An extension of Liouville's theorem on integration in finite terms

Nested (inverse) binomial sums and new iterated integrals for massive Feynman diagrams

“…There are several computer algebra algorithms for different types of integrands f (x,t) which, given f (x,t), compute relations of the form (3.11). They either utilize differential fields [38,49,18,35] or holonomic systems and Ore algebras [13,21,31,19]. After obtaining a differential equation for F(x) we need to solve it explicitly, preferably in terms of iterated integrals.…”

Integration in terms of exponential integrals and incomplete gamma functions I

“…Attempts to do this started quite early. Starting point for us were [14] which forms basis for early work on integration in terms of logarithmic integrals and error functions ( [2], [5] and [6]). We extend previous results allowing incomplete gamma function…”

“…incomplete gamma function is more general than error function. Let us add that Liouville principle in [14] from one point of view is very general and handles large class of special functions, however this class has small intersection with classical special functions. So for the purpose of integration in terms of classical special functions [14] has limited use.…”

“…After writing first version of this paper we learned about closely related paper [7] by U. Leerawat and V. Laohakosol. They define class of extensions called Ei-Gamma extensions which generalised extensions from [14] by allowing directly exponential integrals and incomplete gamma functions with rational first argument. Our work also allows incomplete gamma functions with irrational first argument so is more general in this aspect.…”

“…If we ignore incomplete gamma functions with irrational first argument, then Theorem 3.1 in [7] would be more general than our Theorem 13. Our paper deals only with exponential integrals and incomplete gamma functions which means that we can use trace in straightforward way, avoiding extra arguments in [14] and [7] needed to handle nonclassical special functions. As our work is intended as first step towards integration algorithm we need more structural information about integrals which we give in Theorems 18, 19 and 21.…”