A Symbolic Summation Approach to Feynman Integral Calculus

Given a Feynman parameter integral, depending on a single discrete variable $N$ and a real parameter $\epsilon$, we discuss a new algorithmic framework to compute the first coefficients of its Laurent series expansion in $\epsilon$. In a first step, the integrals are expressed by hypergeometric multi-sums by means of symbolic transformations. Given this sum format, we develop new summation tools to extract the first coefficients of its series expansion whenever they are expressible in terms of indefinite nested product-sum expressions. In particular, we enhance the known multi-sum algorithms to derive recurrences for sums with complicated boundary conditions, and we present new algorithms to find formal Laurent series solutions of a given recurrence relation.

💡 Research Summary

The paper introduces a comprehensive algorithmic framework for the symbolic evaluation of Feynman parameter integrals that depend on a single discrete index N and a continuous regulator ε. The authors start by converting the original integral into a hypergeometric multi‑sum representation using an extended set of symbolic transformation rules. This conversion is non‑trivial because the integration limits and the ε‑dependence intertwine: the upper and lower bounds of the sums become functions of N, while the summand itself contains rational functions of N, the summation variables, and ε. By expressing the integral as a nested sum of the form
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