2F1 - Maple Help
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convert/2F1

convert to special functions admitting a 2F1 hypergeometric representation

 Calling Sequence convert(expr, 2F1)

Parameters

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

Description

 • convert/2F1 converts, when possible, hypergeometric / MeijerG functions into special functions admitting a 2F1 hypergeometric representation; that is, into one of
 > FunctionAdvisor( 2F1 );
 The 26 functions in the "2F1" class are particular cases of the hypergeometric function and are given by:
 $\left[{\mathrm{ChebyshevT}}{,}{\mathrm{ChebyshevU}}{,}{\mathrm{EllipticCE}}{,}{\mathrm{EllipticCK}}{,}{\mathrm{EllipticE}}{,}{\mathrm{EllipticK}}{,}{\mathrm{GaussAGM}}{,}{\mathrm{GegenbauerC}}{,}{\mathrm{JacobiP}}{,}{\mathrm{LegendreP}}{,}{\mathrm{LegendreQ}}{,}{\mathrm{LerchPhi}}{,}{\mathrm{SphericalY}}{,}{\mathrm{arccos}}{,}{\mathrm{arccosh}}{,}{\mathrm{arccot}}{,}{\mathrm{arccoth}}{,}{\mathrm{arccsc}}{,}{\mathrm{arccsch}}{,}{\mathrm{arcsec}}{,}{\mathrm{arcsech}}{,}{\mathrm{arcsin}}{,}{\mathrm{arcsinh}}{,}{\mathrm{arctan}}{,}{\mathrm{arctanh}}{,}{\mathrm{ln}}\right]$ (1)
 • convert/2F1 accepts as optional arguments all those described in convert/to_special_function.

Examples

 > $z\mathrm{hypergeom}\left(\left[\frac{1}{2},1\right],\left[\frac{3}{2}\right],{z}^{2}\right)$
 ${z}{}{\mathrm{hypergeom}}{}\left(\left[\frac{{1}}{{2}}{,}{1}\right]{,}\left[\frac{{3}}{{2}}\right]{,}{{z}}^{{2}}\right)$ (2)
 > $\mathrm{convert}\left(,\mathrm{2F1}\right)$
 ${\mathrm{arctanh}}{}\left({z}\right)$ (3)
 > $\mathrm{hypergeom}\left(\left[-\frac{1}{2},\frac{1}{2}\right],\left[1\right],{z}^{2}\right)$
 ${\mathrm{hypergeom}}{}\left(\left[{-}\frac{{1}}{{2}}{,}\frac{{1}}{{2}}\right]{,}\left[{1}\right]{,}{{z}}^{{2}}\right)$ (4)
 > $\mathrm{convert}\left(,\mathrm{2F1}\right)$
 ${\mathrm{JacobiP}}{}\left(\frac{{1}}{{2}}{,}{0}{,}{-1}{,}{-}{2}{}{{z}}^{{2}}{+}{1}\right)$ (5)
 > $\mathrm{hypergeom}\left(\left[\frac{1a}{2}+\frac{1b}{2}+\frac{1}{2},1+\frac{1a}{2}+\frac{1b}{2}\right],\left[a+\frac{3}{2}\right],\frac{1}{{z}^{2}}\right)$
 ${\mathrm{hypergeom}}{}\left(\left[{1}{+}\frac{{a}}{{2}}{+}\frac{{b}}{{2}}{,}\frac{{a}}{{2}}{+}\frac{{b}}{{2}}{+}\frac{{1}}{{2}}\right]{,}\left[{a}{+}\frac{{3}}{{2}}\right]{,}\frac{{1}}{{{z}}^{{2}}}\right)$ (6)
 > $\mathrm{convert}\left(,\mathrm{2F1}\right)$
 $\frac{{\mathrm{\Gamma }}{}\left({a}{+}\frac{{3}}{{2}}\right){}{\mathrm{\Gamma }}{}\left({-}\frac{{a}}{{2}}{-}\frac{{b}}{{2}}\right){}{\mathrm{JacobiP}}{}\left({-}\frac{{a}}{{2}}{-}{1}{-}\frac{{b}}{{2}}{,}{a}{+}\frac{{1}}{{2}}{,}{b}{,}\frac{{{z}}^{{2}}{-}{2}}{{{z}}^{{2}}}\right)}{{\mathrm{\Gamma }}{}\left(\frac{{a}}{{2}}{+}\frac{{1}}{{2}}{-}\frac{{b}}{{2}}\right)}$ (7)
 > $\mathrm{hypergeom}\left(\left[b+c+a+1,-a\right],\left[1+b\right],\frac{1}{2}-\frac{1z}{2}\right)$
 ${\mathrm{hypergeom}}{}\left(\left[{-}{a}{,}{b}{+}{c}{+}{a}{+}{1}\right]{,}\left[{1}{+}{b}\right]{,}\frac{{1}}{{2}}{-}\frac{{z}}{{2}}\right)$ (8)
 > $\mathrm{convert}\left(,\mathrm{2F1}\right)$
 $\frac{{\mathrm{\Gamma }}{}\left({1}{+}{b}\right){}{\mathrm{\Gamma }}{}\left({a}{+}{1}\right){}{\mathrm{JacobiP}}{}\left({a}{,}{b}{,}{c}{,}{z}\right)}{{\mathrm{\Gamma }}{}\left({a}{+}{b}{+}{1}\right)}$ (9)
 > $\mathrm{MeijerG}\left(\left[\left[0,\frac{1}{2}\right],\left[\right]\right],\left[\left[0\right],\left[-\frac{1}{2}\right]\right],-\frac{1}{{z}^{2}}\right)$
 ${\mathrm{MeijerG}}{}\left(\left[\left[{0}{,}\frac{{1}}{{2}}\right]{,}\left[\right]\right]{,}\left[\left[{0}\right]{,}\left[{-}\frac{{1}}{{2}}\right]\right]{,}{-}\frac{{1}}{{{z}}^{{2}}}\right)$ (10)
 > $\mathrm{convert}\left(,\mathrm{2F1}\right)$
 ${2}{}{z}{}{\mathrm{arctanh}}{}\left(\frac{{1}}{{z}}\right)$ (11)
 > $\mathrm{MeijerG}\left(\left[\left[\frac{1}{2},\frac{1}{2}\right],\left[\right]\right],\left[\left[0\right],\left[0\right]\right],-1+{z}^{2}\right)$
 ${\mathrm{MeijerG}}{}\left(\left[\left[\frac{{1}}{{2}}{,}\frac{{1}}{{2}}\right]{,}\left[\right]\right]{,}\left[\left[{0}\right]{,}\left[{0}\right]\right]{,}{{z}}^{{2}}{-}{1}\right)$ (12)
 > $\mathrm{convert}\left(,\mathrm{2F1}\right)$
 ${\mathrm{\pi }}{}{\mathrm{GegenbauerC}}{}\left({-}\frac{{1}}{{2}}{,}\frac{{1}}{{2}}{,}{2}{}{{z}}^{{2}}{-}{1}\right)$ (13)
 > $\mathrm{MeijerG}\left(\left[\left[-\frac{1a}{2}-\frac{1b}{2},\frac{1}{2}-\frac{1a}{2}-\frac{1b}{2}\right],\left[\right]\right],\left[\left[0\right],\left[-\frac{1}{2}-a\right]\right],-\frac{1}{{z}^{2}}\right)$
 ${\mathrm{MeijerG}}{}\left(\left[\left[{-}\frac{{a}}{{2}}{-}\frac{{b}}{{2}}{,}\frac{{1}}{{2}}{-}\frac{{a}}{{2}}{-}\frac{{b}}{{2}}\right]{,}\left[\right]\right]{,}\left[\left[{0}\right]{,}\left[{-}{a}{-}\frac{{1}}{{2}}\right]\right]{,}{-}\frac{{1}}{{{z}}^{{2}}}\right)$ (14)
 > $\mathrm{convert}\left(,\mathrm{2F1}\right)$
 ${-}\frac{{\mathrm{\pi }}{}{\mathrm{\Gamma }}{}\left({1}{+}\frac{{a}}{{2}}{+}\frac{{b}}{{2}}\right){}{\mathrm{JacobiP}}{}\left({-}\frac{{1}}{{2}}{-}\frac{{a}}{{2}}{-}\frac{{b}}{{2}}{,}{a}{+}\frac{{1}}{{2}}{,}{b}{,}\frac{{{z}}^{{2}}{-}{2}}{{{z}}^{{2}}}\right)}{{\mathrm{\Gamma }}{}\left(\frac{{a}}{{2}}{+}{1}{-}\frac{{b}}{{2}}\right){}{\mathrm{sin}}{}\left({-}\frac{\left({a}{+}{b}{+}{1}\right){}{\mathrm{\pi }}}{{2}}\right)}$ (15)
 > $\mathrm{MeijerG}\left(\left[\left[-a,a+1\right],\left[\right]\right],\left[\left[0\right],\left[b\right]\right],-\frac{1}{2}+\frac{1z}{2}\right)$
 ${\mathrm{MeijerG}}{}\left(\left[\left[{-}{a}{,}{a}{+}{1}\right]{,}\left[\right]\right]{,}\left[\left[{0}\right]{,}\left[{b}\right]\right]{,}{-}\frac{{1}}{{2}}{+}\frac{{z}}{{2}}\right)$ (16)
 > $\mathrm{convert}\left(,\mathrm{2F1}\right)$
 ${-}\frac{{\mathrm{\pi }}{}{\mathrm{csc}}{}\left({\mathrm{\pi }}{}{a}\right){}{\left({-}{1}{+}{z}\right)}^{\frac{{b}}{{2}}}{}{\mathrm{LegendreP}}{}\left({a}{,}{b}{,}{z}\right)}{{\left({1}{+}{z}\right)}^{\frac{{b}}{{2}}}}$ (17)

 See Also