Chebyshev - Maple Help

convert/Chebyshev

convert special functions admitting 2F1 hypergeometric representation into Chebyshev functions

 Calling Sequence convert(expr, Chebyshev)

Parameters

 expr - Maple expression, equation, or a set or list of them

Description

 • convert/Chebyshev converts, when possible, special functions admitting a 2F1 hypergeometric representation into Chebyshev functions (see ?ChebyshevT and ?ChebyshevU). The Chebyshev functions are
 The 2 functions in the "Chebyshev" class are:
 $\left[{\mathrm{ChebyshevT}}{,}{\mathrm{ChebyshevU}}\right]$ (1)

Examples

 > $\left(a+1\right)\mathrm{hypergeom}\left(\left[-a,a+2\right],\left[\frac{3}{2}\right],\frac{1}{2}-\frac{1}{2}z\right)$
 $\left({a}{+}{1}\right){}{\mathrm{hypergeom}}{}\left(\left[{-}{a}{,}{a}{+}{2}\right]{,}\left[\frac{{3}}{{2}}\right]{,}\frac{{1}}{{2}}{-}\frac{{z}}{{2}}\right)$ (2)
 > $\mathrm{convert}\left(,\mathrm{Chebyshev}\right)$
 ${\mathrm{ChebyshevU}}{}\left({a}{,}{z}\right)$ (3)
 > $\mathrm{JacobiP}\left(-a+b,-\frac{1}{2},-\frac{1}{2},\frac{1}{2}z\right)+\mathrm{JacobiP}\left(a-b,\frac{1}{2},\frac{1}{2},\frac{1}{2}z\right)$
 ${\mathrm{JacobiP}}{}\left({-}{a}{+}{b}{,}{-}\frac{{1}}{{2}}{,}{-}\frac{{1}}{{2}}{,}\frac{{z}}{{2}}\right){+}{\mathrm{JacobiP}}{}\left({a}{-}{b}{,}\frac{{1}}{{2}}{,}\frac{{1}}{{2}}{,}\frac{{z}}{{2}}\right)$ (4)
 > $\mathrm{convert}\left(,\mathrm{Chebyshev}\right)$
 $\left(\genfrac{}{}{0}{}{{-}{a}{+}{b}{-}\frac{{1}}{{2}}}{{-}\frac{{1}}{{2}}}\right){}{\mathrm{ChebyshevT}}{}\left({a}{-}{b}{,}\frac{{z}}{{2}}\right){+}\frac{\left(\genfrac{}{}{0}{}{{a}{-}{b}{+}\frac{{1}}{{2}}}{\frac{{1}}{{2}}}\right){}{\mathrm{ChebyshevU}}{}\left({a}{-}{b}{,}\frac{{z}}{{2}}\right)}{{a}{-}{b}{+}{1}}$ (5)
 > $-\frac{1}{{\mathrm{\pi }}^{\frac{1}{2}}}\mathrm{sin}\left(\mathrm{\pi }a\right)a\mathrm{MeijerG}\left(\left[\left[1-a,a+1\right],\left[\right]\right],\left[\left[0\right],\left[\frac{1}{2}\right]\right],-\frac{1}{2}+\frac{1}{2}z\right)$
 ${-}\frac{{\mathrm{sin}}{}\left({\mathrm{\pi }}{}{a}\right){}{a}{}{\mathrm{MeijerG}}{}\left(\left[\left[{1}{-}{a}{,}{a}{+}{1}\right]{,}\left[\right]\right]{,}\left[\left[{0}\right]{,}\left[\frac{{1}}{{2}}\right]\right]{,}{-}\frac{{1}}{{2}}{+}\frac{{z}}{{2}}\right)}{\sqrt{{\mathrm{\pi }}}}$ (6)
 > $\mathrm{simplify}\left(\mathrm{convert}\left(,\mathrm{Chebyshev}\right)\right)$
 ${\mathrm{ChebyshevT}}{}\left({a}{,}{z}\right)$ (7)

When converting to a function class (e.g. Chebyshev) it is possible to request additional conversion rules to be performed. Compare for instance these two different outputs:

 > $\mathrm{GegenbauerC}\left(a,1,z\right)$
 ${\mathrm{GegenbauerC}}{}\left({a}{,}{1}{,}{z}\right)$ (8)
 > $\mathrm{convert}\left(,\mathrm{Chebyshev}\right)$
 ${\mathrm{ChebyshevU}}{}\left({a}{,}{z}\right)$ (9)
 > $\mathrm{convert}\left(,\mathrm{Chebyshev},"raise a"\right)$
 ${\mathrm{ChebyshevU}}{}\left({-}{4}{-}{a}{,}{z}\right){-}{2}{}{z}{}{\mathrm{ChebyshevU}}{}\left({-}{3}{-}{a}{,}{z}\right)$ (10)