HomotopySum - Maple Help
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SumTools[IndefiniteSum]

 HomotopySum
 compute closed forms of indefinite sums of expressions containing unspecified functions

 Calling Sequence HomotopySum(E, k)

Parameters

 E - any algebraic expression k - name, specifies the summation index

Description

 • 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.

Notes

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

Examples

 > $\mathrm{with}\left({\mathrm{SumTools}}_{\mathrm{IndefiniteSum}}\right):$
 > $E≔{u}_{k+1}-{u}_{k}$
 ${E}{≔}{{u}}_{{k}{+}{1}}{-}{{u}}_{{k}}$ (1)
 > $\mathrm{HomotopySum}\left(E,k\right)$
 ${{u}}_{{k}}$ (2)
 > $E≔\frac{1}{{u}_{k+2}+{u}_{k+1}}-\frac{1}{{u}_{k+1}+{u}_{k}}$
 ${E}{≔}\frac{{1}}{{{u}}_{{k}{+}{2}}{+}{{u}}_{{k}{+}{1}}}{-}\frac{{1}}{{{u}}_{{k}{+}{1}}{+}{{u}}_{{k}}}$ (3)
 > $\mathrm{HomotopySum}\left(E,k\right)$
 $\frac{{1}}{{{u}}_{{k}{+}{1}}{+}{{u}}_{{k}}}$ (4)

If no anti-difference is found, HomotopySum minimizes the number of terms remaining unsummed, as well as the order of their summation indices.

 > $E≔2{u}_{k+3}^{2}{u}_{k+2}-{u}_{k+1}^{2}{u}_{k}+{u}_{k+2}$
 ${E}{≔}{-}{{u}}_{{k}}{}{{u}}_{{k}{+}{1}}^{{2}}{+}{2}{}{{u}}_{{k}{+}{3}}^{{2}}{}{{u}}_{{k}{+}{2}}{+}{{u}}_{{k}{+}{2}}$ (5)
 > $\mathrm{HomotopySum}\left(E,k\right)$
 ${2}{}{{u}}_{{k}}{}{{u}}_{{k}{+}{1}}^{{2}}{+}{2}{}{{u}}_{{k}{+}{2}}^{{2}}{}{{u}}_{{k}{+}{1}}{+}{{u}}_{{k}}{+}{{u}}_{{k}{+}{1}}{+}{\sum }_{{k}}{}\left({{u}}_{{k}}{}{{u}}_{{k}{+}{1}}^{{2}}{+}{{u}}_{{k}}\right)$ (6)

The input expression may contain combinations of specified and unspecified functions of the summation index.

 > $E≔\mathrm{expand}\left({\left(k+1\right)}^{3}{u}_{k+2}{v}_{k+1}^{5}+{v}_{k+2}^{3}-{k}^{3}{u}_{k+1}{v}_{k}^{5}\right)$
 ${E}{≔}{-}{{k}}^{{3}}{}{{u}}_{{k}{+}{1}}{}{{v}}_{{k}}^{{5}}{+}{{k}}^{{3}}{}{{u}}_{{k}{+}{2}}{}{{v}}_{{k}{+}{1}}^{{5}}{+}{3}{}{{k}}^{{2}}{}{{u}}_{{k}{+}{2}}{}{{v}}_{{k}{+}{1}}^{{5}}{+}{3}{}{k}{}{{u}}_{{k}{+}{2}}{}{{v}}_{{k}{+}{1}}^{{5}}{+}{{u}}_{{k}{+}{2}}{}{{v}}_{{k}{+}{1}}^{{5}}{+}{{v}}_{{k}{+}{2}}^{{3}}$ (7)
 > $\mathrm{HomotopySum}\left(E,k\right)$
 ${{k}}^{{3}}{}{{u}}_{{k}{+}{1}}{}{{v}}_{{k}}^{{5}}{+}{{v}}_{{k}}^{{3}}{+}{{v}}_{{k}{+}{1}}^{{3}}{+}\left({\sum }_{{k}}{}{{v}}_{{k}}^{{3}}\right)$ (8)

References

 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.