Plot of the Legendre rational functions for n=0,1,2 and 3 for x between 0.01 and 100.
Plot of the Legendre rational functions for n=0,1,2 and 3 for x between 0.01 and 100.

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

Plot of the seventh order (n=7) Legendre rational function multiplied by 1+x for x between 0.01 and 100. Note that there are n zeroes arranged symmetrically about x=1 and if x0 is a zero, then 1/x0 is a zero as well. These properties hold for all orders.
Plot of the seventh order (n=7) Legendre rational function multiplied by 1+x for x between 0.01 and 100. Note that there are n zeroes arranged symmetrically about x=1 and if x0 is a zero, then 1/x0 is a zero as well. These properties hold for all orders.

It can be shown that

and

Orthogonality

where 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". Mat. Apl. Comput. 24 (3). doi:10.1590/S0101-82052005000300002.