Two-fold Mellin-Barnes transforms of Usyukina-Davydychev functions

TL;DR

This paper derives an explicit formula for the coefficients in two-fold Mellin-Barnes transforms of Usyukina-Davydychev functions, using limits of smooth functions and differential operators, enhancing the understanding of their mathematical structure.

Contribution

It provides a new explicit formula for the coefficients in two-fold Mellin-Barnes transforms of UD functions, based on a double-uniform limit and differential operators, which was not previously known.

Findings

Derived explicit formula for coefficients in MB transforms

Established connection between limits of smooth functions and MB integrals

Reproduced original integral representations using the new differential operator approach

Abstract

In our previous paper [Nucl.Phys. B 870 (2013) 243], we showed that multi-fold Mellin-Barnes (MB) transforms of Usyukina-Davydychev (UD) functions may be reduced to two-fold MB transforms. The MB transforms were written there as polynomials of logarithms of ratios of squares of the external momenta with certain coefficients. We also showed that these coefficients have a combinatoric origin. In this paper, we present an explicit formula for these coefficients. The procedure of recovering the coefficients is based on taking the double-uniform limit in certain series of smooth functions of two variables which is constructed according to a pre-determined iterative way. The result is obtained by using basic methods of mathematical analysis. We observe that the finiteness of the limit of this iterative chain of smooth functions should reflect itself in other mathematical constructions, too, since it is not related in any way to the explicit form of the MB transforms. This finite double-uniform limit is represented in terms of a differential operator with respect to an auxiliary parameter which acts on the integrand of a certain two-fold MB integral. To demonstrate that our result is compatible with original representations of UD functions, we reproduce the integrands of these original integral representations by applying this differential operator to the integrand of the simple integral representation of the scalar triangle four-dimensional integral J(1,1,1-e).