In mathematics, the Legendre rational functions are a sequence of orthogonal functions on [0, ∞). They are obtained by composing the Cayley transform with Legendre polynomials. A rational Legendre function of degree n is defined as:
R n ( x ) = 2 x + 1 P n ( x − 1 x + 1 ) {\displaystyle R_{n}(x)={\frac {\sqrt {2}}{x+1}}\,P_{n}\left({\frac {x-1}{x+1}}\right)}
where P n ( x ) {\displaystyle P_{n}(x)} is a Legendre polynomial. These functions are eigenfunctions of the singular Sturm–Liouville problem:
( x + 1 ) d d x ( x d d x [ ( x + 1 ) v ( x ) ] ) + λ v ( x ) = 0 {\displaystyle (x+1){\frac {d}{dx}}\left(x{\frac {d}{dx}}\left[\left(x+1\right)v(x)\right]\right)+\lambda v(x)=0}
with eigenvalues
λ n = n ( n + 1 ) {\displaystyle \lambda _{n}=n(n+1)\,}
Properties Many properties can be derived from the properties of the Legendre polynomials of the first kind. Other properties are unique to the functions themselves.
Recursion
R n + 1 ( x ) = 2 n + 1 n + 1 x − 1 x + 1 R n ( x ) − n n + 1 R n − 1 ( x ) f o r n ≥ 1 {\displaystyle R_{n+1}(x)={\frac {2n+1}{n+1}}\,{\frac {x-1}{x+1}}\,R_{n}(x)-{\frac {n}{n+1}}\,R_{n-1}(x)\quad \mathrm {for\,n\geq 1} }
and
2 ( 2 n + 1 ) R n ( x ) = ( x + 1 ) 2 ( d d x R n + 1 ( x ) − d d x R n − 1 ( x ) ) + ( x + 1 ) ( R n + 1 ( x ) − R n − 1 ( x ) ) {\displaystyle 2(2n+1)R_{n}(x)=\left(x+1\right)^{2}\left({\frac {d}{dx}}R_{n+1}(x)-{\frac {d}{dx}}R_{n-1}(x)\right)+(x+1)\left(R_{n+1}(x)-R_{n-1}(x)\right)}
Limiting behavior
It can be shown that
lim x → ∞ ( x + 1 ) R n ( x ) = 2 {\displaystyle \lim _{x\to \infty }(x+1)R_{n}(x)={\sqrt {2}}}
and
lim x → ∞ x ∂ x ( ( x + 1 ) R n ( x ) ) = 0 {\displaystyle \lim _{x\to \infty }x\partial _{x}((x+1)R_{n}(x))=0}
Orthogonality
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