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 $j\&comma;k<0$.  Then if $\mathrm{f1}-\mathrm{f2}-\mathrm{f3}\&equals;n$, an integer, the product is multiplied by

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

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

Transformation 2: given a product

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

Case 1: $1<\left|r\right|$.  Multiply by   $\frac{\mathrm{f2}\&excl;\mathrm{binomial}\left(\mathrm{f1}\&comma;\mathrm{f2}\right)\left(\mathrm{f1}-\mathrm{f2}\right)\&excl;}{\mathrm{f1}\&excl;}$

Case 2: $\left|r\right|<1$.  Multiply by   $\frac{\mathrm{f2}\&excl;}{\mathrm{f1}\&excl;\mathrm{binomial}\left(\mathrm{f2}\&comma;\mathrm{f1}\right)\left(\mathrm{f2}-\mathrm{f1}\right)\&excl;}$

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

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

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

$\mathrm{binomial}\left(n\&comma;k\right)$

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

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

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

$\frac{n\mathrm{binomial}\left(n^2\&plus;m\&comma;k\right)}{n^2-k\&plus;m\&plus;1}$

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

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

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

$\frac{1}{\mathrm{binomial}\left(3m\&comma;m\right)\mathrm{binomial}\left(2m\&comma;m\right)}$

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

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

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

$m\left(m\&plus;1\right)\mathrm{binomial}\left(m\&plus;\frac{1}{2}\&comma;-\frac{1}{2}\right)$