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# Questions tagged [umbral-calculus]

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### Can't parse a statement in an article on coalgebras and umbral calculus

This question is cross-posted from MSE. I am reading Nigel Ray's "Universal Constructions in Umbral Calculus" (1998, published in "Mathematical Essays in Honor of Gian-Carlo Rota", ...
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### Building representation of an arbitrary umbral calculus

Consider a set of integrable functions on the interval $(0,1)$. Let's introduce an operation $\operatorname{eval}f=\int_0^1 f(x)\,dx$ (which is the mean value of the function). In such system the ...
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3 answers
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### What did Rota mean by "one can define cumulants relative to any sequence of binomial type"?

This question is cross-posted from MSE.$\newcommand{\E}{\mathbb{E}}$ Near the end of "Finite Operator Calculus" (1976), G.C. Rota writes: Note that one can define cumulants relative to any ...
5 votes
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### Are umbral moonshine and umbral calculus connected?

In a 2013 article, Cheng, Duncan and Harvey introduce the concept of umbral moonshine as a generalization of monstrous moonshine. The terminology they use, starting with the title, is common in umbral ...
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1 answer
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### Categorical description of umbral calculus?

The theme of my current research project is related to umbral calculus in the context of more general algebraic structures, like Hopf algebras (and, in particular, shuffle algebras), so I am trying to ...
3 votes
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### Are any interesting classes of polynomial sequences besides Sheffer sequences groups under umbral composition?

This question on math.stackexchange.com has 35 views, three up-votes, and not a word from anybody, so I'm posting it here. Let us understand the term polynomial sequence to mean a sequence \$(p_n(x))_{...
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