Cholewinski developed a version of the umbral calculus for studying differential equations of Bessel type and related topics in [16].
A connection between approximation operators and polynomials of binomial type can be found in [44]. Further papers in this direction are [47,65,121].
Orthogonal polynomials play an important role in analysis. It is therefore important to know whether polynomials are orthogonal. The classification of orthogonal Sheffer polynomials was first found by Meixner [63]; it has been reproved many times (see e.g., [26,48,104,109]).
General papers on orthogonal polynomials and umbral calculus are [29,48,82].
Hypergeometric and related functions are dealt with in an umbral calculus way in [122,123].
Constructing umbral calculi based on the operator 
yields a powerful way to study interpolation theory
[40,96,110,125,126,127,128].
Banach algebras are used by Di Bucchianico [21] to study the convergence properties of the generating function of polynomials of binomial type and by Grabiner [33,34] to extend the umbral calculus to certain classes of entire functions.