Residue current approach to Ehrenpreis-Malgrange type theorem for linear differential equations with constant coefficients and commensurate time lags

We introduce a ring H of partial difference-differential operators with constant coefficients initially defined by H. Glüsing-Lürßen for ordinary difference-differential operators and investigate its cohomological properties. Combining this ring theoretic observation with the integral representation technique developed by M. Andersson, we solve a certain type of division with bounds. In the last section, we deduce from this injectivity properties of various function modules over H as well as the density results of exponential polynomial solutions for partial difference-differential equations.