Charles Graves in the 1850s investigated iterated operators of the form $g(x) \frac {d}{dx}$ (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 $\omega=h(z)$ and $g(z)=1/[dh(z)/dz]$,

$$\exp \left[ {t \cdot g(z)\frac{d}{{dz}}} \right]f(z) = \exp \left[ {t\frac{d}{{d\omega }}} \right]f[{h^{ - 1}}(\omega )] = f[{h^{ - 1}}[t + \omega]] = f[{h^{ - 1}}[t + h(z)]],$$

(see OEIS A145271 and A139605 for more relations and references).

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) \frac {d}{dx} \; $?



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