First of all, I'm not a mathematician, I study physics (so please be gentle). In learning physics (much to my chagrin) the finer mathematical details are often swept under the proverbial “carpet”. It is pretty common to come across some functional:
$$J=\intop F\left[\overrightarrow{X}(\tau)\right]d\tau$$
Where F is an arbitrary (in this case) function and X represents a position vector being parameterized by $\tau$. If we're seeking the minimization, it is common to consider a variation in J with respect to some variation in our parameter tau.
$$\delta J=\intop F\left[\overrightarrow{X}(\tau+\delta\tau)\right]d\tau-\intop F\left[\overrightarrow{X}(\tau)\right]d\tau$$
I'm wondering is it then legitimate to utilize the shift operator which is defined by:
$$F\left[\overrightarrow{X}(\tau+\delta\tau)\right]=e^{\delta\tau\frac{\partial\overrightarrow{X}}{\partial\tau}\cdot\overrightarrow{\nabla_{x}}}F\left[\overrightarrow{X}(\tau)\right]$$
Which is essentially just a concise way of expressing the Taylor/Maclaurin expansion. If one considers utilizing this then we have an expression of the form:
$$\delta J=\intop\left(e^{\delta\tau\frac{\partial\overrightarrow{X}}{\partial\tau}\cdot\overrightarrow{\nabla}}-1\right)F\left[\overrightarrow{X}(\tau)\right]d\tau$$
Or examining just the integrand:
$$\left(e^{\delta\tau\frac{\partial\overrightarrow{X}}{\partial\tau}\cdot\overrightarrow{\nabla}}-1\right)F\left[\overrightarrow{X}(\tau)\right]$$
and expanding out a few terms:
$$=\left(\delta\tau\overrightarrow{U}\cdot\overrightarrow{\nabla}+\left[\delta\tau\overrightarrow{U}\cdot\overrightarrow{\nabla}\right]\left[\delta\tau\overrightarrow{U}\cdot\overrightarrow{\nabla}\right]+\cdots\right)F\left[\overrightarrow{X}(\tau)\right]$$
$$=\left(\delta\tau\overrightarrow{U}\cdot\overrightarrow{\nabla}+\frac{1}{2}\delta\tau\delta\tau\overrightarrow{U}\cdot\left(\overrightarrow{\nabla}\overrightarrow{U}\right)+\delta\tau\delta\tau\overrightarrow{U}\cdot\overrightarrow{U}\overrightarrow{\nabla}\cdot\overrightarrow{\nabla}\cdots\right)F\left[\overrightarrow{X}(\tau)\right]$$
The first term seems to correspond to the first variation (which is then manipulated to obtain the Lagrange equation, are the other terms higher variations?
Now I know this isn't right (as we've taken no limits and specified no boundary conditions); however I was wondering if you can express higher order variations (since they generally only teach us physicists the first variation) in terms of the shift operator (which happens to be linear). I can think of several applications of such a mathematical tool within the realm of physics. Any literature you might be able to steer me towards?
I should probably note that the problems I have in mind will take place on general (pseudo/semi) Riemannian manifolds where derivatives will be covariant ones.
This is only a first reaction, because I don't have a book or a paper in mind to recommend. It seems to me that there are at least 3 levels to consider your calculation
1) The formal level, where nothing is mathematically justified, but it still does have a meaning because formal solutions usually lead to correct solutions far before one is able to prove rigourous theorems. This is a very good reason to perform formal calculations.
2) The regular case where functions such as $F$ and $X$ are supposed to be infinitely smooth and have all the good properties so that the formal calculations become mathematically true. This is ideal for physicists equations. Remark that you are using the exponential of the derivation operator, this is called functional calculus with unbounded operators. Books on functional analysis could tell you about this. Basically however the first tool is that the calculus can be fully justified when functions are infinitely smooth.
3) The 'real' situation where functions are not infinitely smooth. Mathematical legitimacy is usually obtained by establishing limits when smooth functions converge to the real functions (density of smooth functions spaces in other function spaces). The important part is to discover inequalities that apply to the problem, such as some energy is bounded, etc. This is where the real mathematical work takes place.
Edit: it seems to me that some factorials are missing near the end of your question.
Edit2: I forgot to mention that your use of $e^{\delta_\tau \frac{\partial X}{\partial \tau} \nabla}$ can also be viewd as being part of the theory of semi-groups of bounded operators, a field of functional analysis generaly related to evolution equations.