Legendre functions

Remarks

For $\nu = n$ a nonnegative integer, the factor $(-n)_k$ vanishes for $k > n$, the series stops, and $P_n$ is the Legendre polynomial of degree $n$. Check for $n = 1$: $P_1(z) = 1 + \frac{(-1)(2)}{1} \cdot \frac{1 - z}{2} = z$. For $\mu = 0$ the equation is Legendre's equation with $\lambda = \nu(\nu + 1)$. A second, independent solution is the Legendre function of the second kind $Q_\nu$; for example $Q_0(x) = \frac12 \ln \frac{1 + x}{1 - x}$ for $-1 < x < 1$. Non-example: $w(z) = z^2$ is not a Legendre function of order $0$ for any $\nu$, since the equation would require $2 - 6z^2 + \nu(\nu + 1) z^2 = 0$ for all $z$, which fails at $z = 0$. For general order $\mu$ the functions $P_\nu^\mu$ and $Q_\nu^\mu$ are also given by hypergeometric functions, with a factor $\left( \frac{z + 1}{z - 1} \right)^{\mu/2}$ in front.