(Subscript[zip, q_,λ_])[η_,y_] := ((E^(λ η y/(1-λ q)))/(1-λ q));
F[q_,λ_]:=Simplify[(Subscript[zip, q,λ])[q ζ+η,q z+y](E^(q ζ z+y ζ+η z))];
{F[q,λ], F[q,0], F[q,1], F[q,1]/.{z|ζ->0}} // Simplify

{lhs=(∂_λ)F[q,λ],rhs=(∂_ζ)(∂_z)F[q,λ],lhs==rhs}//Simplify