Legendre Rational Functions

In mathematics the Legendre rational functions are a sequence of orthogonal functions on  , ∞). They are obtained by composing the Cayley transform with Legendre polynomials">Cayley_transform.html" ;"title=", ∞). They are obtained by composing the Cayley transform">, ∞). They are obtained by composing the Cayley transform with Legendre polynomials.

A rational Legendre function of degree ''n'' is defined as:

:$R_n(x) = \frac\,P_n\left(\frac\right)$

where $P_n(x)$ is a Legendre polynomial. These functions are eigenfunctions of the singular Sturm–Liouville problem:

:$(x+1)\partial_x(x\partial_x((x+1)v(x)))+\lambda v(x)=0$

with eigenvalues

:$\lambda_n=n(n+1)\,$

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

:$R_(x)=\frac\,\frac\,R_n(x)-\frac\,R_(x)\quad\mathrm$

and

:$2(2n+1)R_n(x)=(x+1)^2(\partial_x R_(x)-\partial_x R_(x))+(x+1)(R_(x)-R_(x))$

Limiting behavior

It can be shown that

:$\lim_(x+1)R_n(x)=\sqrt$

and

:$\lim_x\partial_x((x+1)R_n(x))=0$

Orthogonality

:$\int_^\infty R_m(x)\,R_n(x)\,dx=\frac\delta_$

where $\delta_$ is the Kronecker delta function.

Particular values

:$R_0(x)=1\,$
:$R_1(x)=\frac\,$
:$R_2(x)=\frac\,$
:$R_3(x)=\frac\,$
:$R_4(x)=\frac\,$

References

{{cite journal
, last = Zhong-Qing
, first = Wang
, authorlink =
, author2=Ben-Yu, Guo
, year = 2005
, title = A mixed spectral method for incompressible viscous fluid flow in an infinite strip
, journal = Mat. Apl. Comput.
, volume = 24
, issue = 3
, pages =
, doi = 10.1590/S0101-82052005000300002
, id =
, doi-access = free

Rational functions