Timeline for How are Sheffer polynomials related to Lie theory?
Current License: CC BY-SA 4.0
37 events
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| Aug 29 at 15:15 | comment | added | Tom Copeland | See also my post "Pre-Lie algebras, Cayley’s analytic trees, and mathemagical forests" tcjpn.wordpress.com/2018/07/10/pre-lie-algebra-and-cayley-trees. | |
| Aug 1 at 21:05 | history | edited | Tom Copeland | CC BY-SA 4.0 |
sign/pasting error corrected
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| Aug 1 at 13:48 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Added explicit relations to the infinigen for fractional calculus and connections to the BCH theorem
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| Feb 23 at 21:14 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Missing sign in argument added. Additional link to Kummer functions added.
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| Dec 16, 2023 at 21:06 | comment | added | Andrius Kulikauskas | @Tom_Copeland Thank you! I am returning to this topic and hope to connect with you! | |
| Dec 14, 2023 at 22:27 | comment | added | Tom Copeland | I mentioned Feinsilver above. Here is a ref: "Lie algebras, Representations, and Analytic Semigroups through Dual Vector Fields" chanoir.math.siu.edu/MATH/Merida/PDF/Merida.pdf | |
| Jul 13, 2023 at 22:38 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Edit by another user reverse for a symbol
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| Jul 13, 2023 at 17:11 | history | edited | Michael Hardy | CC BY-SA 4.0 |
added 45 characters in body
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| Jul 2, 2023 at 1:56 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Typo corrected and proof added
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| Apr 12, 2023 at 23:46 | history | edited | Michael Hardy | CC BY-SA 4.0 |
added 295 characters in body
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| Nov 5, 2022 at 6:07 | comment | added | Andrius Kulikauskas | @Tom_Copeland Thank you! I appreciate your help very much. I'm intrigued by the connections, including with the associahedra polynomials. | |
| Nov 3, 2022 at 20:51 | comment | added | Tom Copeland | Btw, the refinement of the Stirling polynomials of the second kind is OEIS A036040, which is the coproduct of the Faa di Bruno Hopf algebra (see the Figueroa et al. paper in the entry and Zeidler in his book QFT I, p. 859 onward ). In some contexts, a more convenient/enlightening basis for composition with the famous associahedra polynomials for the antipode / compositional inverse is the set of refined Lah partition polynomials of A130561. Both of these, of course, are related to Scherk-Graves-Lie infinigins that are umbralizations of infinigins for the coarser polynomials. | |
| Nov 3, 2022 at 20:19 | comment | added | Andrius Kulikauskas | @Tom_Copeland Yes! That is a lovely formula. I found a link to that paper: maths.ed.ac.uk/~v1ranick/papers/rota2.pdf | |
| Nov 2, 2022 at 19:12 | comment | added | Tom Copeland | Then you should also pay particular attention to Gian Carlo Rota on the Dobinski formula (see ref at en.m.wikipedia.org/wiki/Dobi%C5%84ski%27s_formula). | |
| Nov 2, 2022 at 17:35 | comment | added | Andrius Kulikauskas | @Tom_Copeland Thank you! This is all very helpful for me. Especially as I am making a video about the connection between Sheffer polynomials and Bell numbers, which are sums of Stirling numbers of the second kind. | |
| Nov 1, 2022 at 16:47 | comment | added | Tom Copeland | Other related refs are "Boson Normal Ordering via Substitutions and Sheffer-type Polynomials" by Blasiak, Horzela, Penson, Duchamp, and Solomon; "Normal ordering problem and the extensions of the Striling grammar" by Ma, Mansour, and Schork; "Combinatorial Models of Creation-Annihilation" by Blasiak and Flajolet; and the book Commutation Relations, Normal Ordering, and Stirling Numbers by Mansour and Schork with an extensive bibliography. | |
| Oct 17, 2022 at 10:25 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Explicated relation to normal ordering of Witt-Lie ops and fractional calculus
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| Sep 22, 2022 at 17:04 | comment | added | Andrius Kulikauskas | Thank you! I appreciate that. The paper is relevant and very readable, both concrete and intuitive. | |
| Sep 21, 2022 at 3:14 | comment | added | Tom Copeland | See also section XIV: GROUP THEORY AND SPECIAL FUNCTIONS in :Group Theory" by Gilmore (physics.drexel.edu/~bob/GroupTheory/group.pdf) | |
| Feb 6, 2022 at 20:24 | comment | added | Andrius Kulikauskas | Oh, good! @TomCopeland Thank you for the references and the link to the post. I will understand better and also know the extent to which I'm working in a new direction. | |
| Feb 6, 2022 at 16:36 | comment | added | Tom Copeland | @AndriusKulikauskas, I added some refs in comments to the MO-Q on Wick's theorem mathoverflow.net/questions/47350/whats-up-with-wicks-theorem | |
| Feb 6, 2022 at 8:15 | comment | added | Andrius Kulikauskas | Thank you for the links. I look forward to learning from them and also from your blog. Best wishes in your research! Thank you for sharing your knowledge. | |
| Feb 6, 2022 at 1:05 | comment | added | Tom Copeland | @AndriusKulikauskas, the article on causal relations I mentioned is "Causal set actions in various dimensions" by Lisa Glaser with link in oeis.org/A039683. I'm sitting on three long drafts of notes, so it'll be a couple of weeks before I track down the relevant articles on Wick's thm, but see oeis.org/A344678, mathoverflow.net/questions/40268/…, and my two blog posts "The Heat Equation" and "Cycles and Heat". | |
| Feb 5, 2022 at 21:56 | comment | added | Andrius Kulikauskas | Thank you for replying! I do have a lot to learn and more questions to ask some day. I appreciate any links to the relation between Wick's theorem, Hermite polynomials, Gaussian pdf because I haven't found any. Kim and Zeng have a nice formula for the moments in terms of ascents and descents of permutations which I interpret as expanding the raising and lowering operators. Also, if each particle has a clock, then I imagine Minkowski space comes for free. For me, the weight function is just a wrapper expressing space-time, it seems, probabilistically. | |
| Feb 5, 2022 at 21:42 | comment | added | Tom Copeland | @AndriusKulikauskas, thks for the compliment. Yes, connections to QFT are wide and deep. I tried to include several refs on apps in numerous fields of math and physics in more OEIS entries than I can shake a stick at. There is one entry with a reference to a paper on a formalism of causality that I'm not familiar with. I'll try to track it down. (The relation of Wick's theorem to perfect matchings and, therefore, the Hermite polynomials and the Gaussian pdf is nicely presented by some authors. I also, recently came across a paper (?) with the thm. related to free probability.) | |
| Feb 5, 2022 at 21:06 | comment | added | Andrius Kulikauskas | Some notes on my work-in-progress. math4wisdom.com/wiki/Exposition/ResearchProgramFivesome | |
| Feb 5, 2022 at 21:04 | comment | added | Andrius Kulikauskas | Kim and Zeng's paper can be intepreted as yielding trees that exhibit double causality (links and kinks) because the recurrence relation for the polynomials has two inputs $P_{n-1}$, $P_n$ for one output $P_{n+1}$. Alternatively, there are particle-clocks that count steps $a$ from between two events and steps $b$ going back. There are five kinds of Sheffer polynomials: Meixner (a,b), Charlier (a,0), Laguerre (a,a), Hermite (0,0), Meixner-Pollaczek (a,$\bar{a}$). I think the ladder operators behind the Wick contractions could be expanded to mark time between events, in 5 such zones. | |
| Feb 5, 2022 at 20:55 | comment | added | Andrius Kulikauskas | The raising and lowering operators have immediate resonance for me. I am a lifelong philosopher with a Math PhD (algebraic combinatorics) trying now to show how cognitive frameworks show up in math. This includes a fivefold framework for space and time: "Every cause has had its efffect, yet not every effect has had its cause", and there is a fifth perspective where these two causalities meet or not. I noticed the combinatorics of orthogonal polynomials shows up in Schroedinger's equation but also Wick's theorem in quantum field theory. | |
| Feb 5, 2022 at 20:48 | comment | added | Andrius Kulikauskas | Dear @Tom_Copeland, Fantastic! You provided me with multiple answers to study, understand and pursue further. Thank you for the links! I am especially grateful to learn of you and your blog. "Go forward, faith will follow." | |
| Feb 5, 2022 at 20:46 | vote | accept | Andrius Kulikauskas | ||
| Feb 5, 2022 at 20:05 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Added generator for general Sheffer sequence
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| Feb 5, 2022 at 19:03 | history | edited | Tom Copeland | CC BY-SA 4.0 |
link, Comtet ref and more on commutators
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| Feb 5, 2022 at 18:39 | history | edited | Tom Copeland | CC BY-SA 4.0 |
more connections
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| Feb 5, 2022 at 17:14 | history | edited | Tom Copeland | CC BY-SA 4.0 |
corrected variable
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| Feb 5, 2022 at 16:58 | history | edited | LSpice | CC BY-SA 4.0 |
Link to Vilenkin book; TeX dashes -> Unicode (except em-dashes in TeX are ---)
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| Feb 5, 2022 at 16:47 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Typos
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| Feb 5, 2022 at 16:15 | history | answered | Tom Copeland | CC BY-SA 4.0 |