The WZMethod(f,r,n,k,cert) command certifies identities of the form $\sum _{k}f\left(n\,k\right)=r\left(n\right)$.

Let $F\left(n\,k\right)=\frac{f\left(n\,k\right)}{r\left(n\right)}$ if $r\left(n\right)\ne 0$ and $F\left(n\,k\right)=f\left(n\,k\right)$, otherwise. If the method succeeds in certifying the given identity, the output is a list of two elements $\left[F\,G\right]$ representing the WZ-pair $F,G$ such that $F\left(n+1\,k\right)-F\left(n\,k\right)=G\left(n\,k+1\right)-G\left(n\,k\right)$. Otherwise, it returns the error message "WZ method fails".

If the method is successful and if the fifth optional argument cert is given, cert is assigned the WZ certificate $R\left(n\,k\right)=\frac{G\left(n\,k\right)}{F\left(n\,k\right)}$.

It is assumed that for each integer $0\le n$, $\underset{k\to \mathrm{\infty }}{lim}G\left(n\,k\right)=0$ and $\underset{k\to -\mathrm{\infty }}{lim}G\left(n\,k\right)=0$.

$\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{Hypergeometric}\right]\right)\:$

$f≔\frac{\left(n+k\right)!\left(b+k\right)!\left(c-n-1\right)!\left(c-b-1\right)!}{\left(c+k\right)!\left(n-1\right)!\left(c-n-b-1\right)!\left(k+1\right)!\left(b-1\right)!}$

$f≔\frac{\left(n+k\right)!\left(b+k\right)!\left(c-n-1\right)!\left(c-b-1\right)!}{\left(c+k\right)!\left(n-1\right)!\left(c-n-b-1\right)!\left(k+1\right)!\left(b-1\right)!}$

$r≔1$

$r≔1$

$\mathrm{WZpair}≔\mathrm{WZMethod}\left(f\,r\,n\,k\,'\mathrm{cert}'\right)\:$

$F≔\mathrm{WZpair}\left[1\right]$

$F≔\frac{\left(n+k\right)!\left(b+k\right)!\left(c-n-1\right)!\left(c-b-1\right)!}{\left(c+k\right)!\left(n-1\right)!\left(c-n-b-1\right)!\left(k+1\right)!\left(b-1\right)!}$

$G≔\mathrm{WZpair}\left[2\right]$

$G≔-\frac{\left(\left(c-n-2-b\right)!n!\left(n+k\right)!\left(c-n-1\right)!-\left(c-n-2\right)!\left(n+1+k\right)!\left(n-1\right)!\left(c-n-b-1\right)!\right)\left(b+k\right)!\left(c-b-1\right)!\left(c+k\right)\left(k+1\right)}{\left(c-n-b-1\right)!\left(n-1\right)!\left(b-1\right)!\left(k+1\right)!\left(c-n-2-b\right)!n!\left(c+k\right)!\left(bk+bn-ck+nk+b-c+k+n+1\right)}$

$\mathrm{cert}$

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

$F≔{\left(-1\right)}^k\mathrm{binomial}\left(n+b\,n+k\right)\mathrm{binomial}\left(n+c\,c+k\right)\mathrm{binomial}\left(b+c\,b+k\right)$

$F≔{\left(\mathrm{-1}\right)}^k\left(\genfrac{}{}{0ex}{}{n+b}{n+k}\right)\left(\genfrac{}{}{0ex}{}{n+c}{c+k}\right)\left(\genfrac{}{}{0ex}{}{b+c}{b+k}\right)$

$r≔\frac{\left(n+b+c\right)!}{n!b!c!}$

$r≔\frac{\left(n+b+c\right)!}{n!b!c!}$

$\mathrm{WZpair}≔\mathrm{WZMethod}\left(F\,r\,n\,k\,'\mathrm{certificate}'\right)\:$

$F≔\mathrm{WZpair}\left[1\right]$

$F≔\frac{{\left(\mathrm{-1}\right)}^k\left(\genfrac{}{}{0ex}{}{n+b}{n+k}\right)\left(\genfrac{}{}{0ex}{}{n+c}{c+k}\right)\left(\genfrac{}{}{0ex}{}{b+c}{b+k}\right)n!b!c!}{\left(n+b+c\right)!}$

$G≔\mathrm{WZpair}\left[2\right]$

$G≔-\frac{\left(\left(\genfrac{}{}{0ex}{}{n+1+b}{n+1+k}\right)\left(\genfrac{}{}{0ex}{}{n+1+c}{c+k}\right)\left(n+1\right)!\left(n+b+c\right)!-\left(n+1+b+c\right)!\left(\genfrac{}{}{0ex}{}{n+b}{n+k}\right)\left(\genfrac{}{}{0ex}{}{n+c}{c+k}\right)n!\right)c!b!\left(\genfrac{}{}{0ex}{}{b+c}{b+k}\right){\left(\mathrm{-1}\right)}^k\left(c+k\right)\left(n+1+k\right)\left(b+k\right)}{2\left(n+b+c\right)!\left(n+1+b+c\right)!\left(bcn+b{k}^2+c{k}^2+k^{2}n+cb+k^2\right)}$

$\mathrm{certificate}$

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

Wilf, H., and Zeilberger, D. "Rational function certify combinatorial identities." J. Amer. Math. Soc. Vol. 3. (1990): 147-158.