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At Umbral calculus (wikipedia) there is reference to Rota’s work demystifying the so-called ’Umbral calculus’, where shady reasoning (pun completely intended - not due to me) can be used to prove results like identities for Bernoulli functions. To me it seems clear that something to do with species is going on, and the quote
Rota later stated that much confusion resulted from the failure to distinguish between three equivalence relations that occur frequently in this topic, all of which were denoted by “=”.
screams category theory to me. I would think that something like this might interest people around here (John, even, if he’s not too busy saving the world) to at least warrant a mention somewhere on the lab. I’m going to be very bad and not do it myself, for which I humbly apologise.
It may be worth noting that Nige Ray wrote an article called Universal Constructions in Umbral Calculus and that he certainly looks to have put in quite a categorical setting. (I have only just found it. I remember him giving talks on the subject back at the time he wrote it.)
I recorded the Ray’s article into umbral calculus, to start an entry in nlab.
Would anyone like to describe what Ray did? I certainly don’t have easy access to this article.
Todd: I found a copy at http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.66.2434 Does that help. I have not read the paper.
Great! Thanks, Tim. I now have it and would like to take a look, especially since “category of coassociative coalgebras” is mentioned on page 1.
Recorded. And few other references, including the first from 5-article series of Zeilberger on umbral transfer matrix method in enumerative combinatorics, which is somewhat more general than the original methods as Rota put them, including Rota operators. This area is quite close to the Hopf algebraic and operadic methods in algebraic combinatorics; some of the names in that area are Hazewinkel, Lascoux etc.
New article Rota-Baxter algebra and Gian-Carlo Rota. Among the relevant names I forgot to mention Marcelo Aguiar.
Some reference additions and corrections at Gian-Carlo Rota and umbral calculus.
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