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 .
Plot of the Legendre rational functions for n=0,1,2 and 3 for x between 0.01 and 100. 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)}
eigenfunctions of the singular Sturm-Liouville problem :
(
x
+
1
)
∂
x
(
x
∂
x
(
(
x
+
1
)
v
(
x
)
)
)
+
λ
v
(
x
)
=
0
{\displaystyle (x+1)\partial _{x}(x\partial _{x}((x+1)v(x)))+\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
(
∂
x
R
n
+
1
(
x
)
−
∂
x
R
n
−
1
(
x
)
)
+
(
x
+
1
)
(
R
n
+
1
(
x
)
−
R
n
−
1
(
x
)
)
{\displaystyle 2(2n+1)R_{n}(x)=(x+1)^{2}(\partial _{x}R_{n+1}(x)-\partial _{x}R_{n-1}(x))+(x+1)(R_{n+1}(x)-R_{n-1}(x))}
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 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\rightarrow \infty }(x+1)R_{n}(x)={\sqrt {2}}}
and
lim
x
→
∞
x
∂
x
(
(
x
+
1
)
R
n
(
x
)
)
=
0
{\displaystyle \lim _{x\rightarrow \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}}
Kronecker delta function.
Particular values
R
0
(
x
)
=
1
{\displaystyle R_{0}(x)=1\,}
R
1
(
x
)
=
x
−
1
x
+
1
{\displaystyle R_{1}(x)={\frac {x-1}{x+1}}\,}
R
2
(
x
)
=
x
2
−
4
x
+
1
(
x
+
1
)
2
{\displaystyle R_{2}(x)={\frac {x^{2}-4x+1}{(x+1)^{2}}}\,}
R
3
(
x
)
=
x
3
−
9
x
2
+
9
x
−
1
(
x
+
1
)
3
{\displaystyle R_{3}(x)={\frac {x^{3}-9x^{2}+9x-1}{(x+1)^{3}}}\,}
R
4
(
x
)
=
x
4
−
16
x
3
+
36
x
2
−
16
x
+
1
(
x
+
1
)
4
{\displaystyle R_{4}(x)={\frac {x^{4}-16x^{3}+36x^{2}-16x+1}{(x+1)^{4}}}\,}
References 
