  In this paper we proceed to study properties of Mellin-Barnes (MB) transforms
of Usyukina-Davydychev (UD) functions. In our previous papers [Nuclear Physics
B 870 (2013) 243], [Nuclear Physics B 876 (2013) 322] we showed that multi-fold
Mellin-Barnes (MB) transforms of Usyukina-Davydychev (UD) functions may be
reduced to two-fold MB transforms and that the higher-order UD functions were
obtained in terms of a differential operator by applying it to a slightly
modified first UD function. The result is valid in $d=4$ dimensions and its
analog in $d=4-2\varepsilon$ dimensions exits too [Theoretical and Mathematical
Physics 177 (2013) 1515]. In [Nuclear Physics B 870 (2013) 243] the chain of
recurrent relations for analytically regularized UD functions was obtained
implicitly by comparing the left hand side and the right hand side of the
diagrammatic relations between the diagrams with different loop orders. In
turn, these diagrammatic relations were obtained due to the method of loop
reduction for the triangle ladder diagrams proposed in 1983 by Belokurov and
Usyukina. Here we reproduce these recurrent relations by calculating explicitly
via Barnes lemmas the contour integrals produced by the left hand sides of the
diagrammatic relations. In such a way we explicitly calculate a family of
multi-fold contour integrals of certain ratios of Euler gamma functions. We
make a conjecture that similar results for the contour integrals are valid for
a wider family of smooth functions which includes the MB transforms of UD
functions.
