平方根的牛顿迭代和哈雷迭代的部分分式展开
Partial Fraction Expansions for Newton's and Halley's Iterations for Square Roots
摘要：
当用牛顿法或哈雷法来逼近$1-z$的$p$第{th}根时，得到一列有理函数。在本文中，一个美丽的公式，这些有理函数证明在平方根的情况下，使用一个有趣的链接到切比雪夫多项式。它允许确定这些有理函数的幂级数展开的系数的符号。这肯定地回答了郭（2010）提出的猜想的平方根情况。
英文摘要：
When Newton's method, or Halley's method is used to approximate the $p${th} root of $1-z$, a sequence of rational functions is obtained. In this paper, a beautiful formula for these rational functions is proved in the square root case, using an interesting link to Chebyshev's polynomials. It allows the determination of the sign of the coefficients of the power series expansion of these rational functions. This answers positively the square root case of a proposed conjecture by Guo(2010).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
