Timeline for How are Sheffer polynomials related to Lie theory?
Current License: CC BY-SA 4.0
22 events
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| Jul 12, 2023 at 18:31 | comment | added | Tom Copeland | @MichaelHardy, for me the most useful and revealing way to approach (and extend) Sheffer sequences and the associated operational calculus and its relation to umbral manipulations has been via the diff op reps of the umbral substitution operators (some papers by Lenart and Ray first opened the door for me). I posted a condensed set of notes on this at tcjpn.wordpress.com/2023/07/04/…. The delta (lowering) and raising ops are easily derived as conjugations. | |
| Jun 26, 2023 at 22:37 | comment | added | Andrius Kulikauskas | @TomCopeland Thank you! very much! I have made enormous use of a subsequent paper by Kim and Zeng from 2001 pdfs.semanticscholar.org/12fb/… The paper you cite covers a lot of helpful, relevant ground! | |
| Jun 25, 2023 at 21:38 | comment | added | Tom Copeland | Btw, a nice, short intro to some combinatorics of orthogonal Sheffer sequences is "Weighted Derangements And The Linearization Coefficients Of Orthogonal Sheffer Polynomials" by Zeng (researchgate.net/publication/…). | |
| Apr 12, 2023 at 19:48 | comment | added | Tom Copeland | @MichaelHardy, the conditions that the polynomials $P_n(x)$ have order $n$ and that they have a delta op are not sufficient to characterize even an Appell Sheffer sequence. It is necessary that $P_0(x) =1$ in order for the raising op to be of the form of that of the Sheffer formalism. The lack of this last condition has been a source of confusion for some. Best to be familiar with the several ways of characterizing Sheffer sequences. | |
| Apr 12, 2023 at 19:08 | comment | added | Tom Copeland | In addition, every sequence of Sheffer polynomials has an associated lower triangular matrix of coefficients and every sequence has a dual umbral inverse Sheffer sequence, whose coefficient matrix is orthogonal to, i.e., the multiplicative inverse of, that of its umbral dual. To conflate matters further, some use 'reciprocal polynomials' to mean an umbral inverse set of polynomials. So, the use of 'orthogonal', 'inverse', 'reciprocal' is often confusing. | |
| Apr 12, 2023 at 18:52 | comment | added | Tom Copeland | Btw, the use of the phrase 'Sheffer orthogonal polynomials' in "Meixner polynomials of the second kind and quantum algebras representing su(1,1)" by Gábor Hetyei can lead to confusion as well as his use of 'inverse'. The meaning of orthogonal and inverse should always be clearly stated. I use an 'umbral inverse sequence of polynomials' in the sense he uses 'inverse'. I reserve 'orthogonal sequence of polynomials' to mean self-orthogonality of a sequence of polynomials w.r.t. to a weight function. | |
| Dec 7, 2022 at 3:59 | comment | added | Michael Hardy | @TomCopeland : Rather than "one definition" one could say "one characterization", and then which characterization serves as a definition could depend on context. But I think there's something to be said for choosing as the definition a characterization that makes clear the motivation for defining the concept. | |
| Nov 3, 2022 at 21:11 | comment | added | Tom Copeland | @MichaelHardy, there is more than one way to skin a cat. Accordingly saying 'THE definition' is not precise. One should more accurately say 'ONE definition' or 'A definition'. Off-hand I can think of three or so equivalent definitions. I'm sure there are more, underlying the richness of the subject. (Andrius should give some restrictions on $A(t)$ and $B(t)$ to be precise, but this is a question not a treatise, and ambiguity tolerance is essential for a good discussion.) | |
| Nov 3, 2022 at 20:13 | comment | added | Andrius Kulikauskas | @Michael_Hardy thank you for writing the definition of the Sheffer sequence. The Wikipedia article en.wikipedia.org/wiki/Sheffer_sequence further states "A Sheffer sequence {$p_n$} is characterised by its exponential generating function ..." Are the two statements equivalent? I suppose but I don't know. But above I have simply written that they "have a generating function..." | |
| Nov 3, 2022 at 0:09 | comment | added | Michael Hardy | Is your first sentence intended to be understood as a definition of Sheffer polynomials? What I have taken to be the definition is that a sequence $(P_n(x))_{n=0,1,2,\ldots},$ where each $P_n(x)$ is an $n$th-degree polynomial in $x,$ is a Sheffer sequence if the linear operator on the space of all polynomials that takes $P_n(x)$ to $nP_{n-1}(x)$ for all $n,$ is shift-equivariant. | |
| Nov 2, 2022 at 17:38 | history | edited | Andrius Kulikauskas | CC BY-SA 4.0 |
deleting incorrect reference to orthogonal polynomials
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| Feb 5, 2022 at 21:29 | comment | added | Tom Copeland | @MarkWildon, self-containedness, like beauty, is in the eye of the beholder. // Btw, like your notes on comb. and sym. functions at your website. A couple of suggestions for the benefit of your self-motivated students: ref relevant OEIS entries and in addition for students majoring in physics add a supplement on connections to classical and quantum physics. Mathematics can be fun--mathematics + physics, sublime. | |
| Feb 5, 2022 at 20:46 | vote | accept | Andrius Kulikauskas | ||
| Feb 5, 2022 at 20:45 | comment | added | Mark Wildon | I'm delighted the question got such an interesting answer and have retracted my close vote. | |
| Feb 5, 2022 at 20:15 | history | became hot network question | |||
| Feb 5, 2022 at 17:07 | comment | added | Tom Copeland | Very few of the MO-Qs are self-contained, often alluding to concepts in category theory, topology, or algebraic geometry that take months or even years to master. Naturally the OP is seeking someone with the expertise in this field to direct him, and from my experience there are at most a handful of users here that are familiar with both Lie theory and the op calculus underlying Sheffer sequences to even partially address this question. He has a hunch there's a connection, so he asked. I constantly ask myself this Q when addressing the Sheffer and related calculus, often with fruitful results. | |
| Feb 5, 2022 at 16:53 | history | edited | LSpice | CC BY-SA 4.0 |
DOI links
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| Feb 5, 2022 at 16:15 | answer | added | Tom Copeland | timeline score: 15 | |
| Feb 5, 2022 at 13:12 | review | Close votes | |||
| Feb 5, 2022 at 20:46 | |||||
| Feb 5, 2022 at 13:00 | comment | added | Andrius Kulikauskas | Yes, I wish to ask that question. Can anyone say if it is more than an appearance? I myself don't know. Who can I ask? How can I ask them? Their answer, positive or negative, could save me a lot of effort. Thank you. | |
| Feb 5, 2022 at 12:57 | comment | added | Mark Wildon | I've voted to close as I think the question should be more self-contained and focussed. What is the connection with Lie groups and Lie algebras? Is it more than the appearance of things that look a bit like one-parameter semigroups in $e^{xu(t)}$? | |
| Feb 5, 2022 at 12:14 | history | asked | Andrius Kulikauskas | CC BY-SA 4.0 |