Timeline for Cauchy identity, with sum restricted over partitions with first part $\leq n$
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 25, 2021 at 19:30 | comment | added | Sergii Voloshyn | @IraGessel In another case, if we put condition $λ_1<N$ for $ λ=(λ_1,..,λ_l)$ we arrive to another expression at the $h=1$ (unlike the $-κ^2 \log [1−h^2]$ that divergent in that point), and principally we can make expansion around $h=1$ . ($N$ and $N_f$ both tend to ∞ with $κ=N_f/N$ fixed.) First terms appears in another topic mathoverflow.net/questions/406707/… | |
| May 20, 2017 at 15:23 | comment | added | Ira Gessel | I don't think there is a simple product expression. You can check it by just computing the sum for small values of $n$ and small numbers of variables and seeing if it factors. | |
| May 20, 2017 at 11:16 | vote | accept | D. Donnelly | ||
| May 20, 2017 at 11:15 | comment | added | D. Donnelly | Thank you! I was actually wondering if there is also a simpler product expression for this determinant. I know there is one for the determinant in the aforementioned Theorem 16 (namely $\prod_{j,k=1}^{n}(1-x_{j}y_{k})^{-1}$, the one in the question); this is Baxter's identity. There is also such an expression for the determinant resulting from applying the involution $\omega$ in only one of the sets of variables $x$ or $y$ (namely $\prod_{j,k=1}^{n}(1+x_{j}y_{k})$. Do we have an analogue of Baxter's identity for the determinant appearing in your answer? | |
| May 19, 2017 at 15:39 | history | answered | Ira Gessel | CC BY-SA 3.0 |