§18.9 Recurrence Relations and Derivatives

Permalink:: http://dlmf.nist.gov/18.9

§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

Notes:: For (18.9.1), (18.9.2) see Szegö (1975, (4.5.1)). For Table 18.9.1, Rows 2, 9, 10, see Szegö (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).
Keywords:: Jacobi polynomials, classical orthogonal polynomials
Referenced by:: §18.5(iv), §3.5(vi)
Permalink:: http://dlmf.nist.gov/18.9.i

18.9.1 $p_{{n+1}}(x)=(A_{n}x+B_{n})p_{n}(x)-C_{n}p_{{n-1}}(x).$

Symbols:: $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ , $x$ : real variable, $A_{n}$ : coefficient, $B_{n}$ : coefficient and $C_{n}$ : coefficient
Referenced by:: §18.21(ii), §18.9(i), Table 18.9.1
Permalink:: http://dlmf.nist.gov/18.9.E1
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For $p_{n}(x)=\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ ,

18.9.2

$A_{n}=\dfrac{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}{2(n+1)(n+\alpha+\beta+1)},$

$B_{n}=\dfrac{(\alpha^{2}-\beta^{2})(2n+\alpha+\beta+1)}{2(n+1)(n+\alpha+\beta+1)(2n+\alpha+\beta)},$

$C_{n}=\dfrac{(n+\alpha)(n+\beta)(2n+\alpha+\beta+2)}{(n+1)(n+\alpha+\beta+1)(2n+\alpha+\beta)}.$

Symbols:: $n$ : nonnegative integer, $A_{n}$ : coefficient, $B_{n}$ : coefficient and $C_{n}$ : coefficient
Referenced by:: ¶ ‣ §18.30, §18.9(i)
Permalink:: http://dlmf.nist.gov/18.9.E2
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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).

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

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the second kind, $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{\mathit{He}_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\delta _{{j,k}}$ : Kronecker delta, $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\mathop{T^{{*}}_{{n}}\/}\nolimits\!\left(x\right)$ : shifted Chebyshev polynomial of the first kind, $\mathop{U^{{*}}_{{n}}\/}\nolimits\!\left(x\right)$ : shifted Chebyshev polynomial of the second kind, $\mathop{P^{{*}}_{{n}}\/}\nolimits\!\left(x\right)$ : shifted Legendre polynomial, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ , $x$ : real variable, $A_{n}$ : coefficient, $B_{n}$ : coefficient and $C_{n}$ : coefficient
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
Keywords:: Chebyshev polynomials, Hermite polynomials, Laguerre polynomials, Legendre polynomials, ultraspherical polynomials
Referenced by:: §18.21(ii), §18.9(i), §18.9(i), Version 1.0.1 (June 27, 2011)
Permalink:: http://dlmf.nist.gov/18.9.T1
Errata (effective with 1.0.1):: The coefficient $A_{n}$ for $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ 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

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

Notes:: For (18.9.3)–(18.9.5) see Rainville (1960, §138, (17), (16), (14)). For (18.9.6) see Szegö (1975, (4.5.4)). For (18.9.7) see Szegö (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 Szegö (1975, (5.1.13)). For (18.9.14) see Szegö (1975, (5.1.14)).
Keywords:: classical orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.9.ii

¶ Jacobi

18.9.3 $\mathop{P^{{(\alpha,\beta-1)}}_{{n}}\/}\nolimits\!\left(x\right)-\mathop{P^{{(\alpha-1,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)=\mathop{P^{{(\alpha,\beta)}}_{{n-1}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : 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

18.9.4 $(1-x)\mathop{P^{{(\alpha+1,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)+(1+x)\mathop{P^{{(\alpha,\beta+1)}}_{{n}}\/}\nolimits\!\left(x\right)=2\!\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.7.17
Permalink:: http://dlmf.nist.gov/18.9.E4
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18.9.5 $(2n+\alpha+\beta+1)\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)=(n+\alpha+\beta+1)\mathop{P^{{(\alpha,\beta+1)}}_{{n}}\/}\nolimits\!\left(x\right)+(n+\alpha)\mathop{P^{{(\alpha,\beta+1)}}_{{n-1}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.7.19
Referenced by:: ¶ ‣ §18.9(ii), §18.9(ii)
Permalink:: http://dlmf.nist.gov/18.9.E5
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18.9.6 $(n+\tfrac{1}{2}\alpha+\tfrac{1}{2}\beta+1)(1+x)\mathop{P^{{(\alpha,\beta+1)}}_{{n}}\/}\nolimits\!\left(x\right)=(n+1)\mathop{P^{{(\alpha,\beta)}}_{{n+1}}\/}\nolimits\!\left(x\right)+(n+\beta+1)\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.7.16
Referenced by:: ¶ ‣ §18.9(ii), §18.9(ii)
Permalink:: http://dlmf.nist.gov/18.9.E6
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and a similar pair to (18.9.5) and (18.9.6) by symmetry; compare the second row in Table 18.6.1.

¶ Ultraspherical

18.9.7 $(n+\lambda)\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)=\lambda\left(\mathop{C^{{(\lambda+1)}}_{{n}}\/}\nolimits\!\left(x\right)-\mathop{C^{{(\lambda+1)}}_{{n-2}}\/}\nolimits\!\left(x\right)\right),$

Symbols:: $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.9(ii)
Permalink:: http://dlmf.nist.gov/18.9.E7
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18.9.8 $4\lambda(n+\lambda+1)(1-x^{2})\mathop{C^{{(\lambda+1)}}_{{n}}\/}\nolimits\!\left(x\right)=-(n+1)(n+2)\mathop{C^{{(\lambda)}}_{{n+2}}\/}\nolimits\!\left(x\right)+(n+2\lambda)(n+2\lambda+1)\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.9(ii)
Permalink:: http://dlmf.nist.gov/18.9.E8
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¶ Chebyshev

18.9.9 $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)=\tfrac{1}{2}\left(\mathop{U_{{n}}\/}\nolimits\!\left(x\right)-\mathop{U_{{n-2}}\/}\nolimits\!\left(x\right)\right),$

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : 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
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18.9.10 $(1-x^{2})\mathop{U_{{n}}\/}\nolimits\!\left(x\right)=-\tfrac{1}{2}\left(\mathop{T_{{n+2}}\/}\nolimits\!\left(x\right)-\mathop{T_{{n}}\/}\nolimits\!\left(x\right)\right).$

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{U_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the second kind, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.9.E10
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18.9.11 $\mathop{W_{{n}}\/}\nolimits\!\left(x\right)+\mathop{W_{{n-1}}\/}\nolimits\!\left(x\right)=2\mathop{T_{{n}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{W_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the fourth kind, $n$ : nonnegative integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.9.E11
Encodings:: TeX, pMML, png

18.9.12 $\mathop{T_{{n+1}}\/}\nolimits\!\left(x\right)+\mathop{T_{{n}}\/}\nolimits\!\left(x\right)=(1+x)\mathop{W_{{n}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{T_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the first kind, $\mathop{W_{{n}}\/}\nolimits\!\left(x\right)$ : Chebyshev polynomial of the fourth kind, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: ¶ ‣ §18.7(i), §18.9(ii)
Permalink:: http://dlmf.nist.gov/18.9.E12
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¶ Laguerre

18.9.13 $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)=\mathop{L^{{(\alpha+1)}}_{{n}}\/}\nolimits\!\left(x\right)-\mathop{L^{{(\alpha+1)}}_{{n-1}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : 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)
Permalink:: http://dlmf.nist.gov/18.9.E13
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18.9.14 $x\mathop{L^{{(\alpha+1)}}_{{n}}\/}\nolimits\!\left(x\right)=-(n+1)\mathop{L^{{(\alpha)}}_{{n+1}}\/}\nolimits\!\left(x\right)+(n+\alpha+1)\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $n$ : nonnegative integer and $x$ : real variable
A&S Ref:: 22.7.31
Referenced by:: §18.9(ii)
Permalink:: http://dlmf.nist.gov/18.9.E14
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