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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)}
P
n
(
x
)
{\displaystyle P_{n}(x)}
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}
λ
n
=
n
(
n
+
1
)
{\displaystyle \lambda _{n}=n(n+1)\,}
Many properties can be derived from the properties of the Legendre polynomials of the first kind. Other properties are unique to the functions themselves.
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} }
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)}
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 x 0 is a zero, then 1/x 0 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}}}
lim
x
→
∞
x
∂
x
(
(
x
+
1
)
R
n
(
x
)
)
=
0
{\displaystyle \lim _{x\to \infty }x\partial _{x}((x+1)R_{n}(x))=0}
∫
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}}
δ
n
m
{\displaystyle \delta _{nm}}
Kronecker delta function.
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}}}
