Timeline for Do you recognize this sequence of polynomials?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 9, 2021 at 22:07 | comment | added | Peter Luschny | Following on Ira Gessel's comments one might note that this is the simplest case of the super ballot numbers. A generalization of the integral formula is given in OEIS A135573. | |
| Nov 4, 2021 at 20:02 | comment | added | Tom Copeland | Do your students a service by introducing them to the OEIS if you haven't already. If they don't find the numerous connections interesting, I suggest they pursue a major outside of STEM. | |
| Nov 4, 2021 at 18:53 | comment | added | Tom Copeland | Relations among the Gegenbauer, Jacobi, Legendre, and the polynomials highlighted in this post can be found in the OEIS entries to which I linked in my answer. | |
| Nov 4, 2021 at 18:17 | vote | accept | David Richter | ||
| Nov 4, 2021 at 13:11 | answer | added | Tom Copeland | timeline score: 5 | |
| Nov 4, 2021 at 13:11 | comment | added | Ira Gessel | It might be noted that this integral is, up to normalization, a special case of the beta integral. The general beta integral corresponds to Jacobi polynomials (though with a different normalization than usual). | |
| Nov 4, 2021 at 10:35 | comment | added | David Richter | Ira: This looks very promising. This makes me think that one can get a special case of the Jacobi polynomials from the Legendre polynomials (which I realize are also another special case of Jacobi polynomials) using this transformation that I describe in the original post. (I wish I knew more about these transformations in general.) If this works, then this would mean that the roots of my polynomials always lie in the interval (0,4). Thank-you. Now I will tinker around with this some more. | |
| Nov 4, 2021 at 6:15 | history | became hot network question | |||
| Nov 4, 2021 at 0:32 | history | edited | Max Alekseyev | CC BY-SA 4.0 |
corrected some typos
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| Nov 3, 2021 at 23:32 | answer | added | Max Alekseyev | timeline score: 12 | |
| Nov 3, 2021 at 23:13 | comment | added | Ira Gessel | The integral representation $$C_n = \frac{1}{2\pi} \int_0^4 t^n \sqrt{\frac{4-t}{t}}\, dt$$ can be found in cs.uwaterloo.ca/journals/JIS/VOL4/SIXDENIERS/Catalan.html, though I am sure it is much older. | |
| Nov 3, 2021 at 22:56 | answer | added | Peter Taylor | timeline score: 11 | |
| Nov 3, 2021 at 22:10 | history | asked | David Richter | CC BY-SA 4.0 |