The Müntz polynomials are defined by contour integral associated to a complex sequence Λ={λ0,λ1,λ2,⋯}, which are large extensions of the algebraic polynomials. In this paper, we derive new recurrence formulas for Müntz polynomials, aimed at facilitating the computation of these polynomials and their related integrals. Additionally, we construct a novel class of orthogonal polynomials with respect to the logarithmic weight function xλ(−log⁡x)μ on the interval (0,1). We also develop the corresponding Gauss quadrature rules, which serve as powerful techniques for accurately solving integrals involving singular terms.

中文翻译：

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关于 Müntz 多项式的递归公式和应用


Müntz 多项式由与复序列 Λ={λ0，λ1，λ2，⋯} 相关的等值积分定义，这些序列是代数多项式的大扩展。在本文中，我们推导出了 Müntz 多项式的新递归公式，旨在促进这些多项式及其相关积分的计算。此外，我们在区间 （0,1） 上构建了一类关于对数权重函数 xλ（−logx）μ 的新型正交多项式。我们还开发了相应的高斯求积规则，这些规则是精确求解涉及奇异项的积分的强大技术。