Ladder Operators for Laguerre-type and Jacobi-type Orthogonal Polynomials
Shulin Lyu, Yuanfei Lyu
TL;DR
This work extends the ladder-operator framework for Laguerre-type, Jacobi-type, and shifted Jacobi-type orthogonal polynomials from the classical positive-parameter regime to the broader range $>-1$. By combining a Riemann-Hilbert problem (RHP) approach with the traditional ladder-operator method, the authors derive explicit $A_n(z)$ and $B_n(z)$ coefficients and verify the associated compatibility conditions $(S_1),(S_2),(S_2')$ for all three weight families. They show that, for $\lambda,\alpha,\beta> -1$, the resulting ladder operators generalize known results (valid for $>0$) and remain consistent with prior expressions when the parameters are positive. The paper also provides illustrative examples, including classical and semi-classical weights, and extends the analysis to weights with jumps and Fisher-Hartwig singularities, establishing Painlevé-type characterizations (e.g., Painlevé V and Painlevé VI sigma forms) of Hankel determinants in this broader setting. Overall, the results broaden the applicability of ladder-operator techniques to a wider class of orthogonal polynomials and deepen the connections to integrable systems and nonlinear differential equations.
Abstract
In the literature concerning the Laguerre-type weight function $x^λw_0(x), x\in[0,+\infty)$, the Jacobi-type weight function $(1-x)^α(1+x)^βw_0(x),x\in[-1,1]$, and the shifted Jacobi-type weight function $x^α(1-x)^βw_0(x), x\in[0,1]$, with $w_0(x)$ continuously differentiable, the parameters $λ,α,β$ are usually constrained to be strictly positive to ensure the validity of the results. Recently, in [C. Min and P. Fang, Physica D 473 (2025), 134560 (9pp)], the ladder operators for the monic Laguerre-type orthogonal polynomials with $λ>-1$ were derived by exploiting the orthogonality properties. The quantities $A_n$ and $B_n$, which appear as coefficients in the ladder operators, exhibit different expressions compared with the previous ones for $λ>0$. In this paper, we construct an alternative deduction by making use of the Riemann-Hilbert problem satisfied by the orthogonal polynomials. Moreover, we employ both derivation strategies mentioned above to produce the ladder operators for the monic standard and shifted Jacobi-type orthogonal polynomials with $α,β>-1$. When $λ,α,β$ are restricted to positive values, our expressions of $A_n$ and $B_n$ are consistent with those in prior work. We present examples to validate our findings and generalize the existing conclusions, established by using the three compatibility conditions of the ladder operators and differentiating the orthogonality relations for the monic orthogonal polynomials, from $λ,α,β>0$ to $λ,α,β>-1$.