14 Legendre and Related Functions Real Arguments 14.6 Integer Order 14.8 Behavior at Singularities

§14.7 Integer Degree and Order

ⓘ

Keywords:: Ferrers functions, associated Legendre functions, integer degree and order
Permalink:: http://dlmf.nist.gov/14.7
See also:: Annotations for Ch.14

§14.7(i) $\mu=0$
§14.7(ii) Rodrigues-Type Formulas
§14.7(iii) Reflection Formulas
§14.7(iv) Generating Functions

§14.7(i) $\mu=0$

ⓘ

Keywords:: Ferrers functions, Legendre polynomials, associated Legendre functions, relations to other functions
Notes:: See Olver (1997b, pp. 174, 181, and 188).
Referenced by:: §1.17(iii), §14.5(ii), §18.3
Permalink:: http://dlmf.nist.gov/14.7.i
See also:: Annotations for §14.7 and Ch.14

For $n=0,1,2,\dots$ ,

14.7.1		$\mathsf{P}^{0}_{n}\left(x\right)=\mathsf{P}_{n}\left(x\right)=P^{0}_{n}\left(x% \right)=P_{n}\left(x\right),$
$x\in\mathbb{R}$ ,
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\in$ : element of, $\mathbb{R}$ : real line, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $x$ : real variable and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/14.7.E1 Encodings: TeX, pMML, png See also: Annotations for §14.7(i), §14.7 and Ch.14

where $P_{n}\left(x\right)$ is the Legendre polynomial of degree $n$ . For additional properties of $P_{n}\left(x\right)$ see Chapter 18.

14.7.2		$\mathsf{Q}^{0}_{n}\left(x\right)=\mathsf{Q}_{n}\left(x\right)=\frac{1}{2}P_{n}% \left(x\right)\ln\left(\frac{1+x}{1-x}\right)-W_{n-1}(x),$
ⓘ Symbols: $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\ln\NVar{z}$ : principal branch of logarithm function, $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : Ferrers function of the second kind, $x$ : real variable, $n$ : nonnegative integer and $W_{n-1}(x)$ : quantity A&S Ref: 8.6.19 Permalink: http://dlmf.nist.gov/14.7.E2 Encodings: TeX, pMML, png See also: Annotations for §14.7(i), §14.7 and Ch.14

where $W_{-1}(x)=0$ , and for $n\geq 1$ ,

14.7.3		$W_{n-1}(x)=\sum_{s=0}^{n-1}\frac{(n+s)!(\psi\left(n+1\right)-\psi\left(s+1% \right))}{2^{s}(n-s)!(s!)^{2}}{(x-1)^{s}};$
ⓘ Symbols: $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $!$ : factorial (as in $n!$ ), $x$ : real variable, $n$ : nonnegative integer and $W_{n-1}(x)$ : quantity Permalink: http://dlmf.nist.gov/14.7.E3 Encodings: TeX, pMML, png See also: Annotations for §14.7(i), §14.7 and Ch.14

equivalently,

14.7.4		$W_{n-1}(x)=\sum_{k=1}^{n}\frac{1}{k}P_{k-1}\left(x\right)P_{n-k}\left(x\right).$
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $x$ : real variable, $n$ : nonnegative integer and $W_{n-1}(x)$ : quantity A&S Ref: 8.6.19 Permalink: http://dlmf.nist.gov/14.7.E4 Encodings: TeX, pMML, png See also: Annotations for §14.7(i), §14.7 and Ch.14

14.7.5	$\displaystyle W_{0}(x)$	$\displaystyle=1,$
	$\displaystyle W_{1}(x)$	$\displaystyle=\tfrac{3}{2}x,$
	$\displaystyle W_{2}(x)$	$\displaystyle=\tfrac{5}{2}x^{2}-\tfrac{2}{3}.$
ⓘ Symbols: $x$ : real variable and $W_{n-1}(x)$ : quantity Permalink: http://dlmf.nist.gov/14.7.E5 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §14.7(i), §14.7 and Ch.14

Next,

14.7.6		$Q^{0}_{n}\left(x\right)=Q_{n}\left(x\right)=n!\boldsymbol{Q}^{0}_{n}\left(x% \right)=n!\boldsymbol{Q}_{n}\left(x\right),$
ⓘ Symbols: $Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the second kind, $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Olver’s associated Legendre function, $!$ : factorial (as in $n!$ ), $Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}\left(z\right)$ : Legendre function of the second kind, $\boldsymbol{Q}_{\NVar{\nu}}\left(\NVar{z}\right)=\boldsymbol{Q}^{0}_{\nu}\left% (z\right)$ : Olver’s associated Legendre function, $x$ : real variable and $n$ : nonnegative integer Referenced by: §14.21(iii) Permalink: http://dlmf.nist.gov/14.7.E6 Encodings: TeX, pMML, png See also: Annotations for §14.7(i), §14.7 and Ch.14

where

14.7.7		$Q_{n}\left(x\right)=\frac{1}{2}P_{n}\left(x\right)\ln\left(\frac{x+1}{x-1}% \right)-W_{n-1}(x),$
$n=0,1,2,\dots$ .
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\ln\NVar{z}$ : principal branch of logarithm function, $Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}\left(z\right)$ : Legendre function of the second kind, $x$ : real variable, $n$ : nonnegative integer and $W_{n-1}(x)$ : quantity Referenced by: §14.21(iii) Permalink: http://dlmf.nist.gov/14.7.E7 Encodings: TeX, pMML, png See also: Annotations for §14.7(i), §14.7 and Ch.14

§14.7(ii) Rodrigues-Type Formulas

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Keywords:: Ferrers functions, Rodrigues-type formulas, associated Legendre functions
Notes:: See Olver (1997b, pp. 174 and 181–182).
Permalink:: http://dlmf.nist.gov/14.7.ii
See also:: Annotations for §14.7 and Ch.14

For $m=0,1,2,\dots$ , and $n=0,1,2,\dots$ ,

14.7.8	$\displaystyle\mathsf{P}^{m}_{n}\left(x\right)$	$\displaystyle=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{\mathrm{d}}^{m}}{{% \mathrm{d}x}^{m}}\mathsf{P}_{n}\left(x\right),$
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $x$ : real variable, $m$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/14.7.E8 Encodings: TeX, pMML, png See also: Annotations for §14.7(ii), §14.7 and Ch.14
14.7.9	$\displaystyle\mathsf{Q}^{m}_{n}\left(x\right)$	$\displaystyle=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{\mathrm{d}}^{m}}{{% \mathrm{d}x}^{m}}\mathsf{Q}_{n}\left(x\right),$
ⓘ Symbols: $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : Ferrers function of the second kind, $x$ : real variable, $m$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/14.7.E9 Encodings: TeX, pMML, png See also: Annotations for §14.7(ii), §14.7 and Ch.14

14.7.10		$\mathsf{P}^{m}_{n}\left(x\right)=(-1)^{m+n}\frac{\left(1-x^{2}\right)^{m/2}}{2% ^{n}n!}\frac{{\mathrm{d}}^{m+n}}{{\mathrm{d}x}^{m+n}}\left(1-x^{2}\right)^{n}.$
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $!$ : factorial (as in $n!$ ), $x$ : real variable, $m$ : nonnegative integer and $n$ : nonnegative integer Referenced by: §14.30(ii) Permalink: http://dlmf.nist.gov/14.7.E10 Encodings: TeX, pMML, png See also: Annotations for §14.7(ii), §14.7 and Ch.14

14.7.11	$\displaystyle P^{m}_{n}\left(x\right)$	$\displaystyle=\left(x^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}}{{\mathrm{d}x}^% {m}}P_{n}\left(x\right),$
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $x$ : real variable, $m$ : nonnegative integer and $n$ : nonnegative integer Referenced by: §14.21(iii) Permalink: http://dlmf.nist.gov/14.7.E11 Encodings: TeX, pMML, png See also: Annotations for §14.7(ii), §14.7 and Ch.14
14.7.12	$\displaystyle Q^{m}_{n}\left(x\right)$	$\displaystyle=\left(x^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}}{{\mathrm{d}x}^% {m}}Q_{n}\left(x\right),$
ⓘ Symbols: $Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the second kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}\left(z\right)$ : Legendre function of the second kind, $x$ : real variable, $m$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/14.7.E12 Encodings: TeX, pMML, png See also: Annotations for §14.7(ii), §14.7 and Ch.14
14.7.13	$\displaystyle P_{n}\left(x\right)$	$\displaystyle=\frac{1}{2^{n}n!}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}}\left% (x^{2}-1\right)^{n},$
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $!$ : factorial (as in $n!$ ), $x$ : real variable and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/14.7.E13 Encodings: TeX, pMML, png See also: Annotations for §14.7(ii), §14.7 and Ch.14
14.7.14	$\displaystyle P^{m}_{n}\left(x\right)$	$\displaystyle=\frac{\left(x^{2}-1\right)^{m/2}}{2^{n}n!}\frac{{\mathrm{d}}^{m+% n}}{{\mathrm{d}x}^{m+n}}\left(x^{2}-1\right)^{n},$
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $!$ : factorial (as in $n!$ ), $x$ : real variable, $m$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/14.7.E14 Encodings: TeX, pMML, png See also: Annotations for §14.7(ii), §14.7 and Ch.14
14.7.15	$\displaystyle P^{m}_{m}\left(x\right)$	$\displaystyle=\frac{(2m)!}{2^{m}m!}\left(x^{2}-1\right)^{m/2}.$
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $!$ : factorial (as in $n!$ ), $x$ : real variable and $m$ : nonnegative integer Permalink: http://dlmf.nist.gov/14.7.E15 Encodings: TeX, pMML, png See also: Annotations for §14.7(ii), §14.7 and Ch.14

When $m$ is even and $m\leq n$ , $\mathsf{P}^{m}_{n}\left(x\right)$ and $P^{m}_{n}\left(x\right)$ are polynomials of degree $n$ . Also,

14.7.16		$\mathsf{P}^{m}_{n}\left(x\right)=P^{m}_{n}\left(x\right)=0,$
$m>n$ .
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $x$ : real variable, $m$ : nonnegative integer and $n$ : nonnegative integer Referenced by: §14.18(ii), §14.21(iii) Permalink: http://dlmf.nist.gov/14.7.E16 Encodings: TeX, pMML, png See also: Annotations for §14.7(ii), §14.7 and Ch.14

§14.7(iii) Reflection Formulas

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Keywords:: Ferrers functions, reflection formulas
Notes:: These results may be derived from (14.9.8) and (14.9.10).
Permalink:: http://dlmf.nist.gov/14.7.iii
See also:: Annotations for §14.7 and Ch.14

14.7.17	$\displaystyle\mathsf{P}^{m}_{n}\left(-x\right)$	$\displaystyle=(-1)^{n-m}\mathsf{P}^{m}_{n}\left(x\right),$
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $x$ : real variable, $m$ : nonnegative integer and $n$ : nonnegative integer Referenced by: §14.18(iii), §14.30(ii) Permalink: http://dlmf.nist.gov/14.7.E17 Encodings: TeX, pMML, png See also: Annotations for §14.7(iii), §14.7 and Ch.14
14.7.18	$\displaystyle\mathsf{Q}^{\pm m}_{n}\left(-x\right)$	$\displaystyle=(-1)^{n-m-1}\mathsf{Q}^{\pm m}_{n}\left(x\right).$
ⓘ Symbols: $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $x$ : real variable, $m$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/14.7.E18 Encodings: TeX, pMML, png See also: Annotations for §14.7(iii), §14.7 and Ch.14

§14.7(iv) Generating Functions

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Keywords:: Ferrers functions, associated Legendre functions, generating functions, integer degree and order
Notes:: See Erdélyi et al. (1953a, p. 154) and Olver (1997b, pp. 51 and 185).
Referenced by:: §2.10(iv)
Permalink:: http://dlmf.nist.gov/14.7.iv
See also:: Annotations for §14.7 and Ch.14

When $-1<x<1$ and $|h|<1$ ,

14.7.19		$\sum_{n=0}^{\infty}\mathsf{P}_{n}\left(x\right)h^{n}=\left(1-2xh+h^{2}\right)^% {-1/2},$
ⓘ Symbols: $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $x$ : real variable, $n$ : nonnegative integer and $h$ : variable Referenced by: §14.21(iii), §14.7(iv) Permalink: http://dlmf.nist.gov/14.7.E19 Encodings: TeX, pMML, png See also: Annotations for §14.7(iv), §14.7 and Ch.14

14.7.20		$\sum_{n=0}^{\infty}\mathsf{Q}_{n}\left(x\right)h^{n}=\frac{1}{\left(1-2xh+h^{2% }\right)^{1/2}}\*\ln\left(\frac{x-h+\left(1-2xh+h^{2}\right)^{1/2}}{\left(1-x^% {2}\right)^{1/2}}\right).$
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function, $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : Ferrers function of the second kind, $x$ : real variable, $n$ : nonnegative integer and $h$ : variable Permalink: http://dlmf.nist.gov/14.7.E20 Encodings: TeX, pMML, png See also: Annotations for §14.7(iv), §14.7 and Ch.14

When $-1<x<1$ and $|h|>1$ ,

14.7.21		$\sum_{n=0}^{\infty}\mathsf{P}_{n}\left(x\right)h^{-n-1}=\left(1-2xh+h^{2}% \right)^{-1/2}.$
ⓘ Symbols: $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $x$ : real variable, $n$ : nonnegative integer and $h$ : variable Referenced by: §14.21(iii), §14.7(iv) Permalink: http://dlmf.nist.gov/14.7.E21 Encodings: TeX, pMML, png See also: Annotations for §14.7(iv), §14.7 and Ch.14

When $x>1$ , (14.7.19) applies with $|h|<x-\left(x^{2}-1\right)^{1/2}$ . Also, with the same conditions

14.7.22		$\sum_{n=0}^{\infty}Q_{n}\left(x\right)h^{n}=\frac{1}{\left(1-2xh+h^{2}\right)^% {1/2}}\*\ln\left(\frac{x-h+\left(1-2xh+h^{2}\right)^{1/2}}{\left(x^{2}-1\right% )^{1/2}}\right).$
ⓘ Symbols: $\ln\NVar{z}$ : principal branch of logarithm function, $Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}\left(z\right)$ : Legendre function of the second kind, $x$ : real variable, $n$ : nonnegative integer and $h$ : variable Referenced by: §14.21(iii) Permalink: http://dlmf.nist.gov/14.7.E22 Encodings: TeX, pMML, png See also: Annotations for §14.7(iv), §14.7 and Ch.14

Lastly, when $x>1$ , (14.7.21) applies with $|h|>x+\left(x^{2}-1\right)^{1/2}$ .

For other generating functions see Magnus et al. (1966, pp. 232–233) and Rainville (1960, pp. 163–165, 168, 170–171, 184).

14.6 Integer Order 14.8 Behavior at Singularities

NIST

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§14.7 Integer Degree and Order

Contents

§14.7(i) μ=0

§14.7(ii) Rodrigues-Type Formulas

§14.7(iii) Reflection Formulas

§14.7(iv) Generating Functions

§14.7(i) $\mu=0$