�-Species and the �-Mehler Formula

In this paper, we present a bijective proof of the �-Mehler formula. The proof is in the same style as Foata’s proof of the Mehler formula. Since Foata’s proof was extended to show the Kibble-Slepian formula, a very general multilinear extension of the Mehler formula, we hope that the proof provided in this paper helps find some multilinear extension of the �-Mehler formula. The basic idea to obtain this proof comes from generalizing a result by Gessel. The generalization leads to the notion of species on permutations and the �-generating series for these species. The bijective proof is then obtained by applying this new exponential formula to a certain type of species on permutations and a weight preserving bijection relating this species to the �-Mehler formula. Some by-products of the �-exponential formula shall also be derived.