Chebyshev rational functions

---

title: "Chebyshev rational functions" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["rational-functions"] topic_path: "general/rational-functions" source: "https://en.wikipedia.org/wiki/Chebyshev_rational_functions" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::figure[src="https://upload.wikimedia.org/wikipedia/commons/1/16/ChebychevRational1.png" caption="0.01 ≤ ''x'' ≤ 100}}, log scale."] ::

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:

:R_n(x)\ \stackrel{\mathrm{def}}{=}\ T_n\left(\frac{x-1}{x+1}\right)

where Tn(x) is a Chebyshev polynomial of the first kind.

Properties

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

Recursion

:R_{n+1}(x)=2\left(\frac{x-1}{x+1}\right)R_{n}(x)-R_{n-1}(x)\quad\text{for},n\ge 1

Differential equations

:(x+1)^2R_n(x)=\frac{1}{n+1}\frac{\mathrm{d}}{\mathrm{d}x}R_{n+1}(x)-\frac{1}{n-1}\frac{\mathrm{d}}{\mathrm{d}x}R_{n-1}(x) \quad \text{for } n\ge 2

:(x+1)^2x\frac{\mathrm{d}^2}{\mathrm{d}x^2}R_n(x)+\frac{(3x+1)(x+1)}{2}\frac{\mathrm{d}}{\mathrm{d}x}R_n(x)+n^2R_{n}(x) = 0

Orthogonality

::figure[src="https://upload.wikimedia.org/wikipedia/commons/8/8c/ChebychevRational2.png" caption="''x''₀}}}} is a zero as well. The maximum value between the zeros is unity. These properties hold for all orders."] ::

Defining:

:\omega(x) \ \stackrel{\mathrm{def}}{=}\ \frac{1}{(x+1)\sqrt{x}}

The orthogonality of the Chebyshev rational functions may be written:

:\int_{0}^\infty R_m(x),R_n(x),\omega(x),\mathrm{d}x=\frac{\pi c_n}{2}\delta_{nm}

where for and for n ≥ 1; δnm is the Kronecker delta function.

Expansion of an arbitrary function

For an arbitrary function the orthogonality relationship can be used to expand f(x):

:f(x)=\sum_{n=0}^\infty F_n R_n(x)

where

:F_n=\frac{2}{c_n\pi}\int_{0}^\infty f(x)R_n(x)\omega(x),\mathrm{d}x.

Particular values

:\begin{align} R_0(x)&=1\ R_1(x)&=\frac{x-1}{x+1}\ R_2(x)&=\frac{x^2-6x+1}{(x+1)^2}\ R_3(x)&=\frac{x^3-15x^2+15x-1}{(x+1)^3}\ R_4(x)&=\frac{x^4-28x^3+70x^2-28x+1}{(x+1)^4}\ R_n(x)&=(x+1)^{-n}\sum_{m=0}^{n} (-1)^m\binom{2n}{2m}x^{n-m} \end{align}

Partial fraction expansion

:R_n(x)=\sum_{m=0}^{n} \frac{(m!)^2}{(2m)!}\binom{n+m-1}{m}\binom{n}{m}\frac{(-4)^m}{(x+1)^m}

References

• {{cite journal | first1= Ben-Yu | last1= Guo |first2=Jie |last2=Shen |first3=Zhong-Qing |last3=Wang | year = 2002 | title = Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval | journal = Int. J. Numer. Methods Eng. | volume = 53 | issue= 1 | pages = 65–84 | doi = 10.1002/nme.392 | bibcode= 2002IJNME..53...65G | url = http://www.math.purdue.edu/~shen/pub/GSW_IJNME02.pdf | access-date = 2006-07-25 | citeseerx= 10.1.1.121.6069 | s2cid= 9208244

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::