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 is a Legendre polynomial. These functions are eigenfunctions of the singular Sturm–Liouville problem:
with eigenvalues
Properties edit
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 edit
Limiting behavior edit
It can be shown that
Orthogonality edit
Particular values edit
References edit
- Zhong-Qing, Wang; Ben-Yu, Guo (2005). "A mixed spectral method for incompressible viscous fluid flow in an infinite strip". Computational & Applied Mathematics. 24 (3). Sociedade Brasileira de Matemática Aplicada e Computacional. doi:10.1590/S0101-82052005000300002.