Notes on Whittaker & Watson, Chapter XV, part 3

Whittaker & Watson, A Course of Modern Analysis. Chapter XV, Legendre Functions.

15.5 Ferrer’s associated Legendre functions $P_n^m(z)$ and $Q_n^m(z)$​​ are defined as $$ \begin{gather*} P_n^m(z) = (1-z^2)^{\frac12m}\frac{d^m}{dz^m}P_n(z)\\ Q_n^m(z) = (1-z^2)^{\frac12m}\frac{d^m}{dz^m}Q_n(z) \end{gather*} $$ for positive integer $m$​, $z\in(-1,1)$​, and $n\in\mathbb{C}$​​​​​, which satisfy the differential equation $$ (1-z^2)\frac{d^2w}{dz^2}-2\frac{dw}{dz}+\left(n(n+1)-\frac{m^2}{1-z^2}\right)w=0 $$ derived from Legendre’s equation.

15.51 The integral properties of the associated Legendre functions (orthogonality).

For positive integers $n,r>m$ $$ \int_{-1}^1P_n^m(z)P_r^m(z)dz=\frac{2\delta_{rn}}{2n+1}\frac{(n+m)!}{(n-m)!}, $$ which is obtained for $n\ne r$ by multiplying the differential equation for $P_n^m(z)$ by $P_r^m(z)$ and the equation for $P_r^m(z)$ by $P_n^m(z)$; and for $n=r$ through the recurrence relation that $\int_1^1\left(P_n^{m+1}(z)\right)^2dz = (n-m)(n+m+1)\int_1^1\left(P_n^m(z)\right)^2dz$.

15.6 Hobson’s definition of the associated Legendre functions.

For complex $z\in\mathbb{C}$​, and positive ineger $m$​, the associated Legendre functions $P_n^m(z)$​ and $Q_n^m(z)$​ can be defined with $\arg z$​, $\arg(z-1)$​, and $\arg(z+1)$​​​ chosen to have their principal values.

For complex $m$, $$ \begin{gather*} P_n^m(z)=\frac1{\Gamma(1-m)}\left(\frac{z+1}{z-1}\right)^{\frac12m}F\left(-n,n+1;1-m;\frac12-\frac12z\right)\\ Q_n^m(z)=\frac{\sin(n+m)\pi}{\sin n\pi}\frac{\Gamma(n+m+1)\Gamma(\frac12)}{2^{n+1}\Gamma(n+\frac32)}\frac{(z^2-1)^{\frac12m}}{z^{n+m+1}}\\ \times F\left(\frac12n+\frac12m+1,\frac12n+\frac12m+\frac12;n+\frac32;z^{-2}\right). \end{gather*} $$ 15.61 Expression of $P_n^m(z)$​ as an integral of Laplace’s type. $$ \begin{align*} P_n^m(z) &=\frac{(n+1)\cdots(n+m)}{2^n\pi i}(z^2-1)^{\frac12m}\int_A^{(1+,z+)}(t^2-1)^n(t-z)^{-n-m-1}dt &\text{cf. Schläfli}\\ & =\frac{(n+1)\cdots(n+m)}{2^n\pi i}(z^2-1)^{\frac12m}\int_\alpha^{2\pi+\alpha}\frac{(z+(z^2-1)^\frac12\cos\phi)^n}{((z^2-1)^\frac12e^{i\phi})^m}d\phi& t=z+(z^2-1)^\frac12e^{i\phi}\\ &=\frac{(n+1)\cdots(n+m)}{2\pi}\int_\alpha^{2\pi+\alpha}{(z+(z^2-1)^\frac12\cos\phi)^n}e^{-im\phi}d\phi \\ &=\frac{(n+1)\cdots(n+m)}{\pi}\int_0^\pi{(z+(z^2-1)^\frac12\cos\phi)^n}\cos m\phi \ d\phi. \end{align*} $$ 15.7 The addition theorem for the Legendre polynomials.

With $x,x', \omega\in\mathbb{C}$, let $z=xx'+(x^2-1)^\frac12(x'^2-1)^\frac12\cos\omega$​​, we have $$ P_n(z)=\frac1{2\pi}\int_{-\pi}^\pi\frac{\left(x+(x^2-1)^\frac12\cos(\omega-\phi)\right)^n}{\left(x'+(x'^2-1)^\frac12\cos\phi\right)^{n+1}}d\phi $$ via the expansion of the generating function $(1-2hz+h^2)^\frac12$​​​. Thus $P_n(z)$​​ is a polynomial of degree $n$​​ in $\cos\omega$​​​​, and can be expressed as a Fourier series $P_n(z)=\frac12A_0+\sum_{m=1}^nA_m\cos m\omega$​​, with the coefficients determined by $$ \begin{align*} A_m &=\frac1\pi\int_{-\pi}^\pi P_n(z)\cos m\omega\ d\omega \\ &=\frac1{2\pi^2}\int_{-\pi}^\pi\int_{-\pi}^\pi \frac{\left(x+(x^2-1)^\frac12\cos\psi\right)^n\cos m(\phi+\psi)}{\left(x'+(x'^2-1)^\frac12\cos\phi\right)^{n+1}} d\psi d\phi & \omega=\phi+\psi \\ &= \frac{n!}{\pi(n+m)!}\int_{-\pi}^\pi \frac{P_n^m(x)\cos m\phi}{\left(x'+(x'^2-1)^\frac12\cos\phi\right)^{n+1}}d\phi & \int_{-\pi}^\pi\cos^n\psi\sin m\psi\ d\psi=0\\ &=(-1)^m\frac{2(n-m)!}{(n+m)!}P_n^m(x)P_n^m(x'). \end{align*} $$ Thus, we have $$ P_n(z)=P_n(x)P_n(x')+2\sum_{m=1}^n(-1)^m\frac{(n-m)!}{(n+m)!}P_n^m(x)P_n^m(x')\cos m\omega. $$ 15.71 The addition theorem for the Legendre functions.

Consider the elliptic coordinates of $x,x'$ with foci at $t=\pm1$​, we have $x=\cosh\xi$ and $x'=\cosh\xi'$ with $\xi=\alpha+i\beta$​ and $\xi'=\alpha'+i\beta'$​​ , where $\alpha,\alpha'$​ are the semi-major axes, and $\beta,\beta'$​ the eccentric angles. In Schläfli’s integral, let $$ t=\frac{e^{i\phi}(e^{-i\omega}\sinh\xi\cosh\frac12\xi'-\cosh\xi\sinh\frac12\xi')+\cosh\xi\cosh\frac12\xi'-e^{i\omega}\sinh\xi\sinh\frac12\xi'}{\cosh\frac12\xi'+e^{i\phi}\sinh\xi'} $$ with $\phi$​ now the integration variable, it can then be found that $$ P_n(z)=\frac1{2\pi}\int_{-\pi}^\pi\frac{\left(x+(x^2-1)^\frac12\cos(\omega-\phi)\right)^n}{\left(x'+(x'^2-1)^\frac12\cos\phi\right)^{n+1}}d\phi $$ and it follows from the same steps in §15.7 that $$ P_n(z)=P_n(x)P_n(x')+2\sum_{m=1}^n(-1)^m\frac{\Gamma(n-m+1)!}{\Gamma(n+m+1)!}P_n^m(x)P_n^m(x')\cos m\omega. $$ 15.8 The function $C_n^\nu(z)$​​​ (Gugenbauer) statisfies the differential equation $$ \frac{d^2y}{dz^2}+\frac{2\nu+1}{z^2-1}\frac{dy}{dz}-\frac{n(n+\nu)}{z^2-1}y=0, $$ with corresponding integral expression $$ (1-z^2)^{\frac12-\nu}\int_C\frac{(1-t^2)^{n+\nu-\frac12}}{(t-z)^{n+1}}dt $$ on a contour around the branch cut connecting $t=1,z$.

We have the following results.

1º When $n\in\mathbb{Z}$ $$ C_n^{\nu}(z)=\frac{(-2)^n\nu(\nu+1)\cdots(\nu+n-1)}{n!(2n+2\nu-1)(2n+2\nu-2)\cdots(n+2\nu)}(1-z^2)^{\frac12-\nu}\frac{d^n}{dz^n}(1-z^2)^{n+\nu-\frac12}, $$ of which Rodrigues' formula is a particular case with $\nu=\frac12$​.

2º When $r=\nu-\frac12\in\mathbb{Z}$​ $$ C_{n-r}^{r+\frac12}(z)=\frac{1}{(2r-1)!!}\frac{d^r}{dz^r}P_n(z)=\frac{(z^2-1)^{-\frac12r}}{(2r-1)!!}P_n^r(z). $$ 3º Recurrance relations $$ \begin{gather*} zC_{n-1}^{\nu+1}(z)-C_{n-2}^{\nu+1}(z)-\frac{n}{2\nu}C_n^\nu(z)=0,\\ C_n^{\nu+1}(z)-zC_{n-1}^{\nu+1}(z)=\frac{n+2\nu}{2\nu}C_n^\nu(z),\\ \frac{d}{dz}C_n^\nu(z)=2\nu C_{n-1}^{\nu+1}(z),\\ nC_n^\nu(z)=(n-1+2\nu)C_{n-1}^\nu(z)-2\nu(1-z^2)C_{n-2}^{\nu-1}(z). \end{gather*} $$