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 — such as particle physics — give rise to a plethora of problems involving special functions that need to be dealt with efficiently. In this PhD, we concentrated on two such particular problems. In the first part of this PhD thesis, we explored the relation between iterated binomial sums, an extension of general harmonic sums, and their integral representations, in order to compute their asymptotic expansions. To do that in a fully automatic way, we created a dedicated package, RICA. Using Mellin representations, we first formalised and extended a classical recursive method to compute Mellin inverses of such sums, and together with it implemented several methods to compute asymptotic expansions of such integrals. In the process, we introduced and explored a new class of functions related to Mellin convolutions. Those allowed us to automatically compute asymptotic expansions for more general classes of sums in a new and efficient way, while providing a way to get symbolic representations for 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 particle physics computations. Since usually those systems could only be solved up to some order in epsilon, we aimed at developing a method to optimise the solving task of such systems. We studied an approach centered on the minimisation of the epsilon-order in the expansion of the inhomogeneous part. In particular, we proposed a method based on separating the system in smaller subsystems called triangularization, before analysing each of those individually using dffierent uncoupling schemes, selected priorization of equations and through comparisons of epsilon-orders. This method has been implemented in a package called SystemAnalysis.},
year = {2025},
month = {May},
translation = {0},
school = {RISC, Johannes Kepler University Linz},
length = {292},
type = {phdthesis}