Charles Graves in the 1850s investigated iterated operators of the form g(x)ddx (see page 13 in The Theory of Linear Operators … (Principia Press, 1936) by Harold T. Davis). Graves published “A generalization of the symbolic statement of Taylor’s theorem” in the Proceedings of the Royal Irish Academy, Vol. 5, (1850-1853) (when Sophus Lie was a child). Summarizing his conclusions:
with ω=h(z) and g(z)=1/[dh(z)/dz],
Charles’ brother John Graves discovered the octonians (octaves, see Wikipedia) in 1843 and is credited by Hamilton in encouraging his search for the quaternions, so we have a case perhaps of a missed opportunity for scooping Sophus Lie and his work on Lie group manifolds. (John also published work on the operator commutator [p,q]=[d/dq,q]=1.)
Wikipedia in the Shift Operator states that Lagrange published the iconic case with g(z)=1 (in the late 1700s, most likely), but certainly Newton could have been aware of this in Taylor series form.
QUESTION: Is there an earlier publication on iteration of the general op g(x)ddx?