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

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.