Relation between linear recurrence and linear equations that produce the same number sequence

Perhaps the thoughts which I followed recently give a satisfactory answer. I've looked at the "ZETA"-matrix and fiddled out the connection to the integral-representation in the Euler-MacLaurin-Formula. The article is not yet ready, needs some brushing and completing references and such. But as it might be helpful here: here is the link the integral in Euler-MacLaurin in connection with the Pascal-matrix

[update]: added the re-translation into bernoulli-numbers in the final formulae in the text