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

§14.7 Integer Degree and Order

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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) $\mu=0$

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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 $\mathsf{P}^{0}_{n}\left(x\right)=\mathsf{P}_{n}\left(x\right)=P^{0}_{n}\left(x% \right)=P_{n}\left(x\right),$ $x\in\mathbb{R}$ ,

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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, : real variable and : 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 . For additional properties of $P_{n}\left(x\right)$ see Chapter 18.

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

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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, : real variable, : 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 $W_{n-1}(x)=\sum_{s=0}^{n-1}\frac{(n+s)!(\psi\left(n+1\right)-\psi\left(s+1% \right))}{2^{s}(n-s)!(s!)^{2}}{(x-1)^{s}};$

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Symbols:: $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, : factorial (as in ), : real variable, : 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 $W_{n-1}(x)=\sum_{k=1}^{n}\frac{1}{k}P_{k-1}\left(x\right)P_{n-k}\left(x\right).$

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Symbols:: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, : real variable, : 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

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

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

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

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Symbols:: : 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 $Q^{0}_{n}\left(x\right)=Q_{n}\left(x\right)=n!\boldsymbol{Q}^{0}_{n}\left(x% \right)=n!\boldsymbol{Q}_{n}\left(x\right),$

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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 ), $Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}\left(z\right)$ : 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, : real variable and : 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 $Q_{n}\left(x\right)=\frac{1}{2}P_{n}\left(x\right)\ln\left(\frac{x+1}{x-1}% \right)-W_{n-1}(x),$ $n=0,1,2,\dots$ .

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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, : real variable, : 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 $\mathsf{P}^{m}_{n}\left(x\right)=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{% \mathrm{d}}^{m}}{{\mathrm{d}x}^{m}}\mathsf{P}_{n}\left(x\right),$

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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 with respect to , $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, : real variable, : nonnegative integer and : 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 $\mathsf{Q}^{m}_{n}\left(x\right)=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{% \mathrm{d}}^{m}}{{\mathrm{d}x}^{m}}\mathsf{Q}_{n}\left(x\right),$

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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 with respect to , $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : Ferrers function of the second kind, : real variable, : nonnegative integer and : 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 $\mathsf{P}^{m}_{n}\left(x\right)=(-1)^{m+n}\frac{\left(1-x^{2}\right)^{m/2}}{2% ^{n}n!}\frac{{\mathrm{d}}^{m+n}}{{\mathrm{d}x}^{m+n}}\left(1-x^{2}\right)^{n}.$

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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 with respect to , : factorial (as in ), : real variable, : nonnegative integer and : 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 $P^{m}_{n}\left(x\right)=\left(x^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}}{{% \mathrm{d}x}^{m}}P_{n}\left(x\right),$

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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 with respect to , : real variable, : nonnegative integer and : 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 $Q^{m}_{n}\left(x\right)=\left(x^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}}{{% \mathrm{d}x}^{m}}Q_{n}\left(x\right),$

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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 with respect to , $Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}\left(z\right)$ : Legendre function of the second kind, : real variable, : nonnegative integer and : 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 $P_{n}\left(x\right)=\frac{1}{2^{n}n!}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}% }\left(x^{2}-1\right)^{n},$

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Symbols:: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , : factorial (as in ), : real variable and : 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 $P^{m}_{n}\left(x\right)=\frac{\left(x^{2}-1\right)^{m/2}}{2^{n}n!}\frac{{% \mathrm{d}}^{m+n}}{{\mathrm{d}x}^{m+n}}\left(x^{2}-1\right)^{n},$

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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 with respect to , : factorial (as in ), : real variable, : nonnegative integer and : 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 $P^{m}_{m}\left(x\right)=\frac{(2m)!}{2^{m}m!}\left(x^{2}-1\right)^{m/2}.$

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Symbols:: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, : factorial (as in ), : real variable and : 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 is even and $m\leq n$ , $\mathsf{P}^{m}_{n}\left(x\right)$ and $P^{m}_{n}\left(x\right)$ are polynomials of degree . Also,

14.7.16 $\mathsf{P}^{m}_{n}\left(x\right)=P^{m}_{n}\left(x\right)=0,$

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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, : real variable, : nonnegative integer and : 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 $\mathsf{P}^{m}_{n}\left(-x\right)=(-1)^{n-m}\mathsf{P}^{m}_{n}\left(x\right),$

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, : real variable, : nonnegative integer and : 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 $\mathsf{Q}^{\pm m}_{n}\left(-x\right)=(-1)^{n-m-1}\mathsf{Q}^{\pm m}_{n}\left(% x\right).$

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Symbols:: $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, : real variable, : nonnegative integer and : 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 and ,

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

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Symbols:: $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, : real variable, : nonnegative integer and : 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 $\sum_{n=0}^{\infty}\mathsf{Q}_{n}\left(x\right)h^{n}=\frac{1}{\left(1-2xh+h^{2% }\right)^{1/2}}\*\ln\left(\frac{x-h+\left(1-2xh+h^{2}\right)^{1/2}}{\left(1-x^% {2}\right)^{1/2}}\right).$

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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, : real variable, : nonnegative integer and : 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 ,

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

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Symbols:: $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, : real variable, : nonnegative integer and : 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 , (14.7.19) applies with $|h|<x-\left(x^{2}-1\right)^{1/2}$ . Also, with the same conditions

14.7.22 $\sum_{n=0}^{\infty}Q_{n}\left(x\right)h^{n}=\frac{1}{\left(1-2xh+h^{2}\right)^% {1/2}}\*\ln\left(\frac{x-h+\left(1-2xh+h^{2}\right)^{1/2}}{\left(x^{2}-1\right% )^{1/2}}\right).$

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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, : real variable, : nonnegative integer and : 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 , (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

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§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) $\mu=0$