标题： Ladder Operators for Laguerre-type and Jacobi-type Orthogonal Polynomials 显示英文标题

标题： Ladder Operators for Laguerre-type and Jacobi-type Orthogonal Polynomials

摘要： 在关于拉盖尔型权函数$x^\lambda w_0(x), x\in[0,+\infty)$的文献中，雅可比型权函数$(1-x)^{\alpha}(1+x)^{\beta}w_0(x),x\in[-1,1]$和移位雅可比型权函数$x^{\alpha}(1-x)^{\beta}w_0(x), x\in[0,1]$，其中$w_0(x)$连续可微，参数$\lambda,\alpha,\beta$通常被限制为严格正数以确保结果的有效性。最近，在 [C. Min 和 P. Fang, Physica D 473 (2025), 134560 (9pp)] 中，通过利用正交性质推导了带有$\lambda>-1$的首一拉盖尔型正交多项式的升降算子。 量$A_n$和$B_n$在阶梯算子中作为系数出现，与之前的量$\lambda>0$相比表现出不同的表达式。 在本文中，我们通过利用正交多项式满足的黎曼-希尔伯特问题来构建一种替代推导方法。 此外，我们采用上述两种推导策略，以生成单变量标准和移位雅可比型正交多项式的阶梯算子，其中包含$\alpha,\beta>-1$。 当$\lambda,\alpha,\beta$被限制为正值时，我们对$A_n$和$B_n$的表达式与先前工作中的结果一致。 我们提供示例来验证我们的结论，并通过使用阶梯算子的三个相容条件以及对单次正交多项式的正交关系进行微分，将现有的结论从$\lambda,\alpha,\beta>0$推广到$\lambda,\alpha,\beta>-1$。 显示英文摘要

摘要： In the literature concerning the Laguerre-type weight function $x^\lambda w_0(x), x\in[0,+\infty)$, the Jacobi-type weight function $(1-x)^{\alpha}(1+x)^{\beta}w_0(x),x\in[-1,1]$, and the shifted Jacobi-type weight function $x^{\alpha}(1-x)^{\beta}w_0(x), x\in[0,1]$, with $w_0(x)$ continuously differentiable, the parameters $\lambda,\alpha,\beta$ 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 $\lambda>-1$ 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 $\lambda>0$. 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 $\alpha,\beta>-1$. When $\lambda,\alpha,\beta$ 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 $\lambda,\alpha,\beta>0$ to $\lambda,\alpha,\beta>-1$.