Legendre rational functions

Summary

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.

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

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

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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

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  and  

Limiting behavior

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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

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  where   is the Kronecker delta function.

Particular values

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References

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  • 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.