The HomotopySum command allows for the symbolic summation of expressions containing unspecified functions of a discrete variable. A typical example is HomotopySum(u[k+1]-u[k], k), which returns $u_k$.

HomotopySum uses discrete homotopy methods to find an anti-difference of the given expression - see the references at the end.

This command is based on code written by Bernard Deconinck, Michael A. Nivala, and Matthew S. Patterson.

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

$E≔u\left[k+1\right]-u\left[k\right]$

$E≔u_{k+1}-u_k$

$\mathrm{HomotopySum}\left(E\,k\right)$

$u_k$

$E≔\frac{1}{u\left[k+2\right]+u\left[k+1\right]}-\frac{1}{u\left[k+1\right]+u\left[k\right]}$

$E≔\frac{1}{u_{k+2}+u_{k+1}}-\frac{1}{u_{k+1}+u_k}$

$\mathrm{HomotopySum}\left(E\,k\right)$

$\frac{1}{u_{k+1}+u_k}$

$E≔2{u\left[k+3\right]}^{2}u\left[k+2\right]-{u\left[k+1\right]}^{2}u\left[k\right]+u\left[k+2\right]$

$E≔-u_k{u}_{k+1}^2+2u_{k+3}^2{u}_{k+2}+u_{k+2}$

$\mathrm{HomotopySum}\left(E\,k\right)$

$2u_k{u}_{k+1}^2+2u_{k+2}^2{u}_{k+1}+u_k+u_{k+1}+\sum _k\left(u_k{u}_{k+1}^2+u_k\right)$

$E≔\mathrm{expand}\left({\left(k+1\right)}^{3}u\left[k+2\right]{v\left[k+1\right]}^5+{v\left[k+2\right]}^3-k^{3}u\left[k+1\right]{v\left[k\right]}^5\right)$

$E≔-k^3{u}_{k+1}{v}_k^5+k^3{u}_{k+2}{v}_{k+1}^5+3k^2{u}_{k+2}{v}_{k+1}^5+3k{u}_{k+2}{v}_{k+1}^5+u_{k+2}{v}_{k+1}^5+v_{k+2}^3$

$\mathrm{HomotopySum}\left(E\,k\right)$

$k^3{u}_{k+1}{v}_k^5+v_k^3+v_{k+1}^3+\left(\sum _k{v}_k^3\right)$

Hereman, W.; Colagrosso, M.; Sayers, R.; Ringler, A.; Deconinck, B.; Nivala, M.; and Hickman, M. "Continuous and Discrete Homotopy Operators with Applications in Integrability Testing." In Differential Equations with Symbolic computation, pp. 255-290. Edited by D. Wang and Z. Zheng. Birkhauser, 2005.