A toolbox to solve coupled systems of differential and difference equations

by   Jakob Ablinger, et al.

We present algorithms to solve coupled systems of linear differential equations, arising in the calculation of massive Feynman diagrams with local operator insertions at 3-loop order, which do not request special choices of bases. Here we assume that the desired solution has a power series representation and we seek for the coefficients in closed form. In particular, if the coefficients depend on a small parameter (the dimensional parameter), we assume that the coefficients themselves can be expanded in formal Laurent series w.r.t. and we try to compute the first terms in closed form. More precisely, we have a decision algorithm which solves the following problem: if the terms can be represented by an indefinite nested hypergeometric sum expression (covering as special cases the harmonic sums, cyclotomic sums, generalized harmonic sums or nested binomial sums), then we can calculate them. If the algorithm fails, we obtain a proof that the terms cannot be represented by the class of indefinite nested hypergeometric sum expressions. Internally, this problem is reduced by holonomic closure properties to solving a coupled system of linear difference equations. The underlying method in this setting relies on decoupling algorithms, difference ring algorithms and recurrence solving. We demonstrate by a concrete example how this algorithm can be applied with the new Mathematica package SolveCoupledSystem which is based on the packages Sigma, HarmonicSums and OreSys. In all applications the representation in x-space is obtained as an iterated integral representation over general alphabets, generalizing Poincaré iterated integrals.


Calculating Three Loop Ladder and V-Topologies for Massive Operator Matrix Elements by Computer Algebra

Three loop ladder and V-topology diagrams contributing to the massive op...

Extensions of the AZ-algorithm and the Package MultiIntegrate

We extend the (continuous) multivariate Almkvist-Zeilberger algorithm in...

Algorithms to solve coupled systems of differential equations in terms of power series

Using integration by parts relations, Feynman integrals can be represent...

Computer algebra tools for Feynman integrals and related multi-sums

In perturbative calculations, e.g., in the setting of Quantum Chromodyna...

Solving linear difference equations with coefficients in rings with idempotent representations

We introduce a general reduction strategy that enables one to search for...

Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams

We calculate 3-loop master integrals for heavy quark correlators and the...

Discovering and Proving Infinite Pochhammer Sum Identities

We consider nested sums involving the Pochhammer symbol at infinity and ...

Please sign up or login with your details

Forgot password? Click here to reset