In the present work we deal with the quadratic decomposition of symmetric semiclassical polynomial sequences of class 2 orthogonal with respect to the positive definite weight \( | x^2-\frac{1}{2} |^p(1-x^2)^{-\frac{1}{2}}\), \( p > -1\), on \([-1,1]\). The coefficients of the three-term recurrence relation, the structure relation, the differential equation as well as some information about the zeros of the corresponding orthogonal polynomials are given. These results reduce to the Chebyshev case for \(p=0\).

中文翻译：

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$$s=2$$s=2 类的正定线性泛函，切比雪夫多项式的推广

在目前的工作中，我们处理关于正定权重 \( | x^2-\frac{1}{2} |^p(1-x^2) 正交的第 2 类对称半经典多项式序列的二次分解)^{-\frac{1}{2}}\), \( p > -1\), 在 \([-1,1]\) 上。给出了三项递推关系的系数、结构关系、微分方程以及相应正交多项式的零点信息。这些结果简化为 \(p=0\) 的 Chebyshev 情况。