# Questions tagged [umbral-calculus]

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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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### 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 \...

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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 $\{1,0,−1,0,1,\ldots\}$...

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### 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 ...

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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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### 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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### 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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### 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 ...

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### 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 $\...

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### 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 ...

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### 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
$...

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### 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 ...

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### 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))_{...