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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!

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  • $\begingroup$ You might try also MO and, besides the Wikipedia references, this paper. $\endgroup$ Oct 15 '18 at 15:05
  • $\begingroup$ To be honest I forgot to check MathOverflow in the first place. Hence I glanced through the posts containing "umbral calculus" and I only would refer to this one as notable. The others are only refering to "umbral calculus" without really providing examples/applications(it can be that I missed something therefore feel free to add links and related questions/answers). The paper you linked is not freely accessible and I do not want to create an account only for this purpose. Anyway thank you for your response. $\endgroup$
    – mrtaurho
    Oct 15 '18 at 20:50
  • $\begingroup$ I reinvented it as an undergrad compsci for a note on continuity of Bézier curves. $\endgroup$ Oct 17 '18 at 7:36
  • $\begingroup$ Partly because it's hard enough to convince students normal calculus isn't a shell game. If a pro-umbral era comes, its historians may attribute the slow uptake to our time's equivalent of ghosts of departed quantities. $\endgroup$
    – J.G.
    Oct 17 '18 at 8:34
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This is odd, since Rota revived it there is a fairly large community of umbral enthusiasts Di Bucchianico's 1995 survey has a bibliography 506 entries long. American Mathematical Monthly has an accessible History of Blissard's Symbolic Method, with a Sketch of its Inventor's Life by E.T. Bell. Normally I would take anything written by Bell with a grain of salt, see What resources are available for lives of recent mathematicians besides E.T. Bell's Men of Mathematics? But he is a good expositor and in this case he is citing original sources.

"It sometimes happens in the history of mathematics that the credit for a particular method is commonly ascribed to another than its originator. In the interests of historical justice, such oversights should be corrected, where the facts are known. A conspicuous instance is the extremely useful symbolic method expounded and widely applied by Edouard Lucas [3, 5],t and usually attributed to him. The kernel of this symbolic method is described below. This representative notation [1], to use the designation of its inventor, or symbolic method, as Lucas [3, 5] called it, or umbral calculus [8, vol. 31], was fully developed in 1861 by John Blissard [1], in a mathematical journal with a wide circulation among mathematicians, fifteen years before Lucas published his first papers [3, 4] on the subject.

[1] John Blissard, Theory of Generic Equations, Quarterly Journal of Pure and Applied Mathematics, vol. 4, 1861, pp. 279-305; vol. 5, 1862, pp. 58-75, 185-208. In nine further papers, ibid., vols. 5-9, Blissard gave applications of his method. He frequently contributed problems to the Educational Times.

[3] Edouard Lucas, Theorie nouvelle des nombres de Bernoulli et d'Euler, Comptes rendus de l'Academie des Sciences (Paris), vol. 83, 1876, pp. 539-541; Annali di Matematica pura ed applicata, Serie 2, vol. 8, 1877, pp. 56-79.

[4] E. Lucas et E. Catalan, Sur le calcul symbolique des nombres de Bernoulli, Nouvelle Correspondance Math6matique, vol. 2, 1876, pp. 328-336.

[5] Edouard Lucas, Theorie des Nombres, 1891, Chap.

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I think the short answer is that problems for which it is easier to use umbral calculus rather than a more "standard" technique don't arise all that frequently in practice.

My favorite reference for the umbral calculus is Ira Gessel's paper, Applications of the classical umbral calculus, Algebra universalis 49 (2003), 397–434. After a brief explanation of umbral notation, Gessel goes on to say:

When I first encountered umbral notation it seemed to me that this was all there was to it; it was simply a notation for dealing with exponential generating functions, or to put it bluntly, it was a method for avoiding the use of exponential generating functions when they really ought to be used. The point of this paper is that my first impression was wrong: none of the results proved here (with the exception of Theorem 7.1, and perhaps a few other results in section 7) can be easily proved by straightforward manipulation of exponential generating functions. The sequences that we consider here are defined by exponential generating functions, and their most fundamental properties can be proved in a straightforward way using these exponential generating functions. What is surprising is that these sequences satisfy additional relations whose proofs require other methods. The classical umbral calculus is a powerful but specialized tool that can be used to prove these more esoteric formulas.

So the umbral calculus definitely has value as an enumerative tool and is not superseded by other methods. On the other hand, Gessel does use the words specialized and esoteric to describe the situations in which the umbral calculus outperforms more standard techniques.

Corroboration of the claim that the umbral calculus has a rather narrow range of application may be found by examining some of the standard references for enumerative combinatorics. Richard Stanley's encyclopedic two-volume work Enumerative Combinatorics mentions umbral notation just once in passing. I couldn't find any mention of umbral calculus at all in the table of contents or index of the Handbook of Enumerative Combinatorics. The same could be said of just about any other graduate text on combinatorics, such as Goulden and Jackson's classic book Combinatorial Enumeration or Sagan's recent book Combinatorics: The Art of Counting. Evidently, all these authors felt that the umbral calculus was too specialized to deserve space in their books.

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