Symbolic Associated Legendre Polynomials

Hello!

I am trying to create the symbolic associated legendre polynomials of degree n and order m.

I found this forum entry: https://www.mathworks.com/matlabcentral/answers/9485-how-do-i-specify-m-for-orthpoly-legendre but unfortunately MuPad has been removed and I have no idea how I can implement that in the same way.

Is there any possibility? The function legendreP gives me only the unassociated legendre polynomial.

I am currently using Matlab R2018a with the Symbolic Toolbox.

Thank you in advance!

Answer 1

[Masa]

associated Legendre functions could be found from Legendre functions according to the following formula

so in MATLAB you can define a function like this

function plm = assocLegendre(l,m)
    % get symbolic associated legendre function P_lm(x) based on
    % legendre function P_l(x)
    
    syms x;
    % get symbolic form of Legendre function P_l(x)
    leg = legendreP(l,x);
    % differentiate it m times
    legDiff = diff(leg,x,m);
    % calculate associated legendre function P_lm(x)
    plm = ((-1)^m)*((1 - x^2)^(m/2))*legDiff;
end

to get associated Legendre functions in symbolic form.