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convert/Heun converts, when possible, hypergeometric, MeijerG and special functions into Heun functions ; that is, into one of

FunctionAdvisor( Heun );

The 23 functions in the "Heun" class are:

$\left[\mathrm{HeunB}\,\mathrm{HeunBPrime}\,\mathrm{HeunC}\,\mathrm{HeunCPrime}\,\mathrm{HeunD}\,\mathrm{HeunDPrime}\,\mathrm{HeunG}\,\mathrm{HeunGPrime}\,\mathrm{HeunT}\,\mathrm{HeunTPrime}\,\mathrm{MathieuA}\,\mathrm{MathieuB}\,\mathrm{MathieuC}\,\mathrm{MathieuCE}\,\mathrm{MathieuCEPrime}\,\mathrm{MathieuCPrime}\,\mathrm{MathieuExponent}\,\mathrm{MathieuFloquet}\,\mathrm{MathieuFloquetPrime}\,\mathrm{MathieuS}\,\mathrm{MathieuSE}\,\mathrm{MathieuSEPrime}\,\mathrm{MathieuSPrime}\right]$

convert/Heun accepts as optional arguments all those described in convert[to_special_function] .

$\mathrm{functions\_2F1}≔\left[\mathrm{ChebyshevT}\,\mathrm{JacobiP}\,\mathrm{SphericalY}\,\mathrm{EllipticK}\,\mathrm{GaussAGM}\,\arctan \,\arcsin \right]$

$\mathrm{functions\_2F1}≔\left[\mathrm{ChebyshevT}\,\mathrm{JacobiP}\,\mathrm{SphericalY}\,\mathrm{EllipticK}\,\mathrm{GaussAGM}\,\arctan \,\arcsin \right]$

$\mathrm{map2}\left(\mathrm{FunctionAdvisor}\,\mathrm{syntax}\,\mathrm{functions\_2F1}\right)$

$\left[\mathrm{ChebyshevT}\left(a\,z\right)\,\mathrm{JacobiP}\left(a\,b\,c\,z\right)\,\mathrm{SphericalY}\left(\mathrm{\lambda }\,\mathrm{\mu }\,\mathrm{\theta }\,\mathrm{\phi }\right)\,\mathrm{EllipticK}\left(k\right)\,\mathrm{GaussAGM}\left(x\,y\right)\,\arctan \left(y\,x\right)\,\arcsin \left(z\right)\right]$

$\mathrm{map}\left(u\mapsto u=\mathrm{convert}\left(u\,\mathrm{Heun}\right)\,\right)$

$\left[\mathrm{ChebyshevT}\left(a\,z\right)=\mathrm{HeunC}\left(0\,-\frac{1}{2}\,-2a\,0\,a^2+\frac{1}{4}\,\frac{z-1}{z+1}\right){\left(\frac{1}{2}+\frac{z}{2}\right)}^a\,\mathrm{JacobiP}\left(a\,b\,c\,z\right)=\frac{\left(\genfrac{}{}{0ex}{}{a+b}{b}\right)\mathrm{HeunC}\left(0\,b\,b+c+2a+1\,0\,\frac{\left(b+1+2a\right)\left(b+c+a+1\right)}{2}-\frac{b}{2}-\frac{\left(b+1\right)a}{2}\,\frac{z-1}{z+1}\right)}{{\left(\frac{1}{2}+\frac{z}{2}\right)}^{b+c+a+1}}\,\mathrm{SphericalY}\left(\mathrm{\lambda }\,\mathrm{\mu }\,\mathrm{\theta }\,\mathrm{\phi }\right)=\frac{{\left(\mathrm{-1}\right)}^{\mathrm{\mu }}\sqrt{\frac{2\mathrm{\lambda }+1}{\mathrm{\pi }}}\sqrt{\left(\mathrm{\lambda }-\mathrm{\mu }\right)!}{\ⅇ}^{\mathrm{I}\mathrm{\phi }\mathrm{\mu }}{\left(\cos \left(\mathrm{\theta }\right)+1\right)}^{\frac{\mathrm{\mu }}{2}}\mathrm{HeunC}\left(0\,-\mathrm{\mu }\,2\mathrm{\lambda }+1\,0\,{\mathrm{\lambda }}^2+\mathrm{\lambda }+\frac{1}{2}\,\frac{\cos \left(\mathrm{\theta }\right)-1}{\cos \left(\mathrm{\theta }\right)+1}\right)}{2\sqrt{\left(\mathrm{\lambda }+\mathrm{\mu }\right)!}{\left(\cos \left(\mathrm{\theta }\right)-1\right)}^{\frac{\mathrm{\mu }}{2}}\mathrm{\Gamma }\left(1-\mathrm{\mu }\right){\left(\frac{1}{2}+\frac{\cos \left(\mathrm{\theta }\right)}{2}\right)}^{\mathrm{\lambda }+1}}\,\mathrm{EllipticK}\left(k\right)=\frac{\mathrm{\pi }\mathrm{HeunC}\left(0\,0\,0\,0\,\frac{1}{4}\,\frac{k^2}{k^2-1}\right)}{2\sqrt{-k^2+1}}\,\mathrm{GaussAGM}\left(x\,y\right)=\frac{\left(x+y\right)\sqrt{\frac{yx}{{\left(x+y\right)}^2}}}{\mathrm{HeunC}\left(0\,0\,0\,0\,\frac{1}{4}\,-\frac{{\left(x-y\right)}^2}{4yx}\right)}\,\arctan \left(y\,x\right)=-\frac{\mathrm{HeunC}\left(0\,1\,0\,0\,\frac{1}{2}\,\frac{\mathrm{I}y-\sqrt{x^2+y^2}+x}{\sqrt{x^2+y^2}\left(1+\frac{\mathrm{I}y-\sqrt{x^2+y^2}+x}{\sqrt{x^2+y^2}}\right)}\right)\left(-\mathrm{I}y+\sqrt{x^2+y^2}-x\right)}{\mathrm{I}x-y}\,\arcsin \left(z\right)=\frac{z\mathrm{HeunC}\left(0\,\frac{1}{2}\,0\,0\,\frac{1}{4}\,\frac{z^2}{z^2-1}\right)}{\sqrt{-z^2+1}}\right]$

$\mathrm{functions\_1F1}≔\left[\mathrm{erf}\left(z\right)\,\mathrm{dawson}\left(z\right)\,\mathrm{Ei}\left(a\,z\right)\,\mathrm{LaguerreL}\left(a\,b\,z\right)\,\mathrm{hypergeom}\left(\left[a\right]\,\left[b\right]\,z\right)\,\mathrm{MeijerG}\left(\left[\left[a\right]\,\left[\right]\right]\,\left[\left[0\right]\,\left[b\right]\right]\,z\right)\,\cos \left(z\right)\,\sin \left(z\right)\right]$

$\mathrm{functions\_1F1}≔\left[\mathrm{erf}\left(z\right)\,\mathrm{dawson}\left(z\right)\,{\mathrm{Ei}}_a\left(z\right)\,\mathrm{LaguerreL}\left(a\,b\,z\right)\,\mathrm{hypergeom}\left(\left[a\right]\,\left[b\right]\,z\right)\,\mathrm{MeijerG}\left(\left[\left[a\right]\,\left[\right]\right]\,\left[\left[0\right]\,\left[b\right]\right]\,z\right)\,\cos \left(z\right)\,\sin \left(z\right)\right]$

$\mathrm{map}\left(u\mapsto u=\mathrm{convert}\left(u\,\mathrm{Heun}\right)\,\mathrm{functions\_1F1}\right)$

$\left[\mathrm{erf}\left(z\right)=\frac{2z\mathrm{HeunB}\left(1\,0\,1\,0\,\sqrt{-z^2}\right)}{\sqrt{\mathrm{\pi }}}\,\mathrm{dawson}\left(z\right)=\frac{z\mathrm{HeunB}\left(1\,0\,1\,0\,\sqrt{z^2}\right)}{{\ⅇ}^{z^2}}\,{\mathrm{Ei}}_a\left(z\right)=\frac{\mathrm{HeunB}\left(2-2a\,0\,2a\,0\,\sqrt{-z}\right)}{a-1}+z^{a-1}\mathrm{\Gamma }\left(1-a\right)\,\mathrm{LaguerreL}\left(a\,b\,z\right)=\left(\genfrac{}{}{0ex}{}{a+b}{a}\right)\mathrm{HeunB}\left(2b\,0\,2b+2+4a\,0\,\sqrt{z}\right)\,\mathrm{hypergeom}\left(\left[a\right]\,\left[b\right]\,z\right)=\mathrm{HeunB}\left(2b-2\,0\,2b-4a\,0\,\sqrt{z}\right)\,\mathrm{MeijerG}\left(\left[\left[a\right]\,\left[\right]\right]\,\left[\left[0\right]\,\left[b\right]\right]\,z\right)=\frac{\mathrm{\Gamma }\left(1-a\right)\mathrm{HeunB}\left(-2b\,0\,-2-2b+4a\,0\,\sqrt{-z}\right)}{\mathrm{\Gamma }\left(1-b\right)}\,\cos \left(z\right)=\frac{\left(2z+\mathrm{\pi }\right)\mathrm{HeunB}\left(2\,0\,0\,0\,\sqrt{\mathrm{I}\left(2z+\mathrm{\pi }\right)}\right)}{2{\ⅇ}^{\frac{\mathrm{I}}{2}\left(2z+\mathrm{\pi }\right)}}\,\sin \left(z\right)=\frac{z\mathrm{HeunB}\left(2\,0\,0\,0\,\sqrt{2}\sqrt{\mathrm{I}z}\right)}{{\ⅇ}^{\mathrm{I}z}}\right]$

$\mathrm{map}\left(u\mapsto u=\mathrm{convert}\left(u\,\mathrm{HeunC}\right)\,\mathrm{functions\_1F1}\right)$

$\left[\mathrm{erf}\left(z\right)=\frac{2\left(-z^3+z\right)\mathrm{HeunC}\left(1\,\frac{1}{2}\,1\,-\frac{1}{4}\,\frac{3}{4}\,z^2\right)}{\sqrt{\mathrm{\pi }}}\,\mathrm{dawson}\left(z\right)=\frac{z\mathrm{HeunC}\left(1\,\frac{1}{2}\,1\,-\frac{1}{4}\,\frac{3}{4}\,-z^2\right)\left(z^2+1\right)}{{\ⅇ}^{z^2}}\,{\mathrm{Ei}}_a\left(z\right)=\frac{\left(1-z\right)\mathrm{HeunC}\left(1\,1-a\,1\,-\frac{a}{2}\,\frac{1}{2}+\frac{a}{2}\,z\right)}{a-1}+z^{a-1}\mathrm{\Gamma }\left(1-a\right)\,\mathrm{LaguerreL}\left(a\,b\,z\right)=\left(\genfrac{}{}{0ex}{}{a+b}{a}\right)\mathrm{HeunC}\left(1\,b\,1\,-\frac{b}{2}-\frac{1}{2}-a\,\frac{b}{2}+1+a\,-z\right)\left(z+1\right)\,\mathrm{hypergeom}\left(\left[a\right]\,\left[b\right]\,z\right)=\mathrm{HeunC}\left(1\,b-1\,1\,-\frac{b}{2}+a\,\frac{b}{2}-a+\frac{1}{2}\,-z\right)\left(z+1\right)\,\mathrm{MeijerG}\left(\left[\left[a\right]\,\left[\right]\right]\,\left[\left[0\right]\,\left[b\right]\right]\,z\right)=\frac{\mathrm{\Gamma }\left(1-a\right)\mathrm{HeunC}\left(1\,-b\,1\,\frac{b}{2}-a+\frac{1}{2}\,-\frac{b}{2}+a\,z\right)\left(1-z\right)}{\mathrm{\Gamma }\left(1-b\right)}\,\cos \left(z\right)=\frac{\left(2z+\mathrm{\pi }\right)\mathrm{HeunC}\left(1\,1\,1\,0\,\frac{1}{2}\,\mathrm{-I}\left(2z+\mathrm{\pi }\right)\right)\left(\mathrm{I}\mathrm{\pi }+2\mathrm{I}z+1\right)}{2{\ⅇ}^{\frac{\mathrm{I}}{2}\left(2z+\mathrm{\pi }\right)}}\,\sin \left(z\right)=\frac{\left(2\mathrm{I}{z}^2+z\right)\mathrm{HeunC}\left(1\,1\,1\,0\,\frac{1}{2}\,-2\mathrm{I}z\right)}{{\ⅇ}^{\mathrm{I}z}}\right]$