I know that currently umbral calculus is developed as some kind of theory of operators and functionals but were there any attempts to put it on a more solid philosophical grounds as study of functions on some kind of extended reals?

What do I mean? For instance, in Levi-Civita field we can find constants that produce Bernoulli numbers (as their standard part) when raised to natural powers. The constant $p_0=\varepsilon^{-1} -\frac{1}{2}-\frac{\varepsilon }{24}+\frac{3 \varepsilon ^3}{640}-\frac{1525 \varepsilon ^5}{580608}+\dotsb$ would have properties equivalent to the *umbra* in 19-th century terminology.
It seems that studying functions on these arguments will produce a system equal to umbral calculus but with simpler theoretical foundation. Am I correct?