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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.

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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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 certain shadowy techniques used to 'prove'…
Red Banana
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Trying to characterise an "umbral shift"

Consider the function $\;\Phi(A)=\phi A\phi^{-1},\;$ where $\phi\::\:x^n\:\mapsto\:x(x-1)\cdots(x-n+1)$ and $A$ is an arbitrary linear operator over $\mathbb{C}[x]$. It turns out that applying this to the derivative operator gives the forward…
Supware
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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 groundwork of how umbrals were linear functionals before…
Eleven-Eleven
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What's the intuitive meaning of this relation between volumes of $n$-balls and umbral calculus?

The volume of an $n$-ball of radius $1$ is $$V_{n}={\frac {\pi ^{n/2}}{\Gamma {\bigl (}{\tfrac {n}{2}}+1{\bigr )}}}.$$ The functional equation of Riemann zeta function is $${\displaystyle \pi ^{-{s \over 2}}\Gamma \left({s \over 2}\right)\zeta…
Anixx
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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 following operator \begin{equation} \mathcal{S} =…
Jpk
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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 convention is to use $\mathbb N$ or $\mathbb N \cup…
glebovg
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What are the properties of umbra with moments $1,1/2,1/3,1/4,1/5,...$?

If we apply operator $D\Delta^{-1}$ to a function, we will get the (Bernoulli) umbral analog of the function. Particularly, applying it to $x^n$ we will get the Bernoulli polynomials $B_n(x)$. Evaluating them at zero we will get the moments of…
Anixx
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The umbral calculus proof of the higher order product rule

unfortunately, I seem to be quite unable to come up with the correct umbral calculus proof of the identity $$ \frac{\mathrm{d}^{n}\left(fg\right)}{\mathrm{d}x^{n}}\left(x\right) = \sum_{k=0}^{n}{\binom{n}{k} f^{\left(k\right)}\left(x\right)…
Cloudscape
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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 $((p_n\circ q_n)(x))_{n=0}^\infty$) of two polynomial…
Michael Hardy
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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…
Will Orrick
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Can't parse a statement in an article on coalgebras and umbral calculus

I am reading Nigel Ray's "Universal Constructions in Umbral Calculus" (1998, published in "Mathematical Essays in Honor of Gian-Carlo Rota", page 344). The article reads: We define an umbral calculus $(C, r)$ to consist of a coalgebra $C$ in…
Daigaku no Baku
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What are the properties of this new characteristic of mathematical objects?

I will call it "hypermodulus". In simple words, hypermodulus is the exponent of the scalar part of the finite part of the logarithm of the object: $H(A)=\exp (\operatorname{scal} \operatorname{f.p.} \ln A)$. The "finite part" is meant to mean…
Anixx
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Intuition for when a problem may be amenable to the "umbral calculus"?

I've always been interested in situations where we can apply "illegal" operations to objects and still solve problems (as seen here, say), and a common justification for these techniques is the umbral calculus. There are already many questions on…
Chris Grossack
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For the binomial Hopf algebra, what is the group of grouplike elements of the dual algebra?

The linear space of finite polynomials over a field $\mathbb{K}$ of zero characteristic has a structure of a graded connected bialgebra (meaning the zeroth subspace is isomorphic to the base…
Daigaku no Baku
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An equation by the definition of Bernoulli number

I am working on Bernoulli number. I learnt the definition of Bernoulli number on the book by a Japanese mathematician. The name of the book is Number Theory 1: Fermat's dream. The book defines the Bernoulli number by the formula…
TAO CHUAI
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