# §14.7 Integer Degree and Order

## §14.7(i) $\mu=0$

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

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

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

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

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

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: $\mathop{P_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: 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)
 14.7.5 $\displaystyle W_{0}(x)$ $\displaystyle=1,$ $\displaystyle W_{1}(x)$ $\displaystyle=\tfrac{3}{2}x,$ $\displaystyle W_{2}(x)$ $\displaystyle=\tfrac{5}{2}x^{2}-\tfrac{2}{3}.$ 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)

Next,

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

where

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

## §14.7(ii) Rodrigues-Type Formulas

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

 14.7.8 $\displaystyle\mathop{\mathsf{P}^{m}_{n}\/}\nolimits\!\left(x\right)$ $\displaystyle=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{\mathrm{d}}^{m}}{{% \mathrm{d}x}^{m}}\mathop{\mathsf{P}_{n}\/}\nolimits\!\left(x\right),$ 14.7.9 $\displaystyle\mathop{\mathsf{Q}^{m}_{n}\/}\nolimits\!\left(x\right)$ $\displaystyle=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{\mathrm{d}}^{m}}{{% \mathrm{d}x}^{m}}\mathop{\mathsf{Q}_{n}\/}\nolimits\!\left(x\right),$
 14.7.10 $\mathop{\mathsf{P}^{m}_{n}\/}\nolimits\!\left(x\right)=(-1)^{m+n}\frac{\left(1% -x^{2}\right)^{m/2}}{2^{n}n!}\frac{{\mathrm{d}}^{m+n}}{{\mathrm{d}x}^{m+n}}% \left(1-x^{2}\right)^{n}.$
 14.7.11 $\displaystyle\mathop{P^{m}_{n}\/}\nolimits\!\left(x\right)$ $\displaystyle=\left(x^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}}{{\mathrm{d}x}^% {m}}\mathop{P_{n}\/}\nolimits\!\left(x\right),$ 14.7.12 $\displaystyle\mathop{Q^{m}_{n}\/}\nolimits\!\left(x\right)$ $\displaystyle=\left(x^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}}{{\mathrm{d}x}^% {m}}\mathop{Q_{n}\/}\nolimits\!\left(x\right),$ 14.7.13 $\displaystyle\mathop{P_{n}\/}\nolimits\!\left(x\right)$ $\displaystyle=\frac{1}{2^{n}n!}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}}\left% (x^{2}-1\right)^{n},$ 14.7.14 $\displaystyle\mathop{P^{m}_{n}\/}\nolimits\!\left(x\right)$ $\displaystyle=\frac{\left(x^{2}-1\right)^{m/2}}{2^{n}n!}\frac{{\mathrm{d}}^{m+% n}}{{\mathrm{d}x}^{m+n}}\left(x^{2}-1\right)^{n},$ 14.7.15 $\displaystyle\mathop{P^{m}_{m}\/}\nolimits\!\left(x\right)$ $\displaystyle=\frac{(2m)!}{2^{m}m!}\left(x^{2}-1\right)^{m/2}.$

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

## §14.7(iii) Reflection Formulas

 14.7.17 $\displaystyle\mathop{\mathsf{P}^{m}_{n}\/}\nolimits\!\left(-x\right)$ $\displaystyle=(-1)^{n-m}\mathop{\mathsf{P}^{m}_{n}\/}\nolimits\!\left(x\right),$ 14.7.18 $\displaystyle\mathop{\mathsf{Q}^{\pm m}_{n}\/}\nolimits\!\left(-x\right)$ $\displaystyle=(-1)^{n-m-1}\mathop{\mathsf{Q}^{\pm m}_{n}\/}\nolimits\!\left(x% \right).$

## §14.7(iv) Generating Functions

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

 14.7.19 $\sum_{n=0}^{\infty}\mathop{\mathsf{P}_{n}\/}\nolimits\!\left(x\right)h^{n}=% \left(1-2xh+h^{2}\right)^{-1/2},$
 14.7.20 $\sum_{n=0}^{\infty}\mathop{\mathsf{Q}_{n}\/}\nolimits\!\left(x\right)h^{n}=% \frac{1}{\left(1-2xh+h^{2}\right)^{1/2}}\*\mathop{\ln\/}\nolimits\!\left(\frac% {x-h+\left(1-2xh+h^{2}\right)^{1/2}}{\left(1-x^{2}\right)^{1/2}}\right).$

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

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

When $x>1$, (14.7.19) applies with $|h|. Also, with the same conditions

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

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).