In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications.

Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, big q-Legendre polynomials, and associated Legendre functions.

In this approach, the polynomials are defined as an orthogonal system with respect to the weight function ${\displaystyle w(x)=1}$ over the interval ${\displaystyle [-1,1]}$. That is, ${\displaystyle P_{n}(x)}$ is a polynomial of degree ${\displaystyle n}$, such that $${\displaystyle \int _{-1}^{1}P_{m}(x)P_{n}(x)\,dx=0\quad {\text{if }}n\neq m.}$$

With the additional standardization condition ${\displaystyle P_{n}(1)=1}$, all the polynomials can be uniquely determined. We then start the construction process: ${\displaystyle P_{0}(x)=1}$ is the only correctly standardized polynomial of degree 0. ${\displaystyle P_{1}(x)}$ must be orthogonal to ${\displaystyle P_{0}}$, leading to ${\displaystyle P_{1}(x)=x}$, and ${\displaystyle P_{2}(x)}$ is determined by demanding orthogonality to ${\displaystyle P_{0}}$ and ${\displaystyle P_{1}}$, and so on. ${\displaystyle P_{n}}$ is fixed by demanding orthogonality to all ${\displaystyle P_{m}}$ with ${\displaystyle m<n}$. This gives ${\displaystyle n}$ conditions, which, along with the standardization ${\displaystyle P_{n}(1)=1}$ fixes all ${\displaystyle n+1}$ coefficients in ${\displaystyle P_{n}(x)}$. With work, all the coefficients of every polynomial can be systematically determined, leading to the explicit representation in powers of ${\displaystyle x}$ given below.

This definition of the ${\displaystyle P_{n}}$'s is the simplest one. It does not appeal to the theory of differential equations. Second, the completeness of the polynomials follows immediately from the completeness of the powers 1, ${\displaystyle x,x^{2},x^{3},\ldots }$. Finally, by defining them via orthogonality with respect to the Lebesgue measure on ${\displaystyle [-1,1]}$, it sets up the Legendre polynomials as one of the three classical orthogonal polynomial systems. The other two are the Laguerre polynomials, which are orthogonal over the half line ${\displaystyle [0,\infty )}$ with the weight ${\displaystyle e^{-x}}$, and the Hermite polynomials, orthogonal over the full line ${\displaystyle (-\infty ,\infty )}$ with weight ${\displaystyle e^{-x^{2}}}$.

The Legendre polynomials can also be defined as the coefficients in a formal expansion in powers of ${\displaystyle t}$ of the generating function

The coefficient of ${\displaystyle t^{n}}$ is a polynomial in ${\displaystyle x}$ of degree ${\displaystyle n}$ with ${\displaystyle |x|\leq 1}$. Expanding up to ${\displaystyle t^{1}}$ gives $${\displaystyle P_{0}(x)=1\,,\quad P_{1}(x)=x.}$$ Expansion to higher orders gets increasingly cumbersome, but is possible to do systematically, and again leads to one of the explicit forms given below.

It is possible to obtain the higher ${\displaystyle P_{n}}$'s without resorting to direct expansion of the Taylor series, however. Equation 2 is differentiated with respect to t on both sides and rearranged to obtain $${\displaystyle {\frac {x-t}{\sqrt {1-2xt+t^{2}}}}=\left(1-2xt+t^{2}\right)\sum _{n=1}^{\infty }nP_{n}(x)t^{n-1}\,.}$$ Replacing the quotient of the square root with its definition in Eq. 2, and equating the coefficients of powers of t in the resulting expansion gives Bonnet's recursion formula $${\displaystyle (n+1)P_{n+1}(x)=(2n+1)xP_{n}(x)-nP_{n-1}(x)\,.}$$ This relation, along with the first two polynomials P₀ and P₁, allows all the rest to be generated recursively.