  Inspired by ideas from umbral calculus and based on the two types of
integrals occurring in the defining equations for the gamma and the reciprocal
gamma functions, respectively, we develop a multi-variate version of umbral
calculus and of the so-called umbral image technique. Besides providing a class
of new formulae for generalized hypergeometric functions and an implementation
of series manipulations for computing lacunary generating functions, our main
application of these techniques is the study of Sobolev-Jacobi polynomials.
Motivated by applications to theoretical chemistry, we moreover present a deep
link between generalized normal-ordering techniques introduced by Gurappa and
Panigrahi, two-variable Hermite polynomials and our integral-based series
transforms. Notably, we thus calculate all K-tuple L-shifted lacunary
exponential generating functions for a certain family of SJ polynomials
explicitly.
