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.