{(\alpha,\beta)}_{n}\left(x\right)$ ,

18.9.2	$\displaystyle A_{n}$	$\displaystyle=\dfrac{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}{2(n+1)(n+\alpha+% \beta+1)},$
	$\displaystyle B_{n}$	$\displaystyle=\dfrac{(\alpha^{2}-\beta^{2})(2n+\alpha+\beta+1)}{2(n+1)(n+% \alpha+\beta+1)(2n+\alpha+\beta)},$
	$\displaystyle C_{n}$	$\displaystyle=\dfrac{(n+\alpha)(n+\beta)(2n+\alpha+\beta+2)}{(n+1)(n+\alpha+% \beta+1)(2n+\alpha+\beta)}.$
ⓘ Symbols: $n$ : nonnegative integer, $A_{n}$ : coefficient, $B_{n}$ : coefficient and $C_{n}$ : coefficient Referenced by: §18.30(i), §18.9(i) Permalink: http://dlmf.nist.gov/18.9.E2 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §18.9(i), §18.9(i), §18.9 and Ch.18

$A_{0}$ and $B_{0}$ have to be understood for $\alpha+\beta=0$ or $-1$ by continuity in $\alpha$ and $\beta$ , that is, $A_{0}=\tfrac{1}{2}(\alpha+\beta)+1$ and $B_{0}=\tfrac{1}{2}(\alpha-\beta)$ .

For the other classical OP’s see Table 18.9.1; compare also §18.2(iv).

Table 18.9.1: Classical OP’s: recurrence relations (18.9.1).

$p_{n}(x)$	$A_{n}$	$B_{n}$	$C_{n}$
$C^{(\lambda)}_{n}\left(x\right)$	$\frac{2(n+\lambda)}{n+1}$	$0$	$\frac{n+2\lambda-1}{n+1}$
$T_{n}\left(x\right)$	$2-\delta_{n,0}$	$0$	$1$
$U_{n}\left(x\right)$	$2$	$0$	$1$
$T^{*}_{n}\left(x\right)$	$4-2\delta_{n,0}$	$-2+\delta_{n,0}$	$1$
$U^{*}_{n}\left(x\right)$	$4$	$-2$	$1$
$P_{n}\left(x\right)$	$\frac{2n+1}{n+1}$	$0$	$\frac{n}{n+1}$
$P^{*}_{n}\left(x\right)$	$\frac{4n+2}{n+1}$	$-\frac{2n+1}{n+1}$	$\frac{n}{n+1}$
$L^{(\alpha)}_{n}\left(x\right)$	$-\frac{1}{n+1}$	$\frac{2n+\alpha+1}{n+1}$	$\frac{n+\alpha}{n+1}$
$H_{n}\left(x\right)$	$2$	$0$	$2n$
$\mathit{He}_{n}\left(x\right)$	$1$	$0$	$n$

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Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $T^{*}_{\NVar{n}}\left(\NVar{x}\right)$ : shifted Chebyshev polynomial of the first kind, $U^{*}_{\NVar{n}}\left(\NVar{x}\right)$ : shifted Chebyshev polynomial of the second kind, $P^{*}_{\NVar{n}}\left(\NVar{x}\right)$ : shifted Legendre polynomial, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer, $x$ : real variable, $A_{n}$ : coefficient, $B_{n}$ : coefficient and $C_{n}$ : coefficient
Keywords:: Chebyshev polynomials, Hermite polynomials, Laguerre polynomials, Legendre polynomials, recurrence relations, ultraspherical polynomials
A&S Ref:: 22.7.1, 22.7.3, 22.7.4, 22.7.5, 22.7.8, 22.7.9, 22.7.10, 22.7.11, 22.7.12, 22.7.13, 22.7.14
Referenced by:: §18.21(ii), §18.9(i), §18.9(i), Erratum (V1.0.1) for Table 18.9.1
Permalink:: http://dlmf.nist.gov/18.9.T1
Errata (effective with 1.0.1):: The coefficient $A_{n}$ for $C^{(\lambda)}_{n}\left(x\right)$ in the first row of this table originally omitted the parentheses and was given as $\frac{2n+\lambda}{n+1}$ , instead of $\frac{2(n+\lambda)}{n+1}$ .
Reported 2010-09-16 by Kendall Atkinson
See also:: Annotations for §18.9(i), §18.9(i), §18.9 and Ch.18

Second Form

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Keywords:: Jacobi polynomials, classical orthogonal polynomials, recurrence relations
Addition (effective with 1.2.0):: This paragraph was added.
See also:: Annotations for §18.9(i), §18.9 and Ch.18

18.9.2_1		$xp_{n}(x)=a_{n}p_{n+1}(x)+b_{n}p_{n}(x)+c_{n}p_{n-1}(x)$
ⓘ Symbols: $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer and $x$ : real variable Referenced by: Table 18.9.2, §18.9(i), Erratum (V1.2.0) §18.9 Permalink: http://dlmf.nist.gov/18.9.E2_1 Encodings: TeX, pMML, png See also: Annotations for §18.9(i), §18.9(i), §18.9 and Ch.18

with initial values $p_{0}(x)=1$ and $p_{1}(x)=a_{0}^{-1}(x-b_{0})$ .

For $p_{n}(x)=P^{(\alpha,\beta)}_{n}\left(x\right)$ ,

18.9.2_2	$\displaystyle a_{n}$	$\displaystyle=\dfrac{2(n+1)(n+\alpha+\beta+1)}{(2n+\alpha+\beta+1)(2n+\alpha+% \beta+2)},$
	$\displaystyle b_{n}$	$\displaystyle=\dfrac{\beta^{2}-\alpha^{2}}{(2n+\alpha+\beta)(2n+\alpha+\beta+2% )},$
	$\displaystyle c_{n}$	$\displaystyle=\dfrac{2(n+\alpha)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1% )}.$
ⓘ Symbols: $n$ : nonnegative integer Referenced by: §18.9(i), Erratum (V1.2.0) §18.9 Permalink: http://dlmf.nist.gov/18.9.E2_2 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §18.9(i), §18.9(i), §18.9 and Ch.18

$a_{0}$ and $b_{0}$ have to be understood for $\alpha+\beta=0$ or $-1$ by continuity in $\alpha$ and $\beta$ , that is, $a_{0}=\ifrac{2}{(\alpha+\beta+2)}$ and and $b_{0}=\ifrac{(\beta-\alpha)}{(\alpha+\beta+2)}$ .

For the other classical OP’s see Table 18.9.2.

Table 18.9.2: Classical OP’s: recurrence relations (18.9.2_1).

$p_{n}(x)$	$a_{n}$	$b_{n}$	$c_{n}$
$C^{(\lambda)}_{n}\left(x\right)$	$\frac{n+1}{2(n+\lambda)}$	$0$	$\frac{n+2\lambda-1}{2(n+\lambda)}$
$T_{n}\left(x\right)$	$\frac{1}{2}(1+\delta_{n,0})$	$0$	$\frac{1}{2}$
$U_{n}\left(x\right)$	$\frac{1}{2}$	$0$	$\frac{1}{2}$
$V_{n}\left(x\right)$	$\frac{1}{2}$	$-\frac{1}{2}\delta_{n,0}$	$\frac{1}{2}$
$W_{n}\left(x\right)$	$\frac{1}{2}$	$\frac{1}{2}\delta_{n,0}$	$\frac{1}{2}$
$P_{n}\left(x\right)$	$\frac{n+1}{2n+1}$	$0$	$\frac{n}{2n+1}$
$L^{(\alpha)}_{n}\left(x\right)$	$-n-1$	$2n+\alpha+1$	$-n-\alpha$
$H_{n}\left(x\right)$	$\frac{1}{2}$	$0$	$n$

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Symbols:: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $W_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the fourth kind, $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $V_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the third kind, $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer and $x$ : real variable
Keywords:: Chebyshev polynomials, Hermite polynomials, Laguerre polynomials, Legendre polynomials, recurrence relations, ultraspherical polynomials
Referenced by:: §18.9(i), §18.9(i), Erratum (V1.2.0) §18.9
Permalink:: http://dlmf.nist.gov/18.9.T2
Addition (effective with 1.2.0):: This table was added.
See also:: Annotations for §18.9(i), §18.9(i), §18.9 and Ch.18

For the monic versions of the classical OP’s the recurrence coefficients $b_{n}$ and $c_{n}$ (there written as $\alpha_{n}$ and $\beta_{n}$ , respectively) are given in §3.5(vi). They imply the recurrence coefficients for the orthonormal versions of the classical OP’s as well, see again §3.5(vi).

§18.9(ii) Contiguous Relations in the Parameters and the Degree

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Keywords:: classical orthogonal polynomials, contiguous relations
Notes:: For (18.9.3)–(18.9.5) see Rainville (1960, §138, (17), (16), (14)). For (18.9.6) see Szegő (1975, (4.5.4)). For (18.9.7) see Szegő (1975, (4.7.29)). For (18.9.8) substitute (18.7.15) or (18.7.16); the resulting formula is a special case of Rainville (1960, §138, (11)). (18.9.9)–(18.9.12) are rewritings of elementary trigonometric identities in view of (18.5.1)–(18.5.4). For (18.9.13) see Szegő (1975, (5.1.13)). For (18.9.14) see Szegő (1975, (5.1.14)).
Permalink:: http://dlmf.nist.gov/18.9.ii
See also:: Annotations for §18.9 and Ch.18

Jacobi

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See also:: Annotations for §18.9(ii), §18.9 and Ch.18

18.9.3		$P^{(\alpha,\beta-1)}_{n}\left(x\right)-P^{(\alpha-1,\beta)}_{n}\left(x\right)=% P^{(\alpha,\beta)}_{n-1}\left(x\right),$
ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.7.20 Referenced by: §18.9(ii) Permalink: http://dlmf.nist.gov/18.9.E3 Encodings: TeX, pMML, png See also: Annotations for §18.9(ii), §18.9(ii), §18.9 and Ch.18

18.9.4		$(1-x)P^{(\alpha+1,\beta)}_{n}\left(x\right)+(1+x)P^{(\alpha,\beta+1)}_{n}\left% (x\right)=2P^{(\alpha,\beta)}_{n}\left(x\right),$
ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.7.17 Permalink: http://dlmf.nist.gov/18.9.E4 Encodings: TeX, pMML, png See also: Annotations for §18.9(ii), §18.9(ii), §18.9 and Ch.18

18.9.5		$(2n+\alpha+\beta+1)P^{(\alpha,\beta)}_{n}\left(x\right)=(n+\alpha+\beta+1)P^{(% \alpha,\beta+1)}_{n}\left(x\right)+(n+\alpha)P^{(\alpha,\beta+1)}_{n-1}\left(x% \right),$
ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.7.19 Referenced by: §18.9(ii), §18.9(ii), §18.9(ii) Permalink: http://dlmf.nist.gov/18.9.E5 Encodings: TeX, pMML, png See also: Annotations for §18.9(ii), §18.9(ii), §18.9 and Ch.18

18.9.6		$(n+\tfrac{1}{2}\alpha+\tfrac{1}{2}\beta+1)(1+x)P^{(\alpha,\beta+1)}_{n}\left(x% \right)=(n+1)P^{(\alpha,\beta)}_{n+1}\left(x\right)+(n+\beta+1)P^{(\alpha,% \beta)}_{n}\left(x\right),$
ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.7.16 Referenced by: §18.9(ii), §18.9(ii), §18.9(ii) Permalink: http://dlmf.nist.gov/18.9.E6 Encodings: TeX, pMML, png See also: Annotations for §18.9(ii), §18.9(ii), §18.9 and Ch.18

and a similar pair to (18.9.5) and (18.9.6) by symmetry; compare the second row in Table 18.6.1.

Ultraspherical

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See also:: Annotations for §18.9(ii), §18.9 and Ch.18

18.9.7	$\displaystyle(n+\lambda)C^{(\lambda)}_{n}\left(x\right)$	$\displaystyle=\lambda\left(C^{(\lambda+1)}_{n}\left(x\right)-C^{(\lambda+1)}_{% n-2}\left(x\right)\right),$
ⓘ Symbols: $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.9(ii) Permalink: http://dlmf.nist.gov/18.9.E7 Encodings: TeX, pMML, png See also: Annotations for §18.9(ii), §18.9(ii), §18.9 and Ch.18
18.9.8	$\displaystyle 4\lambda(n+\lambda+1)(1-x^{2})C^{(\lambda+1)}_{n}\left(x\right)$	$\displaystyle=-(n+1)(n+2)C^{(\lambda)}_{n+2}\left(x\right)+(n+2\lambda)(n+2% \lambda+1)C^{(\lambda)}_{n}\left(x\right).$
ⓘ Symbols: $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.9(ii) Permalink: http://dlmf.nist.gov/18.9.E8 Encodings: TeX, pMML, png See also: Annotations for §18.9(ii), §18.9(ii), §18.9 and Ch.18

Chebyshev

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See also:: Annotations for §18.9(ii), §18.9 and Ch.18

18.9.9	$\displaystyle T_{n}\left(x\right)$	$\displaystyle=\tfrac{1}{2}\left(U_{n}\left(x\right)-U_{n-2}\left(x\right)% \right),$
ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.5.8 Referenced by: §18.7(i), §18.9(ii) Permalink: http://dlmf.nist.gov/18.9.E9 Encodings: TeX, pMML, png See also: Annotations for §18.9(ii), §18.9(ii), §18.9 and Ch.18
18.9.10	$\displaystyle(1-x^{2})U_{n}\left(x\right)$	$\displaystyle=-\tfrac{1}{2}\left(T_{n+2}\left(x\right)-T_{n}\left(x\right)% \right),$
ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $n$ : nonnegative integer and $x$ : real variable Permalink: http://dlmf.nist.gov/18.9.E10 Encodings: TeX, pMML, png See also: Annotations for §18.9(ii), §18.9(ii), §18.9 and Ch.18

18.9.11	$\displaystyle V_{n}\left(x\right)+V_{n-1}\left(x\right)$	$\displaystyle=2T_{n}\left(x\right),$
ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $W_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the fourth kind, $V_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the third kind, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.9(ii), §18.9(ii), Erratum (V1.0.28) for Table 18.3.1 Permalink: http://dlmf.nist.gov/18.9.E11 Encodings: TeX, pMML, png Correction (effective with 1.0.28): The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_{n}\left(x\right)$ , $W_{n}\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_{n}\left(x\right)$ , $W_{n}\left(x\right)$ , having been interchanged, on the left-hand side we replaced $W_{n}\left(x\right)+W_{n-1}\left(x\right)$ with $V_{n}\left(x\right)+V_{n-1}\left(x\right)$ . For further details see Errata. See also: Annotations for §18.9(ii), §18.9(ii), §18.9 and Ch.18
18.9.12	$\displaystyle T_{n+1}\left(x\right)+T_{n}\left(x\right)$	$\displaystyle=(1+x)V_{n}\left(x\right).$
ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $W_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the fourth kind, $V_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the third kind, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.7(i), §18.9(ii), §18.9(ii), §18.9(ii), Erratum (V1.0.28) for Table 18.3.1 Permalink: http://dlmf.nist.gov/18.9.E12 Encodings: TeX, pMML, png Correction (effective with 1.0.28): The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_{n}\left(x\right)$ , $W_{n}\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_{n}\left(x\right)$ , $W_{n}\left(x\right)$ , having been interchanged, on the right-hand side we replaced $(1+x)W_{n}\left(x\right)$ with $(1+x)V_{n}\left(x\right)$ . For further details see Errata. See also: Annotations for §18.9(ii), §18.9(ii), §18.9 and Ch.18

Identities similar to (18.9.11) and (18.9.12) involving $W_{n}\left(x\right)$ and $T_{n}\left(x\right)$ can be obtained using rows 4 and 7 in Table 18.6.1.

Laguerre

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See also:: Annotations for §18.9(ii), §18.9 and Ch.18

18.9.13	$\displaystyle L^{(\alpha)}_{n}\left(x\right)$	$\displaystyle=L^{(\alpha+1)}_{n}\left(x\right)-L^{(\alpha+1)}_{n-1}\left(x% \right),$
ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.7.30 Referenced by: §18.17(i), §18.9(ii), §18.9(ii) Permalink: http://dlmf.nist.gov/18.9.E13 Encodings: TeX, pMML, png See also: Annotations for §18.9(ii), §18.9(ii), §18.9 and Ch.18
18.9.14	$\displaystyle xL^{(\alpha+1)}_{n}\left(x\right)$	$\displaystyle=-(n+1)L^{(\alpha)}_{n+1}\left(x\right)+(n+\alpha+1)L^{(\alpha)}_% {n}\left(x\right).$
ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.7.31 Referenced by: §18.9(ii), §18.9(ii) Permalink: http://dlmf.nist.gov/18.9.E14 Encodings: TeX, pMML, png See also: Annotations for §18.9(ii), §18.9(ii), §18.9 and Ch.18

Formulas (18.9.5), (18.9.11), (18.9.13) are special cases of (18.2.16). Formulas (18.9.6), (18.9.12), (18.9.14) are special cases of (18.2.17).

§18.9(iii) Derivatives

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Keywords:: classical orthogonal polynomials, degree lowering and raising, derivatives, differentiation formulas
Notes:: For (18.9.15) see Szegő (1975, (4.21.7)). (18.9.16) is an immediate corollary of (18.5.5) and Table 18.5.1, Row 2. For (18.9.17) and (18.9.18) see Koornwinder (2006, §4). For (18.9.19) see Szegő (1975, (4.7.14)). (18.9.20) is an immediate corollary of (18.5.5) and Table 18.5.1, Row 3. (18.9.21) and (18.9.22) are rewritings of elementary trigonometric differentiation formulas. For (18.9.23) see Szegő (1975, (5.1.14)). (18.9.24) is an immediate corollary of (18.5.5) and Table 18.5.1, Row 9. For (18.9.25) see Szegő (1975, (5.5.10)). (18.9.26) is an immediate corollary of (18.5.5) and Table 18.5.1, Row 10.
Referenced by:: item 2., Erratum (V1.2.0) §18.9
Permalink:: http://dlmf.nist.gov/18.9.iii
See also:: Annotations for §18.9 and Ch.18

Jacobi

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Keywords:: Jacobi polynomials, derivatives, structure relation
See also:: Annotations for §18.9(iii), §18.9 and Ch.18

18.9.15		$\frac{\mathrm{d}}{\mathrm{d}x}P^{(\alpha,\beta)}_{n}\left(x\right)=\tfrac{1}{2% }(n+\alpha+\beta+1)P^{(\alpha+1,\beta+1)}_{n-1}\left(x\right),$
ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(i), §18.9(iii), §18.9(iii) Permalink: http://dlmf.nist.gov/18.9.E15 Encodings: TeX, pMML, png See also: Annotations for §18.9(iii), §18.9(iii), §18.9 and Ch.18

18.9.16		$\frac{\mathrm{d}}{\mathrm{d}x}\left((1-x)^{\alpha}(1+x)^{\beta}P^{(\alpha,% \beta)}_{n}\left(x\right)\right)=-2(n+1)(1-x)^{\alpha-1}(1+x)^{\beta-1}P^{(% \alpha-1,\beta-1)}_{n+1}\left(x\right).$
ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(i), §18.9(iii), §18.9(iii) Permalink: http://dlmf.nist.gov/18.9.E16 Encodings: TeX, pMML, png See also: Annotations for §18.9(iii), §18.9(iii), §18.9 and Ch.18

Further $n$ -th derivative formulas relating two different Jacobi polynomials can be obtained from §15.5(i) by substitution of (18.5.7).

Formula (18.9.15) is degree lowering, while it raises the parameters. Formula (18.9.16) is degree raising, while it lowers the parameters. The following three formulas change the degree but preserve the parameters, see (18.2.42)–(18.2.44) for similar formulas for more general OP’s.

18.9.17		$(2n+\alpha+\beta)(1-x^{2})\frac{\mathrm{d}}{\mathrm{d}x}P^{(\alpha,\beta)}_{n}% \left(x\right)=n\left(\alpha-\beta-(2n+\alpha+\beta)x\right)P^{(\alpha,\beta)}% _{n}\left(x\right)+2(n+\alpha)(n+\beta)P^{(\alpha,\beta)}_{n-1}\left(x\right),$
ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.11.1 Referenced by: §18.9(iii) Permalink: http://dlmf.nist.gov/18.9.E17 Encodings: TeX, pMML, png See also: Annotations for §18.9(iii), §18.9(iii), §18.9 and Ch.18

18.9.18		$(2n+\alpha+\beta+2)(1-x^{2})\frac{\mathrm{d}}{\mathrm{d}x}P^{(\alpha,\beta)}_{% n}\left(x\right)=(n+\alpha+\beta+1)\left(\alpha-\beta+(2n+\alpha+\beta+2)x% \right)P^{(\alpha,\beta)}_{n}\left(x\right)-2(n+1)(n+\alpha+\beta+1)P^{(\alpha% ,\beta)}_{n+1}\left(x\right),$
ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.9(iii) Permalink: http://dlmf.nist.gov/18.9.E18 Encodings: TeX, pMML, png See also: Annotations for §18.9(iii), §18.9(iii), §18.9 and Ch.18

and the structure relation

18.9.18_5		$(1-x^{2})\frac{\mathrm{d}}{\mathrm{d}x}P^{(\alpha,\beta)}_{n}\left(x\right)=-% \frac{2n(n+1)(n+\alpha+\beta+1)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}P^{(% \alpha,\beta)}_{n+1}\left(x\right)+\frac{2n(n+\alpha+\beta+1)(\alpha-\beta)}{(% 2n+\alpha+\beta)(2n+\alpha+\beta+2)}P^{(\alpha,\beta)}_{n}\left(x\right)+\frac% {2(n+\alpha)(n+\beta)(n+\alpha+\beta+1)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}% P^{(\alpha,\beta)}_{n-1}\left(x\right).$
ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : Jacobi polynomial, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : nonnegative integer and $x$ : real variable Proved: Koornwinder (2007b, (3.7))(proved) Referenced by: Erratum (V1.2.0) §18.9 Permalink: http://dlmf.nist.gov/18.9.E18_5 Encodings: TeX, pMML, png See also: Annotations for §18.9(iii), §18.9(iii), §18.9 and Ch.18

Ultraspherical

ⓘ

Keywords:: derivatives, ultraspherical polynomials
See also:: Annotations for §18.9(iii), §18.9 and Ch.18

18.9.19		$\frac{\mathrm{d}}{\mathrm{d}x}C^{(\lambda)}_{n}\left(x\right)=2\lambda C^{(% \lambda+1)}_{n-1}\left(x\right),$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable Referenced by: §1.10(xi), §18.17(i), §18.9(iii) Permalink: http://dlmf.nist.gov/18.9.E19 Encodings: TeX, pMML, png See also: Annotations for §18.9(iii), §18.9(iii), §18.9 and Ch.18

18.9.20		$\frac{\mathrm{d}}{\mathrm{d}x}\left((1-x^{2})^{\lambda-\frac{1}{2}}C^{(\lambda% )}_{n}\left(x\right)\right)=-\frac{(n+1)(n+2\lambda-1)}{2(\lambda-1)}{(1-x^{2}% )^{\lambda-\frac{3}{2}}}C^{(\lambda-1)}_{n+1}\left(x\right).$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.9(iii) Permalink: http://dlmf.nist.gov/18.9.E20 Encodings: TeX, pMML, png See also: Annotations for §18.9(iii), §18.9(iii), §18.9 and Ch.18

See also the differentiation formulas in (Erdélyi et al., 1953b, §10.9(15))).

Chebyshev

ⓘ

Keywords:: Chebyshev polynomials, derivatives
See also:: Annotations for §18.9(iii), §18.9 and Ch.18

18.9.21		$\frac{\mathrm{d}}{\mathrm{d}x}T_{n}\left(x\right)=nU_{n-1}\left(x\right),$
ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.9(iii) Permalink: http://dlmf.nist.gov/18.9.E21 Encodings: TeX, pMML, png See also: Annotations for §18.9(iii), §18.9(iii), §18.9 and Ch.18

18.9.22		$\frac{\mathrm{d}}{\mathrm{d}x}\left((1-x^{2})^{\frac{1}{2}}U_{n}\left(x\right)% \right)=-(n+1){(1-x^{2})^{-\frac{1}{2}}}T_{n+1}\left(x\right).$
ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the first kind, $U_{\NVar{n}}\left(\NVar{x}\right)$ : Chebyshev polynomial of the second kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.9(iii) Permalink: http://dlmf.nist.gov/18.9.E22 Encodings: TeX, pMML, png See also: Annotations for §18.9(iii), §18.9(iii), §18.9 and Ch.18

Laguerre

ⓘ

Keywords:: Laguerre polynomials, derivatives
See also:: Annotations for §18.9(iii), §18.9 and Ch.18

18.9.23	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}L^{(\alpha)}_{n}\left(x\right)$	$\displaystyle=-L^{(\alpha+1)}_{n-1}\left(x\right),$
ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(i), §18.39(ii), §18.9(iii) Permalink: http://dlmf.nist.gov/18.9.E23 Encodings: TeX, pMML, png See also: Annotations for §18.9(iii), §18.9(iii), §18.9 and Ch.18
18.9.24	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left({\mathrm{e}}^{-x}x^{\alpha}L^% {(\alpha)}_{n}\left(x\right)\right)$	$\displaystyle=(n+1){\mathrm{e}}^{-x}x^{\alpha-1}L^{(\alpha-1)}_{n+1}\left(x% \right).$
ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.9(iii) Permalink: http://dlmf.nist.gov/18.9.E24 Encodings: TeX, pMML, png See also: Annotations for §18.9(iii), §18.9(iii), §18.9 and Ch.18

Further $n$ -th derivative formulas relating two different Laguerre polynomials can be obtained from §13.3(ii) by substitution of (13.6.19).

Hermite

ⓘ

Keywords:: Hermite polynomials, classical orthogonal polynomials, derivatives
See also:: Annotations for §18.9(iii), §18.9 and Ch.18

18.9.25	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}H_{n}\left(x\right)$	$\displaystyle=2nH_{n-1}\left(x\right),$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.8.7 Referenced by: §18.17(i), §18.9(iii) Permalink: http://dlmf.nist.gov/18.9.E25 Encodings: TeX, pMML, png See also: Annotations for §18.9(iii), §18.9(iii), §18.9 and Ch.18
18.9.26	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left({\mathrm{e}}^{-x^{2}}H_{n}% \left(x\right)\right)$	$\displaystyle=-{\mathrm{e}}^{-x^{2}}H_{n+1}\left(x\right),$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(i), §18.9(iii) Permalink: http://dlmf.nist.gov/18.9.E26 Encodings: TeX, pMML, png See also: Annotations for §18.9(iii), §18.9(iii), §18.9 and Ch.18

18.9.27	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\mathit{He}_{n}\left(x\right)$	$\displaystyle=n\mathit{He}_{n-1}\left(x\right),$
ⓘ Symbols: $\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : nonnegative integer and $x$ : real variable A&S Ref: 22.8.7 Permalink: http://dlmf.nist.gov/18.9.E27 Encodings: TeX, pMML, png See also: Annotations for §18.9(iii), §18.9(iii), §18.9 and Ch.18
18.9.28	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\left({\mathrm{e}}^{-\frac{1}{2}x^{% 2}}\mathit{He}_{n}\left(x\right)\right)$	$\displaystyle=-{\mathrm{e}}^{-\frac{1}{2}x^{2}}\mathit{He}_{n+1}\left(x\right).$
ⓘ Symbols: $\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.17(i) Permalink: http://dlmf.nist.gov/18.9.E28 Encodings: TeX, pMML, png See also: Annotations for §18.9(iii), §18.9(iii), §18.9 and Ch.18

18.8 Differential Equations 18.10 Integral Representations

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§18.9 Recurrence Relations and Derivatives

Contents

§18.9(i) Recurrence Relations

First Form

Second Form

§18.9(ii) Contiguous Relations in the Parameters and the Degree

Jacobi

Ultraspherical

Chebyshev

Laguerre

§18.9(iii) Derivatives

Jacobi

Ultraspherical

Chebyshev

Laguerre

Hermite