This function simplifies an expression f involving powers, factorials, GAMMA function terms, binomial coefficients, and Pochhammer symbols by converting factorials, binomial coefficients, and Pochhammer symbols into GAMMA function terms, and applying simplify to its result. If the output is not rational, it is given in terms of GAMMA functions.

The command with(sumtools,simpcomb) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{sumtools}\right)\:$

$\mathrm{simpcomb}\left(\mathrm{binomial}\left(n\,k\right)\right)$

$\frac{\mathrm{\Gamma }\left(n+1\right)}{\mathrm{\Gamma }\left(n-k+1\right)\mathrm{\Gamma }\left(k+1\right)}$

$\mathrm{simpcomb}\left(\frac{\frac{\mathrm{binomial}\left(n\,k\right)}{2^n}-\frac{\mathrm{binomial}\left(n-1\,k\right)}{2^{n-1}}}{\frac{\mathrm{binomial}\left(n\,k-1\right)}{2^n}-\frac{\mathrm{binomial}\left(n-1\,k-1\right)}{2^{n-1}}}\right)$

$-\frac{\left(2k-n\right)\left(-n+k-1\right)}{k\left(2k-n-2\right)}$

$\mathrm{simpcomb}\left(\frac{\mathrm{binomial}\left(n\,k\right)-\mathrm{binomial}\left(n-2\,k\right)}{\mathrm{binomial}\left(n-3\,k\right)-\mathrm{binomial}\left(n-6\,k\right)}\right)$

$\frac{\left(n-5\right)\left(n-4\right)\left(n-3\right)\left(n-2\right)\left(k-2n+1\right)}{\left(k^2-3kn+3n^2+12k-24n+47\right)\left(k+2-n\right)\left(k+1-n\right)\left(k-n\right)}$