compute closed forms of indefinite sums of expressions containing unspecified functions
any algebraic expression
name, specifies the summation index
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 uk.
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.
E ≔ uk+1−uk
E ≔ 1uk+2+uk+1−1uk+1+uk
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⁢uk+32⁢uk+2−uk+12⁢uk+uk+2
The input expression may contain combinations of specified and unspecified functions of the summation index.
E ≔ expand⁡k+13⁢uk+2⁢vk+15+vk+23−k3⁢uk+1⁢vk5
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.
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