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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