Associated Legendre Polynomial

$P_l^m(x)\equiv (-1{)}^m(1-x^2{)}^{\frac{m}{2}}{\left(\frac{\mathrm{d}}{\mathrm{d}x}\right)}^m{P}_l(x)$

• $P_l(x)$ is the $l$th Legendre polynomial

$P_l^m(x)=\frac{(-1{)}^m}{2^{l}l!}(1-x^2{)}^{\frac{m}{2}}{\left(\frac{\mathrm{d}}{\mathrm{d}x}\right)}^{l+m}(x^2-1{)}^l$

• Full form

$P_l^{-m}(x)=(-1{)}^m\frac{(l-m)!}{(l+m)!}{P}_l^m(x)$

• For $m\le 0$, use this rule

Example Values

(* Set the maximum values for l and m *)
lMax = 5; (* For example, up to l = 5 *)
mMax = 5; (* For example, up to m = 5 *)
 
(* Create a list of headers for l and m *)
lHeaders = Prepend[Table[l, {l, 0, lMax}], "l\m"];
mHeaders = Table[m, {m, 0, mMax}];
 
(* Create the 2D table with associated Legendre polynomials, including the m header *)
assocLegendreTable = Simplify[Table[LegendreP[l, m, x], {l, 0, lMax}, {m, 0, Min[l, mMax]}]];
assocLegendreTableWithHeaders = Prepend[assocLegendreTable, mHeaders];
 
(* Insert the l headers *)
assocLegendreTableWithHeaders = MapThread[Prepend, {assocLegendreTableWithHeaders, lHeaders}];
 
(* Use Grid to display the table with headers and grid lines *)
Grid[assocLegendreTableWithHeaders, Frame -> All]