Legendre rational functions explained

In mathematics, the Legendre rational functions are a sequence of orthogonal functions on . They are obtained by composing the Cayley transform with Legendre polynomials.

A rational Legendre function of degree n is defined as:$$R\_n(x)\; =\; \backslash frac\backslash ,P\_n\backslash left(\backslash frac\backslash right)$$where

P _{n(x)} is a Legendre polynomial. These functions are eigenfunctions of the singular Sturm–Liouville problem:$$(x+1)\; \backslash frac\backslash left(x\; \backslash frac\; \backslash left[\backslash left(x+1\backslash right)\; v(x)\backslash right]\backslash right)\; +\; \backslash lambda\; v(x)\; =\; 0$$with eigenvalues$$\backslash lambda\_n=n(n+1)\backslash ,$$

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

$$R\_(x)=\backslash frac\backslash ,\backslash frac\backslash ,R\_n(x)-\backslash frac\backslash ,R\_(x)\backslash quad\backslash mathrm$$and$$2\; (2n+1)\; R\_n(x)\; =\; \backslash left(x+1\backslash right)^2\; \backslash left(\backslash frac\; R\_(x)\; -\; \backslash frac\; R\_(x)\backslash right)\; +\; (x+1)\; \backslash left(R\_(x)\; -\; R\_(x)\backslash right)$$

Limiting behavior

It can be shown that$$\backslash lim\_(x+1)R\_n(x)=\backslash sqrt$$and$$\backslash lim\_x\backslash partial\_x((x+1)R\_n(x))=0$$

Orthogonality

$$\backslash int\_^\backslash infty\; R\_m(x)\backslash ,R\_n(x)\backslash ,dx=\backslash frac\backslash delta\_$$where

\delta _{nm} is the Kronecker delta function.

Particular values

References

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