Mellin-Barnes representations of Feynman diagrams, linear systems of   differential equations, and polynomial solutions
  We argue that the Mellin-Barnes representations of Feynman diagrams can be used for obtaining linear systems of homogeneous differential equations for the original Feynman diagrams with arbitrary powers of propagators without recourse to the integration-by-parts technique. These systems of differential equations can be used (i) for the differential reductions to sets of basic functions and (ii) for counting the numbers of master integrals. 
Mikhail Yu. Kalmykov, Bernd A. Kniehl
hep-th math-ph math.MP