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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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 field):
$\...
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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 ...
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In umbral calculus, what is the established value of $\operatorname{eval}\ln (B+1)$?
In umbral calculus, what is the established value of evaluation (index-lowering operator) of the logarithm of $B+1$ where $B$ is Bernoulli umbra?
In this preprint the author argues it to be $-\gamma$, ...
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Closed Form for Geometric-like Finite sum of Bell Polynomials
I'm trying to see if there's a nice closed form expression for the following sum:
$\sum_{k=0}^{M} \cos(\pi k t) B_k(x)$
where $M \in \mathbb{N}$, $t \in (0,1)$, and $x \in \mathbb{R}^+$.
Notation: ...
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Basic sequence of $S_{-y}(\delta)$
Let $\delta$ be a delta-operator with associated basic sequence $p_0=1,p_1,p_2,p_3,...$ and consider the shift map $S_{-y}:K[x]\to K[x]$ given by $f(x)\mapsto f(x-y)$, where $f$ is a polynomial whose ...
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What's the intuitive meaning of this relation between volumes of $n$-balls and umbrial calculus? [duplicate]
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 $${\displaystyle \pi ^{-{s \...
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What's the intuitive meaning of this relation between volumes of $n$-balls and umbral calculus?
The volume of an $n$-ball of radius $1$ is $$V_{n}={\frac {\pi ^{n/2}}{\Gamma {\bigl (}{\tfrac {n}{2}}+1{\bigr )}}}.$$
The functional equation of Riemann zeta function is $${\displaystyle \pi ^{-{s \...
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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 $B_n(x)$. ...
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What is Bernoulli umbra philosophically?
Well, Bernoulli umbra is an umbra whose moments are the Bernoulli numbers.
But what is it philosophically?
For instance, we can consider imaginary unit $i$ an umbra with moments $\{1,0,-1,0,1,...\}$, ...
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Is there an accurate representation of Bernoulli umbra?
Bernoulli umbra is some object $B$, an element of a commutative ring, such that there is an “index lowering” linear operator $\operatorname{eval}$ which applied to $B^n$ will give $B_n$, the $n$-th ...
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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 ...
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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, ...
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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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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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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 ...
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