Séminaire Lotharingien de Combinatoire, B48b (2002), 21 pp.
Hung Quang Ngo
P-Species And The q-Mehler Formula
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.
Received: January 18, 2002;
Revised: June 7, 2002
Accepted: September 9, 2002.
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