§14.7 Integer Degree and Order

For $n=0,1,2,\mathrm{\dots }$,

14.7.1		$${\mathsf{P}}_n^0\left(x\right)={\mathsf{P}}_n\left(x\right)=P_n^0\left(x\right)=P_n\left(x\right),$$
$x\in ℝ$,
ⓘ Symbols: : Ferrers function of the first kind, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $P_n\left(x\right)$: Legendre polynomial, $\in$: element of, $ℝ$: real line, ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: 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		$${\mathsf{Q}}_n^0\left(x\right)={\mathsf{Q}}_n\left(x\right)=\frac{1}{2}{P}_n\left(x\right)\ln \left(\frac{1+x}{1-x}\right)-W_{n-1}(x),$$
ⓘ Symbols: : Ferrers function of the second kind, $P_n\left(x\right)$: Legendre polynomial, $\ln z$: principal branch of logarithm function, ${\mathsf{Q}}_{\nu }\left(x\right)={\mathsf{Q}}_{\nu }^0\left(x\right)$: 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 $n\ge 1$,

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: $\psi \left(z\right)$: 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: $P_n\left(x\right)$: 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		$W_0(x)$	$=1,$
		$W_1(x)$	$=\frac{3}{2}x,$
		$W_2(x)$	$=\frac{5}{2}{x}^2-\frac{2}{3}.$
ⓘ 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		$$Q_n^0\left(x\right)=Q_n\left(x\right)=n!{\mathit{Q}}_n^0\left(x\right)=n!{\mathit{Q}}_n\left(x\right),$$
ⓘ Symbols: $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, ${\mathit{Q}}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $!$: factorial (as in $n!$), $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, ${\mathit{Q}}_{\nu }\left(z\right)={\mathit{Q}}_{\nu }^0\left(z\right)$: 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		$$Q_n\left(x\right)=\frac{1}{2}{P}_n\left(x\right)\ln \left(\frac{x+1}{x-1}\right)-W_{n-1}(x),$$
$n=0,1,2,\mathrm{\dots }$.
ⓘ Symbols: $P_n\left(x\right)$: Legendre polynomial, $\ln z$: principal branch of logarithm function, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: 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

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

14.7.8		${\mathsf{P}}_n^m\left(x\right)$	$={(-1)}^m{\left(1-x^2\right)}^{m/2}{\displaystyle \frac{d^m}{{dx}^m}}{\mathsf{P}}_n\left(x\right),$
ⓘ Symbols: : Ferrers function of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: 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		${\mathsf{Q}}_n^m\left(x\right)$	$={(-1)}^m{\left(1-x^2\right)}^{m/2}{\displaystyle \frac{d^m}{{dx}^m}}{\mathsf{Q}}_n\left(x\right),$
ⓘ Symbols: : Ferrers function of the second kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, ${\mathsf{Q}}_{\nu }\left(x\right)={\mathsf{Q}}_{\nu }^0\left(x\right)$: 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		$${\mathsf{P}}_n^m\left(x\right)={(-1)}^{m+n}\frac{{\left(1-x^2\right)}^{m/2}}{2^{n}n!}\frac{d^{m+n}}{{dx}^{m+n}}{\left(1-x^2\right)}^n.$$
ⓘ Symbols: : Ferrers function of the first kind, $\frac{df}{dx}$: 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		$P_n^m\left(x\right)$	$={\left(x^2-1\right)}^{m/2}{\displaystyle \frac{d^m}{{dx}^m}}{P}_n\left(x\right),$
ⓘ Symbols: $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $P_n\left(x\right)$: Legendre polynomial, $\frac{df}{dx}$: 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		$Q_n^m\left(x\right)$	$={\left(x^2-1\right)}^{m/2}{\displaystyle \frac{d^m}{{dx}^m}}{Q}_n\left(x\right),$
ⓘ Symbols: $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: 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		$P_n\left(x\right)$	$={\displaystyle \frac{1}{2^{n}n!}}{\displaystyle \frac{d^n}{{dx}^n}}{\left(x^2-1\right)}^n,$
ⓘ Symbols: $P_n\left(x\right)$: Legendre polynomial, $\frac{df}{dx}$: 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		$P_n^m\left(x\right)$	$={\displaystyle \frac{{\left(x^2-1\right)}^{m/2}}{2^{n}n!}}{\displaystyle \frac{d^{m+n}}{{dx}^{m+n}}}{\left(x^2-1\right)}^n,$
ⓘ Symbols: $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\frac{df}{dx}$: 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		$P_m^m\left(x\right)$	$={\displaystyle \frac{(2m)!}{2^{m}m!}}{\left(x^2-1\right)}^{m/2}.$
ⓘ Symbols: $P_{\nu }^{\mu }\left(z\right)$: 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 $m\le n$, ${\mathsf{P}}_n^m\left(x\right)$ and $P_n^m\left(x\right)$ are polynomials of degree $n$. Also,

14.7.16		$${\mathsf{P}}_n^m\left(x\right)=P_n^m\left(x\right)=0,$$
$m>n$.
ⓘ Symbols: : Ferrers function of the first kind, $P_{\nu }^{\mu }\left(z\right)$: 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

14.7.17		${\mathsf{P}}_n^m\left(-x\right)$	$={(-1)}^{n-m}{\mathsf{P}}_n^m\left(x\right),$
ⓘ Symbols: : 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		${\mathsf{Q}}_n^{\pm m}\left(-x\right)$	$={(-1)}^{n-m-1}{\mathsf{Q}}_n^{\pm m}\left(x\right).$
ⓘ Symbols: : 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

When and ,

14.7.19		$$\sum _{n=0}^{\mathrm{\infty }}{\mathsf{P}}_n\left(x\right)h^n={\left(1-2xh+h^2\right)}^{-1/2},$$
ⓘ Symbols: ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: 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		$$\sum _{n=0}^{\mathrm{\infty }}{\mathsf{Q}}_n\left(x\right)h^n=\frac{1}{{\left(1-2xh+h^2\right)}^{1/2}}\ln \left(\frac{x-h+{\left(1-2xh+h^2\right)}^{1/2}}{{\left(1-x^2\right)}^{1/2}}\right).$$
ⓘ Symbols: $\ln z$: principal branch of logarithm function, ${\mathsf{Q}}_{\nu }\left(x\right)={\mathsf{Q}}_{\nu }^0\left(x\right)$: 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 and $|h|>1$,

14.7.21		$$\sum _{n=0}^{\mathrm{\infty }}{\mathsf{P}}_n\left(x\right)h^{-n-1}={\left(1-2xh+h^2\right)}^{-1/2}.$$
ⓘ Symbols: ${\mathsf{P}}_{\nu }\left(x\right)={\mathsf{P}}_{\nu }^0\left(x\right)$: 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 . Also, with the same conditions

14.7.22		$$\sum _{n=0}^{\mathrm{\infty }}{Q}_n\left(x\right)h^n=\frac{1}{{\left(1-2xh+h^2\right)}^{1/2}}\ln \left(\frac{x-h+{\left(1-2xh+h^2\right)}^{1/2}}{{\left(x^2-1\right)}^{1/2}}\right).$$
ⓘ Symbols: $\ln z$: principal branch of logarithm function, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: 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).