@phdthesis{RISC7201,
author = {Nikolai Fadeev},
title = {{Computer algebra for special functions}},
language = {english},
abstract = {Calculations done in different mathematical areas — such as computer algebra,combinatorics, number theory, differential equations — and physical areas — suchas particle physics — give rise to a plethora of problems involving special functionsthat need to be dealt with efficiently. In this PhD, we concentrated on two suchparticular problems.In the first part of this PhD thesis, we explored the relation between iteratedbinomial sums, an extension of general harmonic sums, and their integral representations, in order to compute their asymptotic expansions. To do that in a fullyautomatic way, we created a dedicated package, RICA. Using Mellin representations,we first formalised and extended a classical recursive method to compute Mellininverses of such sums, and together with it implemented several methods to compute asymptotic expansions of such integrals. In the process, we introduced andexplored a new class of functions related to Mellin convolutions. Those allowed usto automatically compute asymptotic expansions for more general classes of sumsin a new and efficient way, while providing a way to get symbolic representationsfor the constants appearing in the calculation of the Mellin inversions.In the second part of this PhD thesis, we studied first order inhomogeneous systems of differential equations involving an extra parameter epsilon coming from particlephysics computations. Since usually those systems could only be solved up to someorder in epsilon, we aimed at developing a method to optimise the solving task of suchsystems. We studied an approach centered on the minimisation of the epsilon-order in theexpansion of the inhomogeneous part. In particular, we proposed a method basedon separating the system in smaller subsystems called triangularization, beforeanalysing each of those individually using dffierent uncoupling schemes, selectedpriorization of equations and through comparisons of epsilon-orders. This method hasbeen implemented in a package called SystemAnalysis.},
year = {2025},
month = {May},
translation = {0},
school = {RISC, Johannes Kepler University Linz},
length = {292}
}