Legendre rational functions

Sequence of orthogonal functions on [0, ∞)
Plot of the Legendre rational functions for n=0,1,2 and 3 for x between 0.01 and 100.

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

Plot of the seventh order (n=7) Legendre rational function multiplied by 1+x for x between 0.01 and 100. Note that there are n zeroes arranged symmetrically about x=1 and if x0 is a zero, then 1/x0 is a zero as well. These properties hold for all orders.

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

∫ 0 ∞ R m ( x ) R n ( x ) d x = 2 2 n + 1 δ n m {\displaystyle \int _{0}^{\infty }R_{m}(x)\,R_{n}(x)\,dx={\frac {2}{2n+1}}\delta _{nm}}
where δ n m {\displaystyle \delta _{nm}} is the Kronecker delta function.

Particular values

R 0 ( x ) = 2 x + 1 1 R 1 ( x ) = 2 x + 1 x − 1 x + 1 R 2 ( x ) = 2 x + 1 x 2 − 4 x + 1 ( x + 1 ) 2 R 3 ( x ) = 2 x + 1 x 3 − 9 x 2 + 9 x − 1 ( x + 1 ) 3 R 4 ( x ) = 2 x + 1 x 4 − 16 x 3 + 36 x 2 − 16 x + 1 ( x + 1 ) 4 {\displaystyle {\begin{aligned}R_{0}(x)&={\frac {\sqrt {2}}{x+1}}\,1\\R_{1}(x)&={\frac {\sqrt {2}}{x+1}}\,{\frac {x-1}{x+1}}\\R_{2}(x)&={\frac {\sqrt {2}}{x+1}}\,{\frac {x^{2}-4x+1}{(x+1)^{2}}}\\R_{3}(x)&={\frac {\sqrt {2}}{x+1}}\,{\frac {x^{3}-9x^{2}+9x-1}{(x+1)^{3}}}\\R_{4}(x)&={\frac {\sqrt {2}}{x+1}}\,{\frac {x^{4}-16x^{3}+36x^{2}-16x+1}{(x+1)^{4}}}\end{aligned}}}

References

  • Zhong-Qing, Wang; Ben-Yu, Guo (2005). "A mixed spectral method for incompressible viscous fluid flow in an infinite strip". Computational & Applied Mathematics. 24 (3). Sociedade Brasileira de Matemática Aplicada e Computacional. doi:10.1590/S0101-82052005000300002.