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 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 and Limiting behavior It can be shown that and Orthogonality where is the Kronecker delta function. Particular values References Rational functions