Abstract
The article addresses the problem whether indefinite double sums involving a generic sequence can be simplified in terms of indefinite single sums. Depending on the structure of the double sum, the proposed summation machinery may provide such a simplification without exceptions. If it fails, it may suggest a more advanced simplification introducing in addition a single nested sum where the summand has to satisfy a particular constraint. More precisely, an explicitly given parameterized telescoping equation must hold. Restricting to the case that the arising unspecified sequences are specialized to the class of indefinite nested sums defined over hypergeometric, multi-basic or mixed hypergeometric products, it can be shown that this constraint is not only sufficient but also necessary.
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Notes
- 1.
Freely available with password request at http://www.risc.jku.at/research/combinat/software/Sigma/.
- 2.
If a simpler expression exists, Sigma would find it with the same options as described in Remark 1.1.
- 3.
\(\mathbb K\) is a field of characteristic 0.
- 4.
The quotient ring of \(\mathbb K_X[k,\{X_k\}]\) subject to the equivalence relation \(\equiv \); this ring is a subring of \(\mathrm {Seq}(\mathbb K_X)\).
- 5.
This means that \(q_d(k)\) is not equivalent to the 0-sequence \((\dots , 0,0,0,\dots )\in \mathbb K_X^\mathbb Z\).
- 6.
Note that \(s(a)=0\) if \(a<0\).
- 7.
By difference ring theory (see Lemma 4.13 below) the exponent with which F(k) can appear in G(k) is at most 2. As it turns out, exponent 1 suffices here to obtain a solution of the desired form.
- 8.
Note that \(F(-1)=0\) by definition of a generic sum.
- 9.
According to (17): \(G(-1)=0\).
- 10.
By using the option RefinedForwardShift\(\rightarrow \)False, Sigma follows the calculation steps carried out above. Without this option a more complicated (but more efficient) strategy is used that produces a slight variation of the output.
- 11.
Note that \({\text {const}}(\mathbb A,\sigma )\) in general is a subring of \(\mathbb A\).
- 12.
Note that \(\mathbb K\subseteq \mathbb K_X\) and thus the evaluation of a sum has been defined already in (5).
- 13.
In the theorem we require that the set of constants form a field. However, if \({\text {const}}(\mathbb A[s],\sigma )={\text {const}}(\mathbb A,\sigma )\), to prove the non-existence of a telescoping solution one does not need to assume that \({\text {const}}(\mathbb A,\sigma )\) is a field.
- 14.
- 15.
Note that \(\mathbb K\subseteq \mathbb K_X\) and thus the evaluation of a sum has been defined already in (5).
- 16.
Algorithmically, one starts with a base field K (like \(\mathbb Q\) or \(\mathbb Q(n)\)) and constructs —if necessary— a finite algebraic extension of it such that statement (1) is true.
- 17.
This means that \(\tau (\sum _{i=0}^r f_i\,s^i)\equiv \sum _{i=0}^r\tau (f_i)\big (\big (\sum _{k=0}^n \bar{X}_k\big )^i\big )_{n\ge 0}\) for \(f_0,\dots ,f_r\in \mathbb A\).
- 18.
In the q-case (resp. in the mixed case) we also have to replace z by \(q^k\) (resp. \(z_i\) by \(q_i^k\) for \(1\le i\le v\)).
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Acknowledgements
We would like to thank Christian Krattenthaler for inspiring discussions. Special thanks go to Bill Chen and his collaborators at the Center of Applied Mathematics at the Tianjin University for overwhelming hospitality in the endspurt phase of writing up this paper. We are especially grateful for all the valuable and detailed suggestions of the referee that improved substantially the quality of this article.
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Paule, P., Schneider, C. (2019). Towards a Symbolic Summation Theory for Unspecified Sequences. In: Blümlein, J., Schneider, C., Paule, P. (eds) Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-04480-0_15
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