§14.7 Integer Degree and Order

Keywords:: Ferrers functions, associated Legendre functions
Permalink:: http://dlmf.nist.gov/14.7

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

§14.7(i) $\mu=0$

Notes:: See Olver (1997b, pp. 174, 181, and 188).
Keywords:: Ferrers functions, Legendre polynomials, associated Legendre functions
Referenced by:: ¶ ‣ §1.17(iii), §14.5(ii), ¶ ‣ §18.3
Permalink:: http://dlmf.nist.gov/14.7.i

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

14.7.1 $\mathop{\mathsf{P}^{{0}}_{{n}}\/}\nolimits\!\left(x\right)=\mathop{\mathsf{P}_{{n}}\/}\nolimits\!\left(x\right)=\mathop{P^{{0}}_{{n}}\/}\nolimits\!\left(x\right)=\mathop{P_{{n}}\/}\nolimits\!\left(x\right),$ $x\in\Real$ ,

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\in$ : element of, $\Real$ : real line (excluding infinity), $x$ : real variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/14.7.E1
Encodings:: TeX, pMML, png

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

14.7.2 $\mathop{\mathsf{Q}^{{0}}_{{n}}\/}\nolimits\!\left(x\right)=\mathop{\mathsf{Q}_{{n}}\/}\nolimits\!\left(x\right)=\frac{1}{2}\mathop{P_{{n}}\/}\nolimits\!\left(x\right)\mathop{\ln\/}\nolimits\!\left(\frac{1+x}{1-x}\right)-W_{{n-1}}(x),$

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $x$ : 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

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

14.7.3 $W_{{n-1}}(x)=\sum _{{s=0}}^{{n-1}}\frac{(n+s)!(\mathop{\psi\/}\nolimits\!\left(n+1\right)-\mathop{\psi\/}\nolimits\!\left(s+1\right))}{2^{s}(n-s)!(s!)^{2}}{(x-1)^{s}};$

Symbols:: $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function, $!$ : $n!$ : factorial, $x$ : real variable, $n$ : nonnegative integer and $W_{{n-1}}(x)$ : quantity
Permalink:: http://dlmf.nist.gov/14.7.E3
Encodings:: TeX, pMML, png

equivalently,

14.7.4 $W_{{n-1}}(x)=\sum _{{k=1}}^{n}\frac{1}{k}\mathop{P_{{k-1}}\/}\nolimits\!\left(x\right)\mathop{P_{{n-k}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first 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.E4
Encodings:: TeX, pMML, png

14.7.5

$W_{0}(x)=1,$

$W_{1}(x)=\tfrac{3}{2}x,$

$W_{2}(x)=\tfrac{5}{2}x^{2}-\tfrac{2}{3}.$

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

Next,

14.7.6 $\mathop{Q^{{0}}_{{n}}\/}\nolimits\!\left(x\right)=\mathop{Q_{{n}}\/}\nolimits\!\left(x\right)=n!\mathop{\boldsymbol{Q}^{{0}}_{{n}}\/}\nolimits\!\left(x\right)=n!\mathop{\boldsymbol{Q}_{{n}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $!$ : $n!$ : factorial, $x$ : real variable and $n$ : nonnegative integer
Referenced by:: §14.21(iii)
Permalink:: http://dlmf.nist.gov/14.7.E6
Encodings:: TeX, pMML, png

where

14.7.7 $\mathop{Q_{{n}}\/}\nolimits\!\left(x\right)=\frac{1}{2}\mathop{P_{{n}}\/}\nolimits\!\left(x\right)\mathop{\ln\/}\nolimits\!\left(\frac{x+1}{x-1}\right)-W_{{n-1}}(x),$ $n=0,1,2,\dots$ .

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $x$ : 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

§14.7(ii) Rodrigues-Type Formulas

Notes:: See Olver (1997b, pp. 174 and 181–182).
Keywords:: Ferrers functions, associated Legendre functions
Permalink:: http://dlmf.nist.gov/14.7.ii

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

14.7.8 $\mathop{\mathsf{P}^{{m}}_{{n}}\/}\nolimits\!\left(x\right)=(-1)^{m}\left(1-x^{2}\right)^{{m/2}}\frac{{d}^{m}}{{dx}^{m}}\mathop{\mathsf{P}_{{n}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $x$ : real variable, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/14.7.E8
Encodings:: TeX, pMML, png

14.7.9 $\mathop{\mathsf{Q}^{{m}}_{{n}}\/}\nolimits\!\left(x\right)=(-1)^{m}\left(1-x^{2}\right)^{{m/2}}\frac{{d}^{m}}{{dx}^{m}}\mathop{\mathsf{Q}_{{n}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $x$ : real variable, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/14.7.E9
Encodings:: TeX, pMML, png

14.7.10 $\mathop{\mathsf{P}^{{m}}_{{n}}\/}\nolimits\!\left(x\right)=(-1)^{{m+n}}\frac{\left(1-x^{2}\right)^{{m/2}}}{2^{n}n!}\frac{{d}^{m+n}}{{dx}^{m+n}}\left(1-x^{2}\right)^{n}.$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $!$ : $n!$ : factorial, $x$ : 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

14.7.11 $\mathop{P^{{m}}_{{n}}\/}\nolimits\!\left(x\right)=\left(x^{2}-1\right)^{{m/2}}\frac{{d}^{m}}{{dx}^{m}}\mathop{P_{{n}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\frac{df}{dx}$ : 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

14.7.12 $\mathop{Q^{{m}}_{{n}}\/}\nolimits\!\left(x\right)=\left(x^{2}-1\right)^{{m/2}}\frac{{d}^{m}}{{dx}^{m}}\mathop{Q_{{n}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $x$ : real variable, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/14.7.E12
Encodings:: TeX, pMML, png

14.7.13 $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)=\frac{1}{2^{n}n!}\frac{{d}^{n}}{{dx}^{n}}\left(x^{2}-1\right)^{n},$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $!$ : $n!$ : factorial, $x$ : real variable and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/14.7.E13
Encodings:: TeX, pMML, png

14.7.14 $\mathop{P^{{m}}_{{n}}\/}\nolimits\!\left(x\right)=\frac{\left(x^{2}-1\right)^{{m/2}}}{2^{n}n!}\frac{{d}^{m+n}}{{dx}^{m+n}}\left(x^{2}-1\right)^{n},$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $!$ : $n!$ : factorial, $x$ : real variable, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/14.7.E14
Encodings:: TeX, pMML, png

14.7.15 $\mathop{P^{{m}}_{{m}}\/}\nolimits\!\left(x\right)=\frac{(2m)!}{2^{m}m!}\left(x^{2}-1\right)^{{m/2}}.$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $!$ : $n!$ : factorial, $x$ : real variable and $m$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/14.7.E15
Encodings:: TeX, pMML, png

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

14.7.16 $\mathop{\mathsf{P}^{{m}}_{{n}}\/}\nolimits\!\left(x\right)=\mathop{P^{{m}}_{{n}}\/}\nolimits\!\left(x\right)=0,$ $m>n$ .

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : 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

§14.7(iii) Reflection Formulas

Notes:: These results may be derived from (14.9.8) and (14.9.10).
Keywords:: Ferrers functions
Permalink:: http://dlmf.nist.gov/14.7.iii

14.7.17 $\mathop{\mathsf{P}^{{m}}_{{n}}\/}\nolimits\!\left(-x\right)=(-1)^{{n-m}}\mathop{\mathsf{P}^{{m}}_{{n}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : 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

14.7.18 $\mathop{\mathsf{Q}^{{\pm m}}_{{n}}\/}\nolimits\!\left(-x\right)=(-1)^{{n-m-1}}\mathop{\mathsf{Q}^{{\pm m}}_{{n}}\/}\nolimits\!\left(x\right).$

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $x$ : real variable, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/14.7.E18
Encodings:: TeX, pMML, png

§14.7(iv) Generating Functions

Notes:: See Erdélyi et al. (1953a, p. 154) and Olver (1997b, pp. 51 and 185).
Keywords:: Ferrers functions, associated Legendre functions
Referenced by:: ¶ ‣ §2.10(iv)
Permalink:: http://dlmf.nist.gov/14.7.iv

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

14.7.19 $\sum _{{n=0}}^{{\infty}}\mathop{\mathsf{P}_{{n}}\/}\nolimits\!\left(x\right)h^{n}=\left(1-2xh+h^{2}\right)^{{-1/2}},$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : 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

14.7.20 $\sum _{{n=0}}^{{\infty}}\mathop{\mathsf{Q}_{{n}}\/}\nolimits\!\left(x\right)h^{n}=\frac{1}{\left(1-2xh+h^{2}\right)^{{1/2}}}\*\mathop{\ln\/}\nolimits\!\left(\frac{x-h+\left(1-2xh+h^{2}\right)^{{1/2}}}{\left(1-x^{2}\right)^{{1/2}}}\right).$

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $x$ : real variable, $n$ : nonnegative integer and $h$ : variable
Permalink:: http://dlmf.nist.gov/14.7.E20
Encodings:: TeX, pMML, png

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

14.7.21 $\sum _{{n=0}}^{{\infty}}\mathop{\mathsf{P}_{{n}}\/}\nolimits\!\left(x\right)h^{{-n-1}}=\left(1-2xh+h^{2}\right)^{{-1/2}}.$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : 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

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 $\sum _{{n=0}}^{{\infty}}\mathop{Q_{{n}}\/}\nolimits\!\left(x\right)h^{n}=\frac{1}{\left(1-2xh+h^{2}\right)^{{1/2}}}\*\mathop{\ln\/}\nolimits\!\left(\frac{x-h+\left(1-2xh+h^{2}\right)^{{1/2}}}{\left(x^{2}-1\right)^{{1/2}}}\right).$

Symbols:: $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $x$ : 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

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).

§14.7 Integer Degree and Order

Contents

§14.7(i)

§14.7(ii) Rodrigues-Type Formulas

§14.7(iii) Reflection Formulas

§14.7(iv) Generating Functions

§14.7(i) $\mu=0$