Pensieve header: The 1D Zip Algebra.

{(SuperStar[x]),(SuperStar[y]),(SuperStar[z])}={ξ,η,ζ}; {(SuperStar[ξ]),(SuperStar[η]),(SuperStar[ζ])}={x,y,z};
(Subscript[Zip, {}])[P_]:=P;
(Subscript[Zip, {ζ_,ζs___}])[P_] := (Expand[P//(Subscript[Zip, {ζs}])] /. f_.(ζ^(d_.)):>(∂_({(SuperStar[ζ]),d}))f) /. (SuperStar[ζ])->0

{(Subscript[Zip, {ζ}])[(ζ^2)(E^(δ (z^2)))], (Subscript[Zip, {ζ}])[(ζ^4)(E^(δ (z^2)))]}

Kδ/:(Subscript[Kδ, i_,j_]):=If[i===j,1,0];
𝔼 /: (Subscript[Zip, ζs_List])@𝔼[Q_,P_] :=          (* 𝔼[Q,P] means (E^Q)P *)
Module[{ζ,z,zs,c,ys,ηs,qt,zrule,Q1,Q2},
zs=Table[(SuperStar[ζ]),{ζ,ζs}];
c=Q/.Alternatives@@(ζs⋃zs)->0;
ys=Table[(∂_ζ)(Q/.Alternatives@@zs->0),{ζ,ζs}];
ηs=Table[(∂_z)(Q/.Alternatives@@ζs->0),{z,zs}];
qt=Inverse@Table[(Subscript[Kδ, z,(SuperStar[ζ])])-(∂_(z,ζ))Q,{ζ,ζs},{z,zs}];
zrule=Thread[zs->qt.(zs+ys)];
Q1=c+ηs.zs/.zrule; Q2=Q1/.Alternatives@@zs->0;
Simplify /@ 𝔼[Q2,Det[qt](E^(-Q2))(Subscript[Zip, ζs])[(E^Q1)(P/.zrule)]] ];

The Algebra

Table[(Subscript[Zip, {η}])[𝔼[ξ y+η z,(((ξ y))^m)(((η z))^n)]], {m,0,3},{n,0,3}] // MatrixForm

Table[Last@(Subscript[Zip, {η}])[𝔼[ξ y+η z,(((ξ y))^m)(((η z))^n)]]/.ξ->1, {m,0,4},{n,0,4}] // MatrixForm

B[({{n_}, {k_}})]:=Binomial[n,k];
f[m_,n_]:=Simplify[(∑_(k=0)^(Min[m,n]))k!B[({{m}, {k}})]B[({{n}, {k}})](z^(m+n-k))];
Table[f[m,n], {m,0,4},{n,0,4}] // MatrixForm

A Module

Table[(Subscript[Zip, {η}])[𝔼[ξ y,(((ξ y))^m)(((η z))^n)]], {m,0,3},{n,0,3}] // MatrixForm

Table[Last@(Subscript[Zip, {η}])[𝔼[ξ y,(((ξ y))^m)(((η z))^n)]]/.ξ->1, {m,0,4},{n,0,4}] // MatrixForm

Expand[y(y-1)*y(y-1)(y-2)==y(y-1)(y-2)(y-3)(y-4)+6y(y-1)(y-2)(y-3)+6y(y-1)(y-2)]