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convert/binomial

Convert to Binomial Form

 Calling Sequence convert(e, binomial)

Parameters

 e - expression

Description

 • 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

$\dots \mathrm{f1}{!}^{i}\mathrm{f2}{!}^{j}\mathrm{f3}{!}^{k}\dots$

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

$\frac{\mathrm{binomial}\left(\mathrm{f1},\mathrm{f2}\right)c\mathrm{f2}!\mathrm{f3}!}{\mathrm{f1}!}$

 where $c$ is a correction factor depending on $n$ and $\mathrm{f3}$.
 Similarly, CASE 2: where $i,0, $k<0$.  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

$\frac{c\mathrm{f3}!}{\mathrm{f1}!\mathrm{f2}!\mathrm{binomial}\left(\mathrm{f2},\mathrm{f1}\right)}$

 • Transformation 2: given a product

$\dots \mathrm{f1}{!}^{i}\mathrm{f2}{!}^{j}\dots$

 where $i,j$ are integers and $\frac{\mathrm{f1}}{\mathrm{f2}}$ is a rational constant $r$.
 Case 1: $1<\left|r\right|$.  Multiply by   $\frac{\mathrm{f2}!\mathrm{binomial}\left(\mathrm{f1},\mathrm{f2}\right)\left(\mathrm{f1}-\mathrm{f2}\right)!}{\mathrm{f1}!}$
 Case 2: $\left|r\right|<1$.  Multiply by   $\frac{\mathrm{f2}!}{\mathrm{f1}!\mathrm{binomial}\left(\mathrm{f2},\mathrm{f1}\right)\left(\mathrm{f2}-\mathrm{f1}\right)!}$

Examples

 > $a≔\frac{n!}{k!\left(n-k\right)!}$
 ${a}{:=}\frac{{n}{!}}{{k}{!}{}\left({n}{-}{k}\right){!}}$ (1)
 > $\mathrm{convert}\left(a,\mathrm{binomial}\right)$
 ${\mathrm{binomial}}{}\left({n}{,}{k}\right)$ (2)
 > $a≔\frac{n\left({n}^{2}+m-k+2\right)\left({n}^{2}+m\right)!}{k!\left({n}^{2}+m-k+2\right)!}$
 ${a}{:=}\frac{{n}{}\left({{n}}^{{2}}{-}{k}{+}{m}{+}{2}\right){}\left({{n}}^{{2}}{+}{m}\right){!}}{{k}{!}{}\left({{n}}^{{2}}{-}{k}{+}{m}{+}{2}\right){!}}$ (3)
 > $\mathrm{convert}\left(a,\mathrm{binomial}\right)$
 $\frac{{n}{}{\mathrm{binomial}}{}\left({{n}}^{{2}}{+}{m}{,}{k}\right)}{{{n}}^{{2}}{-}{k}{+}{m}{+}{1}}$ (4)
 > $a≔\frac{{m!}^{3}}{\left(3m\right)!}$
 ${a}{:=}\frac{{{m}{!}}^{{3}}}{\left({3}{}{m}\right){!}}$ (5)
 > $\mathrm{convert}\left(a,\mathrm{binomial}\right)$
 $\frac{{1}}{{\mathrm{binomial}}{}\left({3}{}{m}{,}{m}\right){}{\mathrm{binomial}}{}\left({2}{}{m}{,}{m}\right)}$ (6)
 > $a≔\frac{\mathrm{Γ}\left(m+\frac{3}{2}\right)}{\sqrt{\mathrm{π}}\mathrm{Γ}\left(m\right)}$
 ${a}{:=}\frac{{\mathrm{Γ}}{}\left({m}{+}\frac{{3}}{{2}}\right)}{\sqrt{{\mathrm{π}}}{}{\mathrm{Γ}}{}\left({m}\right)}$ (7)
 > $\mathrm{convert}\left(a,\mathrm{binomial}\right)$
 ${m}{}\left({m}{+}{1}\right){}{\mathrm{binomial}}{}\left({m}{+}\frac{{1}}{{2}}{,}{-}\frac{{1}}{{2}}\right)$ (8)