AI 中文总结
研究振荡雅可比型权函数正交多项式的递推系数，利用阶梯算子和相容性条件推导耦合差分方程，该方程结构简单、阶数低，能在确定初始值后计算递推系数并推测其符号形式。
AI 中文摘要
我们研究了两类振荡雅可比型权函数\(x^c(1 - x^2)^{\lambda - 1/2}\exp(i\zeta x)\)，\(x\in[-1,1]\)，\(\lambda > -1/2\)，\(c\in\{0,1\}\)。对\(\lambda\)和\(\zeta\)施加其他限制以保证相关正交多项式的存在。利用近期文献中为与雅可比型权函数相关的首一正交多项式建立的阶梯算子和三个相容性条件，我们推导了三项递推系数满足的两个耦合差分方程。与现有结果相比，这些方程结构更简单且阶数更低。一旦确定初始值，可通过差分方程在任何阶段计算递推系数。所得表达式使我们能够推测递推系数的符号形式。
英文摘要
We study two classes of oscillatory Jacobi-type weight functions $x^c(1-x^2)^{λ-1/2}\exp(iζx)$, $x\in[-1,1], λ>-1/2, c\in\{0,1\}$. Other restrictions are imposed on $λ$ and $ζ$ to guarantee the existence of the associated orthogonal polynomials. By using the ladder operators established in the recent literature for monic orthogonal polynomials associated with Jacobi-type weight functions and three compatibility conditions, we derive two coupled difference equations satisfied by the three-term recurrence coefficients. Compared with the existing results, these equations are structurally simpler and of lower order. Once the initial values are determined, the recurrence coefficients can be computed at any stage through the difference equations. The obtained expressions enable us to conjecture the symbolic forms for the recurrence coefficients.
