Pensieve header: Testing Gaussian pairing, plus g1 logos games.

(Subscript[DP, (Subscript[D, α_]),(Subscript[D, β_])])[P_][f_]:=
Total[CoefficientRules[P,{(Subscript[D, α]),(Subscript[D, β])}]/.({m_,n_}->c_):>c D[f,{α,m},{β,n}]]

With[{n=10},
(Subscript[DP, (Subscript[D, α]),(Subscript[D, β])])[Normal@Series[(E^(δ (Subscript[D, α])(Subscript[D, β]))),{δ,0,n}]][(E^(-α β h + β e+α e))] /. {α|β->0}
]/. {e|f->0}

Series[(1/(1+h δ)),{δ,0,10}]

With[{n=5},(Subscript[DP, (Subscript[D, α]),(Subscript[D, β])])[Normal@Series[(E^(t (Subscript[D, α])(Subscript[D, β]))),{t,0,n}]][(E^(-α β h ))]] // Simplify

Series[((E^(-(((1+h t))^(-1))α β h))/(1+h t)),{t,0,5}] // Normal // Simplify

With[{ψ=((E^(-(((1+h t))^(-1))α β h))/(1+h t))}, {(∂_t)ψ-(∂_(α,β))ψ, ψ /. t->0}]

oldΛ=(1/(2 (μ^4)))(-b (α^2) (β^2)+(u^2) (β^2) δ (2+b δ)+(w^2) δ (α+u δ) (α (2+b δ)+u δ (4+3 b δ))-4 b α β δ μ+4 c α β (μ^2)-2 b (δ^2) (μ^2)+4 c δ (μ^3)+2 u β (α β+2 δ μ (1+c μ))+2 w ((α^2) β+2 α δ (u β (2+b δ)+μ (1+c μ))+u (δ^2) (u β (3+2 b δ)+2 μ (2+b δ+c μ))));

Λl=Simplify[oldΛ /. {δ->0,μ->1,b->h,c->l,u->e,w->f}]

Λq=Factor[oldΛ /. {b->h,c->l,u->e,w->f,α->0,β->0,μ->1+h δ}]

Λp=Factor[Λl+(((1+h δ))^(-1))δ((∂_(α,β))Λl)+(1/2)(((1+h δ))^(-2))(δ^2)((∂_(α,β,α,β))Λl) /. {α->(((1+h δ))^(-1))δ e, β->(((1+h δ))^(-1))δ f}]

Λp==Λq