Could one say, in a sense, that recurrence is essentially differential equations and convolution essentially more complicated operations than those of arithmetic polynomials?

This might sound like nonsense. On the one hand, most trivial convolutions use trivial operators; "polynomials" might include higher operations anyhow, or approximate some of the more important ones, none of which is appealing if simplicity equals efficiency. On the other hand, I never really understood diff-eqs; ODEs seem like polynomials over self similar polynomials, to me, hence "recurrent"; All the other diff-eqs I can't begin to fathom.