In mathematics, the Chebyshev rational functions are a sequence of functions which are both

rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abil ...

and

orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

. They are named after

Pafnuty Chebyshev Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics. Chebysh ...

. A rational Chebyshev function of degree is defined as: :

R_n(x)\ \stackrel\  T_n\left(\frac\right)

where is a

Chebyshev polynomial The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshe ...

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_(x)=2\,\fracR_n(x)-R_(x) \quad \text n\ge 1

Differential equations

(x+1)^2R_n(x)=\frac\fracR_(x)-\frac\fracR_(x) \quad \text n\ge 2

(x+1)^2x\fracR_n(x)+\frac\fracR_n(x)+n^2R_(x) = 0

Orthogonality

Defining: :

\omega(x) \ \stackrel\  \frac

The orthogonality of the Chebyshev rational functions may be written: :

\int_^\infty R_m(x)\,R_n(x)\,\omega(x)\,\mathrmx=\frac\delta_

where for and for ; is the

Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 ...

function.

Expansion of an arbitrary function

For an arbitrary function the orthogonality relationship can be used to expand : :

f(x)=\sum_^\infty F_n R_n(x)

where :

F_n=\frac\int_^\infty f(x)R_n(x)\omega(x)\,\mathrmx.

Particular values

\begin
R_0(x)&=1\\
R_1(x)&=\frac\\
R_2(x)&=\frac\\
R_3(x)&=\frac\\
R_4(x)&=\frac\\
R_n(x)&=(x+1)^\sum_^ (-1)^m\binomx^
\end

Partial fraction expansion

R_n(x)=\sum_^ \frac\binom\binom\frac

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 Rational functions