http://tracker.cse.buffalo.edu/Ticket/Display.html?id=7448 ContactPerson: hungngo@cse.buffalo.edu ### Begin Citation ### Do not delete this line ### %R 2002-02 %U /u0/csestaff/stock/q-Mehler.ps %A Ngo, Hung Q. %T P-Species and the q-Mehler Formula %D January 24, 2002 %I Department of Computer Science and Engineering, SUNY Buffalo %K bijective proof, q-Mehler formula, species, Kibble-Slepian formula %X In this paper, we present a bijective proof of the $q$-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 $q$-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 $q$-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 $q$-Mehler formula. Some by-products of the $q$-exponential formula shall also be derived.