Pensieve header: Comparing exponentials of q-commuting variables.

Series[(E^(β (E^(n ϵ))(Subscript[X, 1]))),{ϵ,0,4}]

Series[((α^n)/(n!))(E^(β (E^(n ϵ))(Subscript[X, 1])))(Subscript[X, 2]^n),{ϵ,0,4}]

Collect[Sum[Normal@Series[((α^n)/(n!))(E^(β (E^(n ϵ))(Subscript[X, 1])))(Subscript[X, 2]^n),{ϵ,0,8}],{n,0,∞}],ϵ,Simplify]