r^m cos(1/2 π (n - m)) P_((n - m)/2)^(m, 0)(1 - 2 r^2)

d/dr(R_n^m(r)) = (r (m + n + 2) R_(n + 1)^(m + 1)(r) - (m + (n + 2) r^2) R_n^m(r))/(r (r^2 - 1))

R_n^m(r) = piecewise | i^(-m + n) r^m P_(1/2 (-m + n))^(m, 0)(1 - 2 r^2) | 1/2 (-m + n) element Z 0 | otherwise for (n element Z and n>=0 and m element Z and m>=0 and n>=m)

R_n^m(r) = r^m cos(1/2 (-m + n) π) sum_(j=0)^(1/2 (-m + n)) ((-1)^j r^(2 j) (j + (m + n)/2)!)/(j! (j + m)! (1/2 (-2 j - m + n))!) for (n element Z and n>=0 and m element Z and m>=0 and n>=m)

R_n^m(r) = piecewise | (r^n n! 2F1(-m/2 - n/2, m/2 - n/2, -n, 1/r^2))/((1/2 (-m + n))! (m + n)/2 !) | 1/2 (-m + n) element Z 0 | otherwise for (n element Z and n>=0 and m element Z and m>=0 and n>=m)

R_n^m(r) = i^(-m + n) cos(1/2 (-m + n) π) sum_(j=0)^(1/2 (-m + n)) ((-1)^j r^(-2 j + n) (-j + n)!)/(j! (1/2 (-2 j - m + n))! (1/2 (-2 j + m + n))!) for (n element Z and n>=0 and m element Z and m>=0 and n>=m)