convert/Chebyshev converts, when possible, special functions admitting a 2F1 hypergeometric representation into Chebyshev functions (see ?ChebyshevT and ?ChebyshevU). The Chebyshev functions are

FunctionAdvisor( Chebyshev );

The 2 functions in the "Chebyshev" class are:

$\left[\mathrm{ChebyshevT}\,\mathrm{ChebyshevU}\right]$

$\left(a+1\right)\mathrm{hypergeom}\left(\left[-a\,a+2\right]\,\left[\frac{3}{2}\right]\,\frac{1}{2}-\frac{1}{2}z\right)$

$\left(a+1\right)\mathrm{hypergeom}\left(\left[-a\,a+2\right]\,\left[\frac{3}{2}\right]\,\frac{1}{2}-\frac{z}{2}\right)$

$\mathrm{convert}\left(\,\mathrm{Chebyshev}\right)$

$\mathrm{ChebyshevU}\left(a\,z\right)$

$\mathrm{JacobiP}\left(-a+b\,-\frac{1}{2}\,-\frac{1}{2}\,\frac{1}{2}z\right)+\mathrm{JacobiP}\left(a-b\,\frac{1}{2}\,\frac{1}{2}\,\frac{1}{2}z\right)$

$\mathrm{JacobiP}\left(-a+b\,-\frac{1}{2}\,-\frac{1}{2}\,\frac{z}{2}\right)+\mathrm{JacobiP}\left(a-b\,\frac{1}{2}\,\frac{1}{2}\,\frac{z}{2}\right)$

$\mathrm{convert}\left(\,\mathrm{Chebyshev}\right)$

$\left(\genfrac{}{}{0ex}{}{-a+b-\frac{1}{2}}{-\frac{1}{2}}\right)\mathrm{ChebyshevT}\left(a-b\,\frac{z}{2}\right)+\frac{\left(\genfrac{}{}{0ex}{}{a-b+\frac{1}{2}}{\frac{1}{2}}\right)\mathrm{ChebyshevU}\left(a-b\,\frac{z}{2}\right)}{a-b+1}$

$-\frac{1}{{\mathrm{\pi }}^{\frac{1}{2}}}\sin \left(\mathrm{\pi }a\right)a\mathrm{MeijerG}\left(\left[\left[1-a\,a+1\right]\,\left[\right]\right]\,\left[\left[0\right]\,\left[\frac{1}{2}\right]\right]\,-\frac{1}{2}+\frac{1}{2}z\right)$

$-\frac{\sin \left(\mathrm{\pi }a\right)a\mathrm{MeijerG}\left(\left[\left[1-a\,a+1\right]\,\left[\right]\right]\,\left[\left[0\right]\,\left[\frac{1}{2}\right]\right]\,-\frac{1}{2}+\frac{z}{2}\right)}{\sqrt{\mathrm{\pi }}}$

$\mathrm{simplify}\left(\mathrm{convert}\left(\,\mathrm{Chebyshev}\right)\right)$

$\mathrm{ChebyshevT}\left(a\,z\right)$

$\mathrm{GegenbauerC}\left(a\,1\,z\right)$

$\mathrm{GegenbauerC}\left(a\,1\,z\right)$

$\mathrm{convert}\left(\,\mathrm{Chebyshev}\right)$

$\mathrm{ChebyshevU}\left(a\,z\right)$

$\mathrm{convert}\left(\,\mathrm{Chebyshev}\,''raise\; a''\right)$

$-2z\mathrm{ChebyshevU}\left(-3-a\,z\right)+\mathrm{ChebyshevU}\left(-a-4\,z\right)$