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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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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The discriminant of a certain Appell sequence of polynomials
Consider the following sequence of polynomials
$$p_n(x) = x^n + \binom{n}{1} a x^{n-1}+ \binom{n}{2} a(a+h) x^{n-2} + \cdots + a(a+h) \cdots ( a+ (n-1) h) = \sum_{k=0}^n \binom{n}{k} a^{(k)} x^{n-k} $...
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Basic sequence of $S_{-y}(\delta)$
Let $\delta$ be a delta-operator with associated basic sequence $p_0=1,p_1,p_2,p_3,...$ and consider the shift map $S_{-y}:K[x]\to K[x]$ given by $f(x)\mapsto f(x-y)$, where $f$ is a polynomial whose ...
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What's the intuitive meaning of this relation between volumes of $n$-balls and umbrial calculus? [duplicate]
The volume of an $n$-ball of radius $1$ is $$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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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 ...
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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}{\...
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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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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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