User Tools
Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
open-problems:family-of-constant-term-identities [2019/10/08 21:58] romik |
open-problems:family-of-constant-term-identities [2019/10/08 22:00] (current) romik |
||
|---|---|---|---|
| Line 7: | Line 7: | ||
| A_n = \frac{1!4!7!\cdots (3n-2)!}{n!(n+1)!\cdots(2n-1)!}. | A_n = \frac{1!4!7!\cdots (3n-2)!}{n!(n+1)!\cdots(2n-1)!}. | ||
| $$ | $$ | ||
| - | Define a polynomial $P_n(z_1,\ldots,z_n)$ by | + | Define a multivariate polynomial $P_n(z_1,\ldots,z_n)$ by |
| $$ | $$ | ||
| P_n(z_1,\ldots,z_n) = \prod_{1\le i<j\le n} (z_j-z_i)(1+z_j+z_i z_j) \prod_{j=2}^n (1+z_j). | P_n(z_1,\ldots,z_n) = \prod_{1\le i<j\le n} (z_j-z_i)(1+z_j+z_i z_j) \prod_{j=2}^n (1+z_j). | ||
| $$ | $$ | ||
| Zinn-Justin and Di Francesco (2008) discovered and proved (with a little help from Zeilberger (2007)) the constant term identity | Zinn-Justin and Di Francesco (2008) discovered and proved (with a little help from Zeilberger (2007)) the constant term identity | ||
| - | \begin{equation} | + | $$ |
| - | [z_1^0 z_2^2 \cdots z_n^{2n-2}] P_n(z_1,\ldots,z_n) = A_n, \tag{*} | + | [z_1^0 z_2^2 \cdots z_n^{2n-2}] P_n(z_1,\ldots,z_n) = A_n, \qquad\qquad (*) |
| - | \end{equation} | + | $$ |
| - | where the notation $[z_1^{j_1} \cdots z_n^{j_n}] Q(z_1,\ldots,z_n)$ refers to the coefficient in a multivariate polynomial $Q(z_1,\ldots,z_n)$ of the monomial $z_1^{j_1} \cdots z_n^{j_n}$. | + | where the notation $[z_1^{j_1} \cdots z_n^{j_n}] Q(z_1,\ldots,z_n)$ refers to the coefficientof the monomial $z_1^{j_1} \cdots z_n^{j_n}$ in a multivariate polynomial $Q(z_1,\ldots,z_n)$ . |
| - | Romik (2014) discovered a family of identities generalizing (*). The idea is that, fixing $k\ge 1$ and a "deformation" vector $(a_1,\ldots,a_k)$ of integers, we deform the polynomial $P_n(z_1,\ldots,z_n)$ by replacing it with the slightly modified version | + | Romik (2014) discovered a family of identities generalizing $(*)$. The idea is that, fixing $k\ge 1$ and a "deformation" vector $(a_1,\ldots,a_k)$ of integers, we deform the polynomial $P_n(z_1,\ldots,z_n)$ by replacing it with the slightly modified version |
| $$ | $$ | ||
| P_{n,k}(z_1,\ldots,z_n) = \prod_{1\le i<j\le n} (z_j-z_i)(1+z_j+z_i z_j) \prod_{j=k+2}^n (1+z_j) \qquad (n\ge k+2), | P_{n,k}(z_1,\ldots,z_n) = \prod_{1\le i<j\le n} (z_j-z_i)(1+z_j+z_i z_j) \prod_{j=k+2}^n (1+z_j) \qquad (n\ge k+2), | ||