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For a sequence $P=(p_n(x))_{n=0}^{\infty}$ of polynomials $p_n(x)$, we studythe $K$-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 analgorithm for efficiently computing $\mathcal{G}_{K,L}(\lambda;x)$ for genericpolynomial sequences $P$. This procedure is exemplified by application to thestudy of Hermite polynomials, whereby we obtain closed-form expressions for$\mathcal{G}_{K,L}(\lambda;x)$ for arbitrary $K$ and $L$, in the form ofinfinite series involving generalized hypergeometric functions. The basis ofour method is provided by certain resummation techniques, supplemented byoperational formulae. Our approach also reproduces all the results previouslyknown in the literature.

Explicit formulae for all higher order exponential lacunary generating functions of Hermite polynomials