Geodesic
P-Species And The q-Mehler Formula
Séminaire lotharingien de combinatoire, Tome 48 (2002-2003)

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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. 

@article{SLC_2002-2003_48_a1,
     author = {Hung Quang Ngo},
     title = {P-Species {And} {The} {q-Mehler} {Formula}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {48},
     year = {2002-2003},
     url = {http://geodesic.mathdoc.fr/item/SLC_2002-2003_48_a1/}
}
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Hung Quang Ngo. P-Species And The q-Mehler Formula. Séminaire lotharingien de combinatoire, Tome 48 (2002-2003). http://geodesic.mathdoc.fr/item/SLC_2002-2003_48_a1/