Séminaire Lotharingien de Combinatoire, B48b (2002), 21 pp.

Hung Quang Ngo

P-Species And The q-Mehler Formula

Abstract. 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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