In physical science and mathematics, the Legendre functions P_λ, Q_λ and associated Legendre functions P^μ
_λ, Q^μ
_λ, and Legendre functions of the second kind, Q_n, are all solutions of Legendre's differential equation. The Legendre polynomials and the associated Legendre polynomials are also solutions of the differential equation in special cases, which, by virtue of being polynomials, have a large number of additional properties, mathematical structure, and applications. For these polynomial solutions, see the separate Wikipedia articles.

The general Legendre equation reads $${\displaystyle \left(1-x^{2}\right)y''-2xy'+\left[\lambda (\lambda +1)-{\frac {\mu ^{2}}{1-x^{2}}}\right]y=0,}$$ where the numbers λ and μ may be complex, and are called the degree and order of the relevant function, respectively. The polynomial solutions when λ is an integer (denoted n), and μ = 0 are the Legendre polynomials P_n; and when λ is an integer (denoted n), and μ = m is also an integer with |m| < n are the associated Legendre polynomials. All other cases of λ and μ can be discussed as one, and the solutions are written P^μ
_λ, Q^μ
_λ. If μ = 0, the superscript is omitted, and one writes just P_λ, Q_λ. However, the solution Q_λ when λ is an integer is often discussed separately as Legendre's function of the second kind, and denoted Q_n.

This is a second order linear equation with three regular singular points (at 1, −1, and ∞). Like all such equations, it can be converted into a hypergeometric differential equation by a change of variable, and its solutions can be expressed using hypergeometric functions.

Since the differential equation is linear, homogeneous (the right hand side =zero) and of second order, it has two linearly independent solutions, which can both be expressed in terms of the hypergeometric function, ${\displaystyle _{2}F_{1}}$. With ${\displaystyle \Gamma }$ being the gamma function, the first solution is $${\displaystyle P_{\lambda }^{\mu }(z)={\frac {1}{\Gamma (1-\mu )}}\left[{\frac {z+1}{z-1}}\right]^{\mu /2}\,_{2}F_{1}\left(-\lambda ,\lambda +1;1-\mu ;{\frac {1-z}{2}}\right),\qquad {\text{for }}\ |1-z|<2,}$$ and the second is $${\displaystyle Q_{\lambda }^{\mu }(z)={\frac {{\sqrt {\pi }}\ \Gamma (\lambda +\mu +1)}{2^{\lambda +1}\Gamma (\lambda +3/2)}}{\frac {e^{i\mu \pi }(z^{2}-1)^{\mu /2}}{z^{\lambda +\mu +1}}}\,_{2}F_{1}\left({\frac {\lambda +\mu +1}{2}},{\frac {\lambda +\mu +2}{2}};\lambda +{\frac {3}{2}};{\frac {1}{z^{2}}}\right),\qquad {\text{for}}\ \ |z|>1.}$$

These are generally known as Legendre functions of the first and second kind of noninteger degree, with the additional qualifier 'associated' if μ is non-zero. A useful relation between the P and Q solutions is Whipple's formula.

For positive integer ${\displaystyle \mu =m\in \mathbb {N} ^{+}}$ the evaluation of ${\displaystyle P_{\lambda }^{\mu }}$ above involves cancellation of singular terms. We can find the limit valid for ${\displaystyle m\in \mathbb {N} _{0}}$ as

$${\displaystyle P_{\lambda }^{m}(z)=\lim _{\mu \to m}P_{\lambda }^{\mu }(z)={\frac {(-\lambda )_{m}(\lambda +1)_{m}}{m!}}\left[{\frac {1-z}{1+z}}\right]^{m/2}\,_{2}F_{1}\left(-\lambda ,\lambda +1;1+m;{\frac {1-z}{2}}\right),}$$

with ${\displaystyle (\lambda )_{n}}$ the (rising) Pochhammer symbol.