In physical science and mathematics, the Legendre functions Pλ, Qλ and associated Legendre functions Pμλ, Qμλ, and Legendre functions of the second kind, Qn, 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.
Legendre's differential equation The general Legendre equation reads
( 1 − x 2 ) y ″ − 2 x y ′ + [ λ ( λ + 1 ) − μ 2 1 − x 2 ] y = 0 , {\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 Pn; 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 Pμλ, Qμλ. If μ = 0, the superscript is omitted, and one writes just Pλ, Qλ. However, the solution Qλ when λ is an integer is often discussed separately as Legendre's function of the second kind, and denoted Qn. 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.
Solutions of the differential equation 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, 2 F 1 {\displaystyle _{2}F_{1}} . With Γ {\displaystyle \Gamma } being the gamma function, the first solution is
P λ μ ( z ) = 1 Γ ( 1 − μ ) [ z + 1 z − 1 ] μ / 2 2 F 1 ( − λ , λ + 1 ; 1 − μ ; 1 − z 2 ) , for | 1 − z | < 2 , {\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
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