Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Using integration by parts relations, Feynman integrals can be represented in terms of coupled systems of differential equations. In the following  suppose that the unknown Feynman integrals can be given in por series representations, and that sufficiently many initial values of the integrals are given. Then there exist algorithms that decide constructively if the coefficients of their por series representations can be given within the class of nested sums over hypergeometric products. In this article  will work out the calculation steps that solve this problem. First,  will present a successful tactic that has been applied recently to challenging problems coming from massive 3-loop Feynman integrals. Here our main tool is to solve scalar linear recurrences within the class of nested sums over hypergeometric products. Second,  will present a new variation of this tactic which relies on more involved summation technologies but succeeds in reducing the problem to solve scalar recurrences with lor recurrence orders. The article will work out the different challenges of this new tactic and demonstrates how they can be treated efficiently with our existing summation technologies.
['J. Ablinger', 'A. Behring', 'J. Bluemlein', 'Abilio De Freitas', 'Carsten Schneider']
I am writing a research paper on a idea Algorithms to solve coupled systems of differential equations in terms of power series
[{"from": "human", "value": "I am writing a research paper on a idea Algorithms to solve coupled systems of differential equations in terms of power series"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Using integration by parts relations, Feynman integrals can be represented in terms of coupled systems of differential equations. In the following  suppose that the unknown Feynman integrals can be given in por series representations, and that sufficiently many initial values of the integrals are given. Then there exist algorithms that decide constructively if the coefficients of their por series representations can be given within the class of nested sums over hypergeometric products. In this article  will work out the calculation steps that solve this problem. First,  will present a successful tactic that has been applied recently to challenging problems coming from massive 3-loop Feynman integrals. Here our main tool is to solve scalar linear recurrences within the class of nested sums over hypergeometric products. Second,  will present a new variation of this tactic which relies on more involved summation technologies but succeeds in reducing the problem to solve scalar recurrences with lor recurrence orders. The article will work out the different challenges of this new tactic and demonstrates how they can be treated efficiently with our existing summation technologies."}]