I would be interested in seeing an insightful proof, or really, any alternative proof of the identity $$ \begin{aligned} &\sum_{j=0}^h(x+1)^j\binom{h}{j}\sum_{k=0}^r\binom{r}{k}x^k(r-k+h-j)!=\sum_{j=0}^h\binom{h}{j}(r+j)!\sum_{i=0}^{j+r}\frac{x^i}{i!}. \end{aligned} $$ The only proof I've managed to come up with is surprisingly cumbersome. It can be seen in this answer by searching for the line, "This is the $x=-2$ case of the sum" in the section "Alternative formula of Wyman and Moser".
I ran across this identity in the course of proving the equality of the two expressions, $$ \varphi(h; n)=\sum_{i=0}^n(-1)^i\frac{2n}{2n-i}\binom{2n-i}{i}\nu(h,h+n-i), $$ where $$ \nu(h,h+n)=\sum_{k=0}^h(-1)^k\binom{h}{k}(n+h-k)!, $$ and $$ \varphi(h;n)=\sum_{i\ge0}(-1)^i\frac{n}{n-i}\binom{n-i}{i}\sum_{j=0}^h\binom{h}{j}k_{n-2i+j}, $$ where $$ k_r=r!\sum_{i=0}^r\frac{(-2)^i}{i!}. $$ The former is a formula of Touchard, related to double derangements and the ménage problem, and the latter is an, empirically discovered, generalization of a formula of Wyman and Moser for the ménage problem.
My feeling that this is connected with umbral calculus is rather vague. It comes from the observation that $$ \sum_{i\ge0}(-1)^i\frac{n}{n-i}\binom{n-i}{i}x^{n-2i} $$ is a rescaled Chebyshev polynomial of the first kind, and that the formula for $\varphi(h;n)$ comes from the umbral-style replacement of $x^{n-2i}$ with $\sum_{j=0}^h\binom{h}{j}k_{n-2i+j}$, while $$ \sum_{i=0}^n(-1)^i\frac{2n}{2n-i}\binom{2n-i}{i}x^{\frac{1}{2}(2n-2i)} $$ is a rescaled Chebyshev polynomial of the first kind of twice the index (in the variable $x^{1/2}$), with $\varphi(h;n)$ arising from the different umbral-style replacement of $x^{n-i}$ with $\nu(h,h+n-i)$. I don't know much about umbral calculus, and don't know whether there's any sort of transformation theory that would shed light on how the replacement of $x^n$ by $x_n$ has to change when polynomial identities are used (such as the identity relating Chebyshev polynomials to Chebyshev polynomials of twice the index). Any comments about umbral calculus would be a bonus, but my main question is about proof of the identity.