 CreativeTelescoping - Maple Help

SumTools[DefiniteSum]

 CreativeTelescoping
 compute closed forms of definite sums using creative telescoping method Calling Sequence CreativeTelescoping(f, k=m..n) Parameters

 f - hypergeometric term k - name; summation index m, n - expressions or integers Description

 • The CreativeTelescoping(f, k=m..n) command computes a closed form of the definite sum of f over the specified range of k using creative telescoping method. Examples

 > $\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{DefiniteSum}\right]\right):$
 > $F≔{\mathrm{binomial}\left(2n,2k\right)}^{2}$
 ${F}{≔}{\left(\genfrac{}{}{0}{}{{2}{}{n}}{{2}{}{k}}\right)}^{{2}}$ (1)
 > $\mathrm{Sum}\left(F,k=0..n\right)=\mathrm{CreativeTelescoping}\left(F,k=0..n\right)$
 ${\sum }_{{k}{=}{0}}^{{n}}{}{\left(\genfrac{}{}{0}{}{{2}{}{n}}{{2}{}{k}}\right)}^{{2}}{=}\frac{{\left({-1}\right)}^{{n}}{}\left(\genfrac{}{}{0}{}{{2}{}{n}}{{n}}\right)}{{2}}{+}\frac{\left(\genfrac{}{}{0}{}{{4}{}{n}}{{2}{}{n}}\right)}{{2}}$ (2) References

 van Hoeij, M. "Finite Singularities and Hypergeometric Solutions of Linear Recurrence Equations." Journal of Pure and Applied Algebra. Vol. 139. (1999): 109-131.
 Zeilberger, D. "The Method of Creative Telescoping." Journal of Symbolic Computing. Vol. 11. (1991): 195-204.