# Questions tagged [umbral-calculus]

Umbral calculus refers to a method of formal computation which can be used to prove certain polynomial identities. The term "umbral", meaning "shadowy" in Latin, describes the manner in which the terms in discrete equations (e.g. difference equations) are similar to (or are "shadows of") related terms in power series expansions.

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### Umbral calculus/Pochhammer - invert falling factorial of binomial in term of falling factorial of monomial

Consider the variables $x,n \in \mathbb{Z}^+$ and define the following falling factorial operator: \begin{equation} L[x^n] = (x)_n = \prod_{k=0}^{n-1}(n-k) \end{equation} now from consider the ...
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### Umbral calculus - eigenfunctions of operator

I'm very new to umbral caluclus and I have come across a paper that makes use of some results in this area, which I do not quite understand. The problem I have is the following. Consider the ...
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### Question about "baffling" umbral calculus result

I am reading a paper here and I've come to a particular passage that is confusing me. It comes on page 2 of the attached paper and it deals with the binomial theorem... The passage lays the ...
Can we do umbral calculus with negative indices (and powers)? Can we write $a_{-n} \equiv a^{-n}$ or $L[a_{-n}] = a^{-n}$ where $L$ is a linear functional and $n$ need not be negative? The common ...