An Algorithmic Approach to the q -Summability Problem of Bivariate Rational Functions
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An Algorithmic Approach to the q-Summability Problem of Bivariate Rational Functions∗ WANG Rong-Hua
DOI: 10.1007/s11424-020-9391-6 Received: 17 September 2019 / Revised: 25 November 2019 c The Editorial Office of JSSC & Springer-Verlag GmbH Germany 2020 Abstract In 2014, Chen and Singer solved the summability problem of bivariate rational functions. Later an algorithmic proof was presented by Hou and the author. In this paper, the algorithm will be simplified and adapted to the q-case. Keywords
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Gosper’s algorithm, polynomial residues, q-summability.
Introduction
Symbolic summation is a classical topic in combinatorics and mathematical physics. One of the central problems in symbolic summation is to decide whether a given sum can be expressed in a closed form, which was fully answered by Gosper’s algorithm[1] for indefinite summations of hypergeometric terms. Based on Gosper’s algorithm, Zeilberger[2, 3] designed a new algorithm to find recurrence relations for single sums of hypergeometric terms, which is known as Zeilberger’s algorithm or the method of creative telescoping. Over the past two decades, more efficient algorithms have been developed for (q-)hypergeometric terms[4–6] . Gosper’s and Zeilberger’s algorithms occupy a central position in the study of mechanical proofs of combinatorial identities. The crucial step of both algorithms is to decide whether a given term T (n) can be written as the difference of another term. If such a term exists, T (n) is said to be summable. Deciding whether a given term is summable or not is the so-called summability problem. For univariate functions, the summability problem has been solved rather successfully. For example, Abramov[7, 8] solved the summability problem for rational functions. Gosper’s algorithm[1] settles the summability problem for hypergeometric terms and was later generalized to the D-finite case by Abramov and van Hoeij[9] and to the difference-field setting by Karr[10, 11] . WANG Rong-Hua School of Mathematical Sciences, Tiangong University, Tianjin 300387, China. Email: [email protected]. ∗ This paper was supported by the National Natural Science Foundation of China under Grant No. 11871067 and the Natural Science Foundation of Tianjin under Grant No. 19JCQNJC14500. This paper was recommended for publication by Editor-in-Chief GAO Xiao-Shan.
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WANG RONG-HUA
Passing from the univariate case to the multivariate case, the summability problem becomes much more complicated. Significant progress has been made by Apagodu and Zeilberger[12] , Koutschan[13] , Schneider[14] and Chen, et al.[15] . However, they did not provide a complete answer to the summability problem of bivariate functions. The first necessary and sufficient condition for the summability of bivariate functions was presented by Chen and Singer[16] for the rational case, and later extended to the remaining mixed cases by Chen in [17]. Based on the theoretical criterion given in [16], Hou and the author[18] presented a new criterion and an algorithm for deciding the summability of bivari
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