Chebyshev rational functions

In mathematics, the Chebyshev rational functions are a sequence of functions which are both rational and orthogonal. They are named after Pafnuty Chebyshev. A rational Chebyshev function of degree n is defined as:

where ${\displaystyle T_{n}(x)}$ is a Chebyshev polynomial of the first kind.

Many properties can be derived from the properties of the Chebyshev polynomials of the first kind. Other properties are unique to the functions themselves.

Defining:

The orthogonality of the Chebyshev rational functions may be written:

where ${\displaystyle c_{n}}$ equals 2 for n = 0 and ${\displaystyle c_{n}}$ equals 1 for ${\displaystyle n\geq 1}$ and ${\displaystyle \delta _{nm}}$ is the Kronecker delta function.

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