Recurrence relations of the associated Legendre Polynomials

The recurrence relations of the associated Legendre polynomials describe how each polynomial in the sequence can be obtained from its predecessors.

Some useful recurrence relations of the associated Legendre polynomials in quantum chemistry are:

$(2l+1)xP_l^{\vert m_l\vert}(x)=(l+\vert m_l\vert)P_{l-1}^{\vert m_l\vert}(x)+(l-\vert m_l\vert+1)P_{l+1}^{\vert m_l\vert}(x)\;\;\;\;\;\;\;\;370$

$(2l+1)(1-x^2)^{\frac{1}{2}}P_l^{\vert m_l\vert}(x)=P_{l+1}^{\vert m_l\vert+1}(x)-P_{l-1}^{\vert m_l\vert+1}(x)\;\;\;\;\;\;\;\;371$

To derive eq370, we differentiate eq366 with respect to $t$ and rearrange the result to give

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

Substituting eq366 in the RHS of the above equation yields

$\sum_{l=0}^{\infty}(1-2tx+t^2)P_l^{\vert m_l\vert}(x)(l-\vert m_l\vert)t^{l-1}=(2\vert m_l\vert+1)(x-t)\sum_{l=0}^{\infty}P_l^{\vert m_l\vert}(x)t^l$

which can be rearranged to

$\sum_{l=0}^{\infty}P_l^{\vert m_l\vert}(x)(l-\vert m_l\vert)t^{l-1}-\sum_{l=0}^{\infty}(2l+1)P_l^{\vert m_l\vert}(x)t^{l}+\sum_{l=0}^{\infty}(\vert m_l\vert+1+l)P_l^{\vert m_l\vert}(x)t^{l+1}=0$

Collecting coefficients with the same powers of $t$ ,

$t^{-2}$	$P_0^{\vert m_l\vert}(x)(-\vert m_l\vert)=0$
$t^{-1}$	$P_0^{\vert m_l\vert}(x)(-\vert m_l\vert)=0$
$t^0$	$P_1^{\vert m_l\vert}(x)(1-\vert m_l\vert)=xP_0^{\vert m_l\vert}(x)$
$t^1$	$P_2^{\vert m_l\vert}(x)(2-\vert m_l\vert)=3xP_1^{\vert m_l\vert}(x)-(\vert m_l\vert+1)P_0^{\vert m_l\vert}(x)$
$t^2$	$P_3^{\vert m_l\vert}(x)(3-\vert m_l\vert)=5xP_2^{\vert m_l\vert}(x)-(\vert m_l\vert+2)P_1^{\vert m_l\vert}(x)$

Comparing the equations, we have

$P_{l+1}^{\vert m_l\vert}(x)(l+1-\vert m_l\vert)=(2l+1)xP_l^{\vert m_l\vert}(x)-(\vert m_l\vert+l)P_{l-1}^{\vert m_l\vert}(x)\;\;\;\;\;l=0,1,2,\cdots$

which is eq370, where $P_{-1}^{\vert m_l\vert}(x)=P_{-2}^{\vert m_l\vert}(x)=0$ .

To derive eq371, we differentiate eq346c $\vert m_l\vert$ times with respect to $x$ to give $(2l+1)\frac{d^{\vert m_l\vert}}{dx^{\vert m_l\vert}}P_{l}=\frac{d^{\vert m_l\vert+1}}{dx^{\vert m_l\vert+1}}P_{l+1}-\frac{d^{\vert m_l\vert+1}}{dx^{\vert m_l\vert+1}}P_{l-1}$ . Multiplying through by $(1-x^2)^{\frac{\vert m_l\vert+1}{2}}$ and substituting eq363a in the resultant equation gives eq371.