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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10 votes
1 answer
156 views

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 ...
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0 votes
0 answers
22 views

Is There a Finite Ratio Operator $_2\Delta$ so that $_2\Delta_n f(n) = \frac{f(n + 1)}{f(n)}$?

In mathematics, there is a finite difference operator $\Delta$ defined by $\Delta_n f(n) = f(n + 1) - f(n)$. This operator shares many properties with the continuous derivative $\mathcal{D}$. However, ...
1 vote
0 answers
94 views

Is there any intuition of why the both, regularized logarithm of zero is $-\gamma$ and the regularized logarithm of Bernoulli umbra is $-\gamma$?

If we take the MacLaurin series for $\ln(x+1)$ and evaluate it at $x=-1$, we will get the Harmonic series with the opposite sign: $-\sum_{k=1}^\infty \frac1x$. Since the regularized sum of the ...
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0 votes
0 answers
11 views

Intuitively, what makes Bernoulli umbra so similar to the zero divisors in split-complex numbers?

Notation. Here I will denote Bernoulli umbra (its moments are Bernoulli numbers $B_n$) as $B_-$, $B_-+1$ as $B_+$ (an umbra with moments being Bernoulli numbers except $B_1=1/2$). I will denote the ...
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3 votes
0 answers
150 views

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} \...
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0 votes
0 answers
21 views

Different rings of divergent integrals and parallels with finite-dimensional algebras over $\mathbb{R}$

It seems that an operation of multiplication can be defined on the set of divergent integrals in different ways. To me this seems analogous to how different (hyper)complex number systems can be ...
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3 votes
0 answers
73 views

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 ...
2 votes
1 answer
99 views

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 ...
2 votes
0 answers
68 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}{\...
0 votes
2 answers
62 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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4 votes
0 answers
85 views

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\...
3 votes
2 answers
212 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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1 vote
1 answer
102 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 ...
  • 317
6 votes
2 answers
200 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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5 votes
1 answer
262 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 votes
2 answers
197 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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31 votes
2 answers
6k 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 ...
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