Formal Power Series as Linear Operators

Introductionary reference:

• Operator Calculus

So why not replace $\,t\,$ by the differential operator $\,d/dx\,$ everywhere it occurs? Next reference:

• Exponential of a function times derivative

Therefore we have in general and especially: $$ e^{y\,d/dx} f(x) = f(x+y) \quad \Longrightarrow \quad e^{y\,d/dx} x^n = (x+y)^n $$ Indeed the operator $1/(d/dx)$ is the inverse of differentiation, which is integration. So let's take a look at your perhaps not as obvious expression: $$ \left(\frac{e^{y\,d/dx}-1}{d/dx}\right)x^n=\frac{1}{d/dx}\left(e^{y\,d/dx}-1\right)x^n\\ =\int\left[(x+y)^n - x^n\right] dx = \frac{(x+y)^{n+1}-x^{n+1}}{n+1} + C $$ with $C$ an arbitrary constant. Evaluation is not ambiguous, because the operators $1/(d/dx)$ and $\left(e^{y\,d/dx}-1\right)$ do indeed commute. Which means that we can do algebra with these operators as with common numbers. The same in, last but not least: $$ \left(\frac{e^{y\,d/dx}-1-y\,d/dx}{(d/dx)^2}\right)x^n=\int\left\{\int\left[(x+y)^n-x^n\right]dx\right\}dx-\int y\,x^n\,dx \\=\frac{(x+y)^{n+2}-x^{n+2}}{(n+1)(n+2)}-\frac{y\,x^{n+1}}{n+1} + Cx + D $$ with $C$ and $D$ arbitrary constants. This result is deviant from your last formula. It is noticed that: $$ \frac{d^2}{dx^2} \left[\frac{(x+y)^{n+2}-x^{n+2}}{(n+1)(n+2)}-\frac{y\,x^{n+1}}{n+1} + C\,x + D\right] = (x+y)^n - x^n - y\frac{dx^n}{dx} $$ EDIT. I almost forgot to mention the stuff where it's all about: the formal power series (Wikipedia). $$ (y+x)^{n}=\sum _{k=0}^{n}{n \choose k}y^{n-k}x^{k} \quad \Longrightarrow \\ \frac{(x+y)^{n+2}-x^{n+2}}{(n+1)(n+2)}-\frac{y\,x^{n+1}}{n+1} =\\ \frac{1}{(n+1)(n+2)}\left[\sum _{k=0}^{n+2}{n+2 \choose k}y^{k}x^{n+2-k} - x^{n+2} - (n+2)y\,x^{n+1}\right]=\\ \frac{1}{(n+1)(n+2)}\left[\sum _{k=2}^{n+2}{n+2 \choose k}y^{k}x^{n+2-k} + {n+2 \choose 0}y^0 x^{n+2} - x^{n+2} + {n+2 \choose 1}y^1 x^{n+1} - (n+2)y\,x^{n+1} \right]\\ =\frac{1}{(n+1)(n+2)}\left[\sum _{k=2}^{n+2}{n+2 \choose k}y^{k}x^{n+2-k}\right] $$ So here is your fix on the lower bound of the summation.

Note: OPs calculations are quite ok and it shows the operators are closely related, but different. I don't think there is a necessity to fix anything.

I skimmed through the classic The Umbral Calculus by Steven Roman, but there was no indication that something more is going on regarding OPs question. Another source I've checked without success was The Calculus of Finite Differences by C. Jordan.

It might be helpful to list a few higher powers of the operators under consideration.

In the following I use OPs notation which is precisely the same used by Steven Roman.

Translation operator: $e^{yt}$

Since this operator satisfies: \begin{align*} e^{yt}x^n&=\sum_{k=0}^\infty \frac{y^k}{k!}t^kx^n =\sum_{k=0}^n \frac{y^k}{k!}(n)_kx^{n-k}=\sum_{k=0}^n\binom{n}{k}y^kx^{n-k}\\ &=(x+y)^n \end{align*} we obtain \begin{align*} \left(e^{yt}\right)^2 x^n=e^{yt}(x+y)^n=(x+2y)^n \end{align*} and in general for $j\geq 1$ \begin{align*} e^{jyt}x^n=(x+jy)^n \end{align*}

Since $x^n, n\geq 0$ form a basis of the vector space of all polynomials $p$ in a single variable $x$ and the translation operator is linear, we obtain \begin{align*} e^{jyt}p(x)=p(x+jy)\tag{1} \end{align*}

hence the name translation operator.

Forward difference operator: $e^{yt}-1$

Here we obtain for polynomials $p$ using (1)

\begin{align*} \left(e^{yt}-1\right)p(x) = p(x+y)-p(x) \end{align*}

The next one is

Operator: $\frac{\exp(yt)-1}{t}$

We obtain \begin{align*} \left(\frac{e^{yt}-1}{t}\right)x^n&=\sum_{k=1}^\infty \frac{y^k}{k!}t^{k-1}x^n\\ &=\sum_{k=1}^{n+1}\frac{y^k}{k!}(n)_{k-1}x^{n-(k-1)}\\ &=\frac{1}{n+1}\sum_{k=1}^{n+1}\binom{n+1}{k}y^kx^{n+1-k}\\ &=\frac{1}{n+1}\left((x+y)^{n+1}-x^{n+1}\right)\tag{2}\\ &=\frac{1}{n+1}\int_x^{x+y}u^n\,du \end{align*}

Similarly we obtain from (2) by linearity \begin{align*} \left(\frac{e^{yt}-1}{t}\right)^2x^n &=\left(\frac{e^{yt}-1}{t}\right)\frac{1}{n+1}\left((x+y)^{n+1}-x^{n+1}\right)\\ &=\frac{1}{n+1}\left[\frac{1}{n+2}\left((x+2y)^{n+2}-(x+y)^{n+2}\right)\right.\\ &\qquad\qquad\quad\left.-\frac{1}{n+2}\left((x+y)^{n+2}-x^{n+2}\right)\right]\\ &=\frac{1}{(n+1)(n+2)}\left((x+2y)^{n+2}-2(x+y)^{n+1}+x^{n+2}\right)\\ &=\frac{1}{(n+2)_2}\left(\int_{x+y}^{x+2y}u\, du-\int_x^{x+y}u\,du\right)\tag{3} \end{align*}

Here at (2) and (3) we can see quite nicely how the operator $\frac{\exp(yt)-1}{t}$ is connected with the integral operator. It can be extended to higher powers without too much effort and the relationship with the integral operator looks plausible.

Operator: $\frac{\exp(yt)-1-t}{t^2}$

Now we take a look at the operator which is on the focus of OP and its generalisation.

\begin{align*} \left(\frac{e^{yt}-1-t}{t^{2}}\right)x^n &=\sum_{k=2}^\infty\frac{y^k}{k!}t^{k-2}x^n\\ &=\sum_{k=2}^{n}\frac{y^{k}}{k!}(n)_{k-2}x^{n-(k-2)}\\ &=\frac{1}{(n+2)_2}\sum_{k=2}^n\binom{n+2}{k}y^kx^{n+2-k}\\ &=\frac{1}{(n+2)_2}\left((x+y)^{n+2}-x^{n+2}-nyx^{n+1}\right) \end{align*}

Comparing the final expression with (3) we do not see a plausible representation via integrals since the term $nyx^{n+1}$ don't provide anything nicely of the form \begin{align*} \text{integrated expression (end point) - integrated expression (starting point)} \end{align*}

This impression becomes more strongly when looking at the general case. We obtain for $j\geq 1$ \begin{align*} \left(\frac{e^{yt}-1-\frac{t^2}{2}-\cdots-\frac{t^{j-1}}{(j-1)!}}{t^j}\right)x^n &=\sum_{k=j}^\infty\frac{y^k}{k!}t^{k-j}x^n\\ &=\frac{1}{(n+j)_j}\sum_{k=j}^\infty\binom{n+j}{k}y^kx^{n+j-k}\\ &=\frac{1}{(n+j)_j}\left((x+y)^{n+j}-\sum_{k=0}^{j-1}\binom{n+j}{k}y^kx^{n+j-k}\right) \end{align*}