  For a sequence $P=(p_n(x))_{n=0}^{\infty}$ of polynomials $p_n(x)$, we study
the $K$-tuple and $L$-shifted exponential lacunary generating functions
$\mathcal{G}_{K,L}(\lambda;x):=\sum_{n=0}^{\infty}\frac{\lambda^n}{n!}
p_{n\cdot K+L}(x)$, for $K=1,2\dotsc$ and $L=0,1,2\dotsc$. We establish an
algorithm for efficiently computing $\mathcal{G}_{K,L}(\lambda;x)$ for generic
polynomial sequences $P$. This procedure is exemplified by application to the
study of Hermite polynomials, whereby we obtain closed-form expressions for
$\mathcal{G}_{K,L}(\lambda;x)$ for arbitrary $K$ and $L$, in the form of
infinite series involving generalized hypergeometric functions. The basis of
our method is provided by certain resummation techniques, supplemented by
operational formulae. Our approach also reproduces all the results previously
known in the literature.
