This option to convert converts the GAMMA function and factorials in an expression e to binomial coefficients. The code performs the following two transformations on products of factorials and the GAMMA function.

Transformation 1: given a product

where $i,j,k$ are integers. Note, the code handles the case where $\mathrm{f1}$, $\mathrm{f2}$, and $\mathrm{f3}$ are GAMMA functions, and also the special case $\sqrt{\mathrm{\pi }}=\mathrm{\Gamma }\left(\frac{1}{2}\right)$.

Case 1: $0<i$ and $j,k<0$.  Then if $\mathrm{f1}-\mathrm{f2}-\mathrm{f3}=n$, an integer, the product is multiplied by

where $c$ is a correction factor depending on $n$ and $\mathrm{f3}$.

Similarly, CASE 2: where $i,0<j$, $k<0$.  This is the case where the binomial appears in the denominator.  Then if $\mathrm{f3}-\mathrm{f1}-\mathrm{f2}=n$, an integer, the product is multiplied by

Transformation 2: given a product

where $i,j$ are integers and $\frac{\mathrm{f1}}{\mathrm{f2}}$ is a rational constant $r$.

Case 1: $1<\left|r\right|$.  Multiply by   $\frac{\mathrm{f2}!\left(\genfrac{}{}{0ex}{}{\mathrm{f1}}{\mathrm{f2}}\right)\left(\mathrm{f1}-\mathrm{f2}\right)!}{\mathrm{f1}!}$

Case 2: $\left|r\right|<1$.  Multiply by   $\frac{\mathrm{f2}!}{\mathrm{f1}!\left(\genfrac{}{}{0ex}{}{\mathrm{f2}}{\mathrm{f1}}\right)\left(\mathrm{f2}-\mathrm{f1}\right)!}$

$a≔\frac{n\&excl;}{k\&excl;\left(n-k\right)\&excl;}$

$a≔\frac{n!}{k!\left(n-k\right)!}$

$\mathrm{convert}\left(a\&comma;\mathrm{binomial}\right)$

$\left(\genfrac{}{}{0ex}{}{n}{k}\right)$

$a≔\frac{n\left(n^2\&plus;m-k\&plus;2\right)\left(n^2\&plus;m\right)\&excl;}{k\&excl;\left(n^2\&plus;m-k\&plus;2\right)\&excl;}$

$a≔\frac{n\left(n^2-k+m+2\right)\left(n^2+m\right)!}{k!\left(n^2-k+m+2\right)!}$

$\mathrm{convert}\left(a\&comma;\mathrm{binomial}\right)$

$\frac{n\left(\genfrac{}{}{0ex}{}{n^2+m}{k}\right)}{n^2-k+m+1}$

$a≔\frac{{m\&excl;}^3}{\left(3m\right)\&excl;}$

$a≔\frac{{m!}^3}{\left(3m\right)!}$

$\mathrm{convert}\left(a\&comma;\mathrm{binomial}\right)$

$\frac{1}{\left(\genfrac{}{}{0ex}{}{3m}{m}\right)\left(\genfrac{}{}{0ex}{}{2m}{m}\right)}$

$a≔\frac{\mathrm{GAMMA}\left(m\&plus;\frac{3}{2}\right)}{\sqrt{\mathrm{Pi}}\mathrm{GAMMA}\left(m\right)}$

$a≔\frac{\mathrm{\Gamma }\left(m+\frac{3}{2}\right)}{\sqrt{\mathrm{\pi }}\mathrm{\Gamma }\left(m\right)}$

$\mathrm{convert}\left(a\&comma;\mathrm{binomial}\right)$

$m\left(m+1\right)\left(\genfrac{}{}{0ex}{}{m+\frac{1}{2}}{-\frac{1}{2}}\right)$