Legendre rational functions

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:

where ${\displaystyle L_{n}(x)}$ is a Legendre polynomial. These functions are eigenfunctions of the singular Sturm-Liouville problem:

with eigenvalues

Many properties can be derived from the properties of the Legendre polynomials of the first kind. Other properties are unique to the functions themselves.

and

It can be shown that

and

where ${\displaystyle \delta _{nm}}$ is the Kronecker delta function.

Zhong-Qing, Wang; Ben-Yu, Guo (2005). "A mixed spectral method for incompressible viscous fluid flow in an infinite strip" (PDF). Mat. apl. comput. 24 (3). doi:10.1590/S0101-82052005000300002. 

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