  We obtain a series transformation formula involving the classical Hermite
polynomials. We then provide a number of applications using appropriate
binomial transformations. Several of the new series involve Hermite polynomials
and harmonic numbers, Lucas sequences, exponential and geometric numbers. We
also obtain a series involving both Hermite and Laguerre polynomials, and a
series with Hermite polynomials and Stirling numbers of the second kind.
