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Symbolic Summation in Difference Rings and Applications

Symbolic summation started with Abramov (1971) for rational sequences and has been pushed forward by Gosper (1978), Zeilberger (1991), Petkovsek (1992) and Paule (1995) to tackle indefinite and definite sums for hypergeometric expressions. In the last decade the class of input sums has been extended significantly and covers, for instance, hypergeometric multi-sums, holonomic sequences, unspecified sequences, radical expressions, Stirling numbers, etc. In this talk we will focus on a new difference ring approach. The foundation was led by Karr's summation algorithm (1981) which enables one to rephrase indefinite nested sums and products in the setting of difference fields. Many new ideas have been incorporated into a strong summation theory which led to new algorithms for the summation paradigms of telescoping, creative telescoping and recurrence solving. However, this elegant difference field approach has one central drawback. Alternating signs cannot be represented in such a field: zero-divisors are introduced which can be formulated only within a ring. We will present a class of difference rings in which one can represent algorithmically indefinite nested sums and products together with the alternating sign, and more generally products over primitive roots of unity. In this setting we can represent all indefinite nested sums defined over hypergeometric expressions. In particular, this construction produces expressions in terms of sums that are all algebraically independent over each other. As a consequence, the derived output of a nested product-sum expression solves the zero-recognition problem: the computed expression evaluates to the zero-sequence if and only if the expression has been simplified to zero. In combination with improved parameterized telescoping algorithms and recurrence solvers within such difference rings we obtain an efficient summation machinery that has been built into the summation package Sigma. We will illustrate the different summation techniques by large scale problems coming from the field of particle physics.