All Questions
Tagged with differential-operators co.combinatorics
9 questions
2
votes
0
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Transform connecting powers of integration and differentiation operators
Just by a chance, I found the following power series identity, which holds for any analytic function $F(\cdot)$, nonnegative integer $m$, and constants $u,v$ not depending on indeterminates $z,t$:
$$\...
- 34.3k
2
votes
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106
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Evaluate action of $f(\frac{d}{dx})$ using the Fourier/Laplace transform
Consider a function $f(x)$ that is numerically defined in $-1 \leq x \leq 1$ interval (assume $N$ samples). I am trying to compute the action of $f(d/dx)$ on a function $g(x)$ using the Fourier ...
- 350
3
votes
0
answers
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Shift Operators and the Weyl Algebra
I have a question about the action of a shift operator $E$ on polynomials $Ep(x) = p(x+1)$ in the context of linear differential operators in one variable with polynomial coefficients, i.e. ...
3
votes
3
answers
637
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Expansions of iterated, or nested, derivatives, or vectors--conjectured matrix computation
The entry OEIS A139605 (also related OEIS A145271) has a matrix computation for the partition polynomials that represent the expansions of iterated derivatives, or vectors in differential geometry,
$...
- 10.5k
4
votes
2
answers
210
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permutations rescuing chain/product rules?
Let $\mathfrak{S}_n$ be the permutation group on $[n]$. Denote the cardinality of $\{\pi\in\mathfrak{S}_n: \pi^2=id\}$, the set of involutions, by $I(n)$.
It is well-known that these numbers have the ...
- 43.2k
7
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124
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in search of intepretations and connections for $k$-central binomials
Fix a positive integer $k$. Then, the sequences
$$c(n,k)=\frac{k^n}{n!}\prod_{m=1}^{n-1}(1+km)=[x^n]\left(\frac1{1-k^2x}\right)^{1/k}$$
are referred to as "$k$-central binomial coefficients",...
- 43.2k
1
vote
1
answer
316
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Combinatorics: Product Rules.
I couldn't find a way to figure this out, though it is a somewhat basic question that came up when studying the stationary phase expansion of an integral. The abstract version is the following:
I ...
- 9,853
5
votes
0
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273
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root system generalizations of Sekiguchi-Debiard (aka Laplace-Beltrami) operators
For the root system $A_n$, taking the limit $q = t^\alpha$ and $t \to 1$, and letting $Y = (t-1) X -1$ one obtains from the Macdonald operator the so-called Sekiguchi-Debiard operator:
$$D_\alpha(X) =...
- 4,466
5
votes
1
answer
584
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differential operator power coefficients
Let $(F(x)\frac{d}{dx})^n=\sum_{i=1}^n H_{n,i}(F, F', F^{(2)}, \ldots , F^{(n)})\frac{d^i}{dx^i}$. I'm curious about the exact formula for $H_{n,i}(y_0, y_1, \ldots , y_n)$. What is known about it?
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