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Thanks, Eric_G. I am interested to help out. Is there a page with Sage volunteering process? I will sign up. I have computer science degrees but new to math (but seriously interested to learn).

Correct definition:

def legendre_function(l, m, f):
    return ((-1)^m * (1 - x^2)^(m/2) * diff(legendre_P(l, x), x, m)).subs(x = f)

sage: legendre_function(2, 2, x)
-3*x^2 + 3

sage: legendre_function(2, 2, cos(x))
-3*cos(x)^2 + 3