Generating function for the associated Legendre Polynomials

The generating function $G(x,t)$ for the associated Legendre polynomials is a mathematical tool that, when expanded as a power series, produces associated Legendre polynomials as its coefficients in terms of a variable.

$G(x,t)$ can be derived from the generating function for the Legendre polynomials. It begins with differentiating eq345 $\vert m_l\vert$ times using Leibniz’s theorem to give

$G(x,t)=\sum_{l=0}^{\infty}t^l\frac{d^{\vert m_l\vert}}{dx^{\vert m_l\vert}}P_l(x)=\frac{d^{\vert m_l\vert}}{dx^{\vert m_l\vert}}\frac{1}{\sqrt{1-2tx+t^2}}\;\;\;\;\;\;\;\;365$

Substituting eq363 in eq365 yields

$G(x,t)=\sum_{l=0}^{\infty}P_l^{\vert m_l\vert}(x)t^l=(1-x^2)^{\frac{\vert m_l\vert}{2}}\frac{d^{\vert m_l\vert}}{dx^{\vert m_l\vert}}\frac{1}{\sqrt{1-2tx+t^2}}$

The derivatives on the RHS have the following pattern:

$\boldsymbol{\vert m_l\vert}$	Derivatives
$1$	$t\frac{d^{\vert m_l\vert-1}}{dx^{\vert m_l\vert-1}}(1-2tx+t^2)^{-\frac{3}{2}}$
$2$	$(t)(3t)\frac{d^{\vert m_l\vert-2}}{dx^{\vert m_l\vert-2}}(1-2tx+t^2)^{-\frac{5}{2}}$
$3$	$(t)(3t)(5t)\frac{d^{\vert m_l\vert-3}}{dx^{\vert m_l\vert-3}}(1-2tx+t^2)^{-\frac{7}{2}}$
$\vdots$	$\vdots$
$\vert m_l\vert$	$(1)(3)(5)\cdots(2\vert m_l\vert-1)t^{\vert m_l\vert}(1-2tx+t^2)^{-(\vert m_l\vert+\frac{1}{2})}$

Since $(1)(3)(5)\cdots(2\vert m_l\vert-1)=\frac{(1)(2)(3)\cdots 2\vert m_l\vert}{(2)(4)(6)\cdots 2\vert m_l\vert}=\frac{(2\vert m_l\vert)!}{2^{\vert m_l\vert}(\vert m_l\vert)!}$

$G(x,t)=\sum_{l=0}^{\infty}P_l^{\vert m_l\vert}(x)t^l=(1-x^2)^{\frac{\vert m_l\vert}{2}}\frac{(2\vert m_l\vert)!}{2^{\vert m_l\vert}(\vert m_l\vert)!}\frac{t^{\vert m_l\vert}}{(1-2tx+t^2)^{\vert m_l\vert +\frac{1}{2}}}\;\;\;\;\;\;\;\;366$

Eq366 is the generating function for the associated Legendre polynomials. It can also be expressed as:

$G(x,t)=\sum_{l=0}^{\infty}P_{l+\vert m_l\vert}^{\vert m_l\vert}(x)t^l=\frac{(2\vert m_l\vert)!}{2^{\vert m_l\vert}(\vert m_l\vert)!}\frac{(1-x^2)^{\frac{\vert m_l\vert}{2}}}{(1-2tx+t^2)^{\vert m_l\vert +\frac{1}{2}}}\;\;\;\;\;\;\;\;367$

which is obtained by letting $l=l+\vert m_l\vert$ .

Question

How do we use eq366 to generate $P_0^0(x)$ and $P_1^0(x)$ ?

Answer

For $\vert m_l\vert=0$ ,

$\sum_{l=0}^{\infty}P_{l}^{0}(x)t^l=\frac{1}{(1-2tx+t^2)^{\frac{1}{2}}$

Expanding $(1-2tx+t^2)^{-\frac{1}{2}}$ using the binomial series gives

$\begin{align}\sum_{l=0}^{\infty}P_{l}^{0}(x)t^l&=(1-2tx+t^2)^{\frac{1}{2}}=\sum_{k=0}^{\infty}\begin{pmatrix}-\frac{1}{2}\\k\end{pmatrix}(-2tx+t^2)^k\\&=\sum_{k=0}^{\infty}\frac{(-\frac{1}{2})!}{k!(-\frac{1}{2}-k)!}(-2tx+t^2)^k=1+\frac{1}{2}(2tx-t^2)+\cdots\;\;\;\;\;\;\;\;368\end{align}$

The polynomial $P_0^0(x)$ corresponds to the coefficient of $t^{l=0}$ on the LHS of eq368. Comparing the coefficients of $t^{l=0}$ on both sides of eq368, $P_0^0(x)=1$ . Using the same logic, $P_1^0(x)=x$ .

Generating function for the associated Legendre Polynomials

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