Legendre rational functions
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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 the Legendre rational functions are a sequence of functions which are both rational and orthogonal
Orthogonal functions
In mathematics, two functions f and g are called orthogonal if their inner product \langle f,g\rangle is zero for f ≠ g. Whether or not two particular functions are orthogonal depends on how their inner product has been defined. A typical definition of an inner product for functions is...

. 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

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