Chebyshev polynomial

Chebyshev polynomial \(T\) is defined with the recursive relation \(T_0(x)=1\) \(T_1(x)=x\) \(T_{n+1}(x)=2 x T_n(x)-T_{n-1}(x)\)

Mathematica
In Mathematica, the Chebyshev polynomials are built-in functions; \(T_n(x)\) is denoted with ChebyshevT[n,x]

Some properties
Relation to the Bessel function (Gradstein, Ryzhik. p.850, formula 7.355)

\(\displaystyle \int_0^1 T_{2n+1}(x)\, \sin(x\, y)\, \frac{\mathrm d x}{\sqrt{1\!-\!x^2}} = (-1)^n \frac{\pi}{2}\, J_{2n+1}(y) \)

\(\displaystyle \int_0^1 T_{2n}(x)\, \cos(x\, y)\, \frac{\mathrm d x}{\sqrt{1\!-\!x^2}} = (-1)^n \frac{\pi}{2}\, J_{2n}(y) \)

Semigroup property:

\(T_n(T_m(x))=T_{mn}(x)\)

However, these polynomials do not form a group, as the inverse elements cannot be provided.

Applications
Chebyshev polynomial is important tool of research since century 19 and has many applications.

In TORI, the Chebyshev polynomial is expected be used for the efficient numerical implementation of the Bessel transform. Mainly, the properties that are important for this application are collected above.