Recently I have encountered the so-called Umbral calculus. The main idea of this field is to treat indices as exponents, applying simpler techniques available to exponents and switching everything back when the work is done. The Wikipedia article on Umbral calculus states that
These techniques were introduced by John Blissard (1861) and are sometimes called Blissard's symbolic method. They are often attributed to Édouard Lucas (or James Joseph Sylvester), who used the technique extensively.
Further there is a brief historical outline of the development of this technique and even a look on the modern usage of Umbral calculus. Hence I am quite fascinated by this topic I would like to learn more about it; especially its applications for proofs.
First of all the Wikipedia side offers a proof of the derivatives of the Bernoulli polynomials by using Umbral calculus. Furthermore I am aware of this question on MSE which was answered by Tom Copeland with some interesting examples. Furthermore within this answer on MSE the user Count Iblis provided an elegant proof of Ramanujan's Master Theorem also using umbral calculus. Anyway I have not found that much further one MSE or somewhere else at first sight.
Refering to the quote from Wikipedia I was not able to find some proofs by neither É. Lucas nor by J. J. Sylvester actually using this field of mathematics. By saying "I was not able to find some proofs" I mean I have found nothing detached from a book of something similiar. It seems like Umbral calculus does not gain that much attention overall.
What I would like to know: $(1)$ Are you aware of proofs using Umbral calculus beside the few I mentioned? I would be interested in some historical examples - for instance the first proof using this technique, some proofs by the named mathematician and other, etc. - as well as mondern examples. $(2)$ Is there a reason why it seems like there are not that much applications of this techniques? Yes, of course the possibility that a proof can be done with Umbral calculus is not that high but however I would say there are enough problems cover something so simple as indices.
Thanks in advance!