Term Algebras, Canonical Representations and Difference Ring Theory for Symbolic Summation

Analytic integration methods in quantum field theory: an Introduction

“…During the following years more and more of these structures have been revealed. Here the difference ring techniques [12,[51][52][53] played an essential role, since the corresponding structures were found constructively, mostly in massive calculations. The Mellin transform of these quantities allowed then to find the associated iterated integrals.…”

“…The topics of the workshop included both techniques to reduce the number of Feynman diagrams by physical relations, such as the integration by parts relations [5][6][7] as well as the mathematical methods to compute these integrals analytically. The latter include the method of generalized hypergeometric functions [8] and the general theory of contiguous relations [9], the methods of integer relations [10], guessing methods of one-dimensional quantities, hyperlogarithms [11], the solution of master-integrals using difference and differential equations [12][13][14][15], Risch algorithms on nested integrals and rationalization algorithms [16], holonomic integration [17], the multivalued Almkvist-Zeilberger algorithm [18], expansion by regions [19], elliptic integrals and related topics [20,21], cutting techniques [22], and special multi-leg applications [23][24][25]. In different precision calculations these technologies are applied.…”

The SAGEX Review on Scattering Amplitudes, Chapter 4: Multi-loop Feynman Integrals

“…Basically all these tools have been generalized to the setting of difference fields [55] and rings [56] (utilizing results from above and [57][58][59][60][61]) that allows one to find such solutions for difference equations (6) where the coefficients a i (n) and the inhomogeneous part r(n) are not just rational functions but can be built again by indefinite nested sums over hypergeometric products. E.g., using the summation package Sigma [62][63][64], that contains this general toolbox, one can compute for the recurrence…”

Error bounds for the asymptotic expansion of the partition function

“…We note that for 0 ≤ m ≤ 10; n ≥ 1 > n 0 (2m + 1), and therefore, Using the symbolic summation package Sigma [27] and its underlying machinery in the setting of difference rings [29] the inner sum S 3 (t, u) can be simplified as follows. Recall from (5.33) that As a result we get a homogeneous linear recurrence of order 2 for S 3 (t, u) = SUM[u](=mySum3).…”

“…In an effort to estimate the inner sum u s=0 ψ(s), the use of the symbolic summation tool Sigma [27] was essential. Schneider considered [27][28][29] a broader algorithmic framework that subsumes the theory of difference field and ring extensions together with the method of creative telescoping. This algorithmic tool began to be aimed at a wider class of multi-sums, most frequently encountered in problems of enumerative combinatorics.…”