Timeline for Why is umbral calculus not used more widely?
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8 events
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| Jun 28, 2023 at 2:56 | comment | added | Tom Copeland | Gessel's comment in mathoverflow.net/questions/328469/eulerian-number-identity/… : "Here's a proof of the identity. Unfortunately, it doesn't show how anyone would come up with this formula." If you understand how the Stirling numbers of the first and second kinds, the Bernoulli numbers, and the Eulerian numbers are related via umbral calculus, you can't help but come up with the identity the question addresses. | |
| Jun 28, 2023 at 1:15 | comment | added | Tom Copeland | 3) Look at the simple presentation of the Todd op in mathoverflow.net/questions/380142/… and the intuitive extension of the Bernoulli numbers to the Hurwitz zeta function. | |
| Jun 28, 2023 at 1:09 | comment | added | Tom Copeland | 2) tcjpn.wordpress.com/2022/08/02/… contains (pdf reader frame 17) an umbral identity for raising the reciprocal polynomials defined by $1/(1+u_1 x + u_2 x^2 + \cdots) = 1 + \sum_{n \geq 1} R_n(u_1,...,u_n) x^n$; that is $R_n(u_1,...,u_n) = b. R_{n-1}(u_1-b.,u_2,...,u_{n-1})$ where $b_k$ are the partition polynomials of oeis.org/A355201. I'd like to see a conventional derivation. | |
| Jun 28, 2023 at 0:51 | comment | added | Tom Copeland | 1) Contrast my easy self-contained answer to mathoverflow.net/questions/172955/… with the two other answers that invoke theorems from external references and are less general in scope. // | |
| Jun 27, 2023 at 22:58 | comment | added | Timothy Chow | @TomCopeland You are correct that I have not made serious use of the umbral calculus in my own work. Can you give more examples where the umbral calculus succeeds where other methods fail, or at least encounter serious difficulties? I do know Riordan's book and I don't recall any convincing examples from there, since his applications of the umbral calculus were pretty tame. | |
| Jun 27, 2023 at 21:21 | comment | added | Tom Copeland | Many of Rota's identities can easily be extended and made more useful with introduction of the diff and integral ops, such as the Mellin transform. // Math is done by clans (tribus / écoles, Doron Zeilberger might say) of researchers, some more insular and biased than others. The combinatorialist John Riordan in his book Combinatorial Identities published in the 1960s made extensive use of umbral techniques, and noting, e.g., the binomial transform at a trivial level is $e^x e^{b.x} = e^{(1+b.)x}$, where $(b.)^n = b_n$, e.g.f.s arguments can typically easily be subsumed under umbral calculus. | |
| Jun 27, 2023 at 21:00 | comment | added | Tom Copeland | Suspect you haven't seriously explored the umbral calculus / finite operator calculus / Sheffer sequence formalism and that the hearsay you present allows you the comfort of not having to do so. From what I've read of Gessel's and Stanley's works, they tend to approach problems from the perspective of linear algebra and don't appear so familiar with the diff / matrix ops and integral transforms that underlie the power of the umbral calculus to suggest and derive new identities. Even the revisionist Rota acknowledged the relative awkwardness of the linear functional approach he championed. | |
| Jul 11, 2021 at 18:29 | history | answered | Timothy Chow | CC BY-SA 4.0 |