Timeline for A question on the real root of a polynomial
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Feb 20, 2023 at 2:16 | vote | accept | Connor | ||
| Feb 28, 2023 at 10:43 | |||||
| Feb 19, 2023 at 18:20 | comment | added | Timothy Chow | In the process of deleting his incorrect comment, Ira Gessel also deleted the math.SE link, Algorithm for computing Hadamard product of two rational generating functions. This is another way to prove the correctness of the empirically derived formulas: take Hadamard products of (suitably rescaled) generating functions for the Chebyshev polynomials, and confirm that they give you the generating function of the $f$ polynomials. | |
| Feb 19, 2023 at 15:47 | comment | added | Ira Gessel | The OEIS page for the Lucas polynomials uses a different normalization. Here is a corrected version of my comment on them. The polynomials $h_n(x)$ in Tim's answer have the generating function $\sum_{n=0}^\infty h_n(x) y^n = (2-xy)/(1-xy-xy^2)$. They are given by $h_0(x)=2$ and $$h_n(x) = \sum_{k=0}^{\left\lfloor n/2\right\rfloor}\frac{n}{n-k}\binom{n-k}{k}x^{n-k}$$ for $n>0$. They are related to the Chebyshev polynomials of the first kind $T_n(x)$ by $$h_n(x) = 2(-\sqrt{-x})^{n} T_n(\sqrt{-x}/2)$$ so their roots can be determined from the roots of the Chebyshev polynomials. | |
| Feb 19, 2023 at 8:26 | comment | added | Connor | Thanks for Timothy Chow and Ira Gessel's answers, amazing construction! At first, we tried to use the Sturm sequence to prove the real zeros, but it seem to be impossible :( | |
| Feb 19, 2023 at 8:10 | vote | accept | Connor | ||
| Feb 20, 2023 at 2:16 | |||||
| Feb 19, 2023 at 4:18 | comment | added | Timothy Chow | I also just discovered that the connection between the Lucas polynomials and $T_n$ is given in Section 6.4 of On the characteristic polynomial of Cartan matrices and Chebyshev polynomials by Pantelis A. Damianou. | |
| Feb 19, 2023 at 3:16 | history | edited | Timothy Chow | CC BY-SA 4.0 |
added 4 characters in body
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| Feb 19, 2023 at 3:08 | history | edited | Timothy Chow | CC BY-SA 4.0 |
Added empirically guessed formulas for the remaining polynomials.
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| Feb 19, 2023 at 0:22 | comment | added | Ira Gessel | So $f_{2n}(x) = (-x)^n U_n(\sqrt{-x}/2)^2$, where $U_n$ is the Chebyshev polynomial of the second kind. Most likely there is some sort of expression for $f_{2n+1}(x)$ in terms of Chebyshev polynomials but it will not be as simple. | |
| Feb 18, 2023 at 14:46 | history | answered | Timothy Chow | CC BY-SA 4.0 |