From equations in coordinate space on Feynman integrals to Picard-Fuchs and back

One of the way of studying analytical structure of some function is deriving the of minimal linear differential equation on it. This task is of utmost interest in the case of some family of Feynman integrals, which are banana diagrams in this case. From coordinate space point of view they are the simplest diagrams to consider however in momentum space they become rather challenging. But momentum space has its advantages too as it is convenient for "serial" connection of two diagrams. That's why it is crucial to connect these representations. From naive point of view Fourier transform should establish this connection, however it does not give the desired result and leads to a lot of problems which I would like to present.