# Questions tagged [umbral-calculus]

The umbral-calculus tag has no usage guidance.

18
questions

1
vote

0
answers

127
views

### Can't parse a statement in an article on coalgebras and umbral calculus

This question is cross-posted from MSE.
I am reading Nigel Ray's "Universal Constructions in Umbral Calculus" (1998, published in "Mathematical Essays in Honor of Gian-Carlo Rota", ...

0
votes

0
answers

125
views

### Building representation of an arbitrary umbral calculus

Consider a set of integrable functions on the interval $(0,1)$.
Let's introduce an operation $\operatorname{eval}f=\int_0^1 f(x)\,dx$ (which is the mean value of the function).
In such system the ...

10
votes

3
answers

1k
views

### What did Rota mean by "one can define cumulants relative to any sequence of binomial type"?

This question is cross-posted from MSE.$\newcommand{\E}{\mathbb{E}}$
Near the end of "Finite Operator Calculus" (1976), G.C. Rota writes:
Note that one can define cumulants relative to any ...

5
votes

2
answers

1k
views

### Are umbral moonshine and umbral calculus connected?

In a 2013 article, Cheng, Duncan and Harvey introduce the concept of umbral moonshine as a generalization of monstrous moonshine. The terminology they use, starting with the title, is common in umbral ...

5
votes

1
answer

206
views

### Categorical description of umbral calculus?

The theme of my current research project is related to umbral calculus in the context of more general algebraic structures, like Hopf algebras (and, in particular, shuffle algebras), so I am trying to ...

3
votes

0
answers

184
views

### What's the 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 \...

0
votes

2
answers

375
views

### 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 $B_n(x)$. ...

-2
votes

1
answer

400
views

### 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 $\{1,0,−1,0,1,\ldots\}$...

1
vote

0
answers

111
views

### Crazy conjecture about Bernoulli umbra and reference request

For years umbral calculus have fascinated me. Bernoulli numbers (which represent powers of Bernoulli umbra) are involved in many classic power series expansions.
Yet, it still remains mistery what ...

4
votes

0
answers

133
views

### 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 ...

6
votes

0
answers

308
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 ...

1
vote

0
answers

100
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 ...

2
votes

1
answer

141
views

### Hadamard product of linear recurrences with umbral calculus

Let $R$ be a ring, $d_0, d_1, d_2, \dots \in R$ and $e_0, e_1, e_2, \dots \in R$ be linear recurrence sequences, such that
$d_m = a_1 d_{m-1} + a_2 d_{m-2} + \dots + a_k d_{m-k}$ for $m \geq k$,
$e_m ...

1
vote

0
answers

109
views

### What is some algebraic intuition behind the fact that the (real part) of the logarithm of Bernoulli umbra plus $1$, is $-\gamma$?

Bernoulli umbra is defined in classical umbral calculus as in Taylor - Difference equations via the classical umbral calculus.
Yu - Bernoulli Operator and Riemann's Zeta Function shows that $\...

3
votes

0
answers

142
views

### Interpreting umbral calculus in terms of some kind of extended numbers

I know that currently umbral calculus is developed as some kind of theory of operators and functionals but were there any attempts to put it on a more solid philosophical grounds as study of functions ...

7
votes

3
answers

1k
views

### Ramanujan's Master Formula: A proof and relation to umbral calculus

The Ramanujan's master theorem states that:
$$
\int_0^{\infty}x^{s-1}\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}a_nx^ndx=\Gamma(s)a_{-s}
$$
I found a really strange proof recently on a personal blog:
Define
$...

2
votes

0
answers

90
views

### Rational zeta series and differential-difference equations

In an earlier question, I mentioned I was looking for generalizations of $$\sum_{n=k+2}^{\infty} \binom{n-1}{k} (\zeta(n) -1) =1. \qquad \qquad (1) $$
A variation of the above identity arises by ...

8
votes

1
answer

382
views

### Are any interesting classes of polynomial sequences besides Sheffer sequences groups under umbral composition?

This question on math.stackexchange.com has 35 views, three up-votes, and not a word from anybody, so I'm posting it here.
Let us understand the term polynomial sequence to mean a sequence $(p_n(x))_{...