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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48 views

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}{\...
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Lagrange inverse formula in positive characteristic

Does it exist an inverse Lagrange formula formula in positive characteristic (maybe with Hasse derivative). M ARILENA BARNABEI in "Lagrange Inversion in Infinitely Many Variables" said such ...
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2answers
55 views

Inversion of linear combination of discrete shift operators

I have recently tackled the following problem and I'm seeking for some help. Let me define the shift operator \begin{equation} T_h[\cdot], h \in \mathbb{Z} \end{equation} such that \begin{equation} ...
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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
157 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
74 views

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
168 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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1answer
189 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
182 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 ...
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5k 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 ...