Converts the functions found in expr, including sum and Sum, to their hypergeometric forms when possible.

For sums, no attempt is made to ensure that the resulting hypergeometric function is convergent or terminates.

convert/hypergeom reacts to the setting of the environment variable _EnvFormal (for more information, see sum/details). When that variable is set to $\mathrm{true}$, the conversion will be attempted regardless of whether the original sum may be divergent or not.

$\mathrm{Sum}\left(\frac{1}{k\mathrm{binomial}\left(k+n\,k\right)}\,k=1..\mathrm{\infty }\right)$

$\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k\left(\genfrac{}{}{0ex}{}{k+n}{k}\right)}$

$\mathrm{convert}\left(\&comma;\mathrm{hypergeom}\right)\mathbf{assuming}0<n$

$\frac{1}{n}$

$\mathrm{BesselJ}\left(a\&comma;z\right)$

$\mathrm{BesselJ}\left(a\&comma;z\right)$

$=\mathrm{convert}\left(\&comma;\mathrm{hypergeom}\right)$

$\mathrm{BesselJ}\left(a\&comma;z\right)=\frac{z^a\mathrm{hypergeom}\left(\left[\right]\&comma;\left[a+1\right]\&comma;-\frac{z^2}{4}\right)}{\mathrm{\Gamma }\left(a+1\right)2^a}$

$\mathrm{LegendreP}\left(a\&comma;b\&comma;z\right)$

$\mathrm{LegendreP}\left(a\&comma;b\&comma;z\right)$

$=\mathrm{convert}\left(\&comma;\mathrm{hypergeom}\right)$

$\mathrm{LegendreP}\left(a\&comma;b\&comma;z\right)=\frac{{\left(z+1\right)}^{\frac{b}{2}}\mathrm{hypergeom}\left(\left[-a\&comma;a+1\right]\&comma;\left[1-b\right]\&comma;\frac{1}{2}-\frac{z}{2}\right)}{{\left(z-1\right)}^{\frac{b}{2}}\mathrm{\Gamma }\left(1-b\right)}$

$\mathrm{KummerU}\left(a\&comma;b\&comma;z\right)$

$\mathrm{KummerU}\left(a\&comma;b\&comma;z\right)$

$=\mathrm{convert}\left(\&comma;\mathrm{hypergeom}\right)$

$\mathrm{KummerU}\left(a\&comma;b\&comma;z\right)=\frac{\mathrm{\Gamma }\left(-1+b\right)\mathrm{hypergeom}\left(\left[a-b+1\right]\&comma;\left[2-b\right]\&comma;z\right)}{z^{-1+b}\mathrm{\Gamma }\left(a\right)}+\frac{\mathrm{\Gamma }\left(1-b\right)\mathrm{hypergeom}\left(\left[a\right]\&comma;\left[b\right]\&comma;z\right)}{\mathrm{\Gamma }\left(a-b+1\right)}$

$=\mathrm{convert}\left(\&comma;\mathrm{hypergeom}\right)\mathbf{assuming}a::\mathrm{negint}$

$\mathrm{KummerU}\left(a\&comma;b\&comma;z\right)=\mathrm{pochhammer}\left(a-b+1\&comma;-a\right)\mathrm{hypergeom}\left(\left[a\right]\&comma;\left[b\right]\&comma;z\right)$

$\mathrm{convert}\left(\sin \left(z\right)\&comma;\mathrm{hypergeom}\right)$

$\sin \left(z\right)$

$\sin \left(z\right)$

$\sin \left(z\right)$

$=\mathrm{convert}\left(\&comma;\mathrm{hypergeom}\&comma;\mathrm{include}=\mathrm{all}\right)$

$\sin \left(z\right)=z\mathrm{hypergeom}\left(\left[\right]\&comma;\left[\frac{3}{2}\right]\&comma;-\frac{z^2}{4}\right)$

$S≔\mathrm{Sum}\left(\frac{x^n}{n}\&comma;n=1..\mathrm{\infty }\right)$

$S≔\sum _{n=1}^{\mathrm{\infty }}\frac{x^n}{n}$

$\mathrm{convert}\left(S\&comma;\mathrm{hypergeom}\right)$

$\sum _{n=1}^{\mathrm{\infty }}\frac{x^n}{n}$

$\mathrm{convert}\left(S\&comma;\mathrm{hypergeom}\right)\mathbf{assuming}\mathrm{abs}\left(x\right)<1$

$-\ln \left(1-x\right)$

$\mathrm{\_EnvFormal}≔\mathrm{true}$

$\mathrm{\_EnvFormal}≔\mathrm{true}$

$\mathrm{convert}\left(S\&comma;\mathrm{hypergeom}\right)$

$-\ln \left(1-x\right)$