Series with Hermite polynomials and applications (Q5891783)

A transformation formula is presented in the paper: if an appropriate sequence \(a_n\) is given, then its generating function twisted by Hermite polynomials is presented. More concretely, if the sequence \(a_n\) has a generating function \(\sum_{n=0}^\infty a_nt^n\) which is analytical in a neighbourhood of 0 in the variable \(t\), then NEWLINE\[NEWLINE \sum_{n=0}^\infty a_nH_n(x)\frac{t^n}{n!}=e^{2xt-t^2}\sum_{n=0}^\infty (-1)^nH_n(x-t)\frac{t^n}{n!}\left(\sum_{k=0}^n\binom{n}{k}(-1)^ka_k\right).NEWLINE\]NEWLINE A large number of application of this transformation formula is given for Laguerre polynomials, generalized harmonic numbers, binomial coefficients, Stirling numbers of the second kind, exponential numbers (aka. Bell numbers), geometric numbers (aka. Fubini numbers), Fibonacci-, and Lucas numbers.