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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Are any interesting classes of polynomial sequences besides Sheffer sequences groups under umbral composition?

Let us understand the term polynomial sequence to mean a sequence $(p_n(x))_{n=0}^\infty$ in which the degree of $p_n(x)$ is $n.$ The umbral composition $((p_n\circ q)(x))_{n=0}^\infty$ (not $((p_n\...
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2answers
96 views

Double sum identity involving binomial coefficients, possibly connected to umbral calculus

I would be interested in seeing an insightful proof, or really, any alternative proof of the identity $$ \begin{aligned} &\sum_{j=0}^h(x+1)^j\binom{h}{j}\sum_{k=0}^r\binom{r}{k}x^k(r-k+h-j)!=\sum_{...
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1answer
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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 ...
6
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2answers
134 views

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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151 views

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 ...
6
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1answer
171 views

Umbral calculus with negative indices (and powers)

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 ...
29
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4k views

What's umbral calculus about?

I've read Wikipedia about it and it says: In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and ...