Painlevé IV, bi-confluent Heun equations and the Hankel determinant generated by a discontinuous semi-classical Laguerre weight
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arXiv
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| Natura: | Preprint |
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2026
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| author | Zhu, Mengkun Yu, Jianduo |
| author_facet | Zhu, Mengkun Yu, Jianduo |
| contents | We consider the discontinuous semi-classical Laguerre weight function with a jump $w(x;t,s)=\mathrm{e}^{-x^2+tx}(A+Bθ(x-s))$, where $x\in\mathbf{R}$, $t,s\ge0$, $A\ge0$, $A+B\ge0$, where $θ(x)$ is 1 for $x > 0$ and 0 otherwise. Based on the ladder operator approach, we obtain some important difference and differential equations about the auxiliary quantities and the recurrence coefficients. By proper tranformation, It is shown that $R_{n}(t,s)$ is related to Painlevé IV equations and $r_{n}(t,s)$ satisfies the Chazy II equations. With the aid of Dyson's Coulomb fluid approach, we derive the asymptotic expansions for $α_{n}$ and $β_{n}$ as $n\rightarrow\infty$. Furthermore, This enables us to obtain the lagre $n$ behavior of the orthogonal polynomials and derive that they satisfy the biconfluent Heun equation. We also consider the Hankel determinant $D_{n}(t,s)$ generated by the discountinuous semi-classical Laguerre weight. We find that the quantity $σ_{n}(t,s)$, allied to the logarithmic derivative of $D_{n}(t,s)$, satisfies the Jimbo-Miwa-Okamoto $σ$-form of Painlevé IV. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16548 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Painlevé IV, bi-confluent Heun equations and the Hankel determinant generated by a discontinuous semi-classical Laguerre weight Zhu, Mengkun Yu, Jianduo Classical Analysis and ODEs 33C47, 34M55, 35C20, 65Q99 We consider the discontinuous semi-classical Laguerre weight function with a jump $w(x;t,s)=\mathrm{e}^{-x^2+tx}(A+Bθ(x-s))$, where $x\in\mathbf{R}$, $t,s\ge0$, $A\ge0$, $A+B\ge0$, where $θ(x)$ is 1 for $x > 0$ and 0 otherwise. Based on the ladder operator approach, we obtain some important difference and differential equations about the auxiliary quantities and the recurrence coefficients. By proper tranformation, It is shown that $R_{n}(t,s)$ is related to Painlevé IV equations and $r_{n}(t,s)$ satisfies the Chazy II equations. With the aid of Dyson's Coulomb fluid approach, we derive the asymptotic expansions for $α_{n}$ and $β_{n}$ as $n\rightarrow\infty$. Furthermore, This enables us to obtain the lagre $n$ behavior of the orthogonal polynomials and derive that they satisfy the biconfluent Heun equation. We also consider the Hankel determinant $D_{n}(t,s)$ generated by the discountinuous semi-classical Laguerre weight. We find that the quantity $σ_{n}(t,s)$, allied to the logarithmic derivative of $D_{n}(t,s)$, satisfies the Jimbo-Miwa-Okamoto $σ$-form of Painlevé IV. |
| title | Painlevé IV, bi-confluent Heun equations and the Hankel determinant generated by a discontinuous semi-classical Laguerre weight |
| topic | Classical Analysis and ODEs 33C47, 34M55, 35C20, 65Q99 |
| url | https://arxiv.org/abs/2601.16548 |