非对称高斯与阿基米德复合均值的渐近分析
Asymptotic Analysis of Nonsymmetric Gaussian and Archimedean Compound Means
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中文总结 AI 辅助
本文针对非对称双变量均值，推导了高斯与阿基米德复合均值渐近展开的系数递归算法，关联了Farhi度量并验证了算法有效性，扩展方法至带符号依赖系数的展开并应用于两类均值。
中文摘要 AI 辅助
本文研究两个任意双变量均值的高斯复合均值与阿基米德复合均值的渐近展开，这两个均值无需对称。此前，此类展开的系数递归算法仅在对称情形下被求得，本文消除了该限制，针对一般非对称情形推导了递归关系。所有系数计算均在形式幂级数代数中进行，这为递归引入了额外情形，以及形式不变性方程无法递归确定系数的奇异构型。本文将形式问题与迭代程序的存在性及收敛性分开处理；还将首个形式系数与Farhi度量关联，回顾其已知的全局收敛准则，并推导相应的局部收敛速率。该算法通过加权幂均值及若干已知恒等式说明；最后，该方法被扩展至带符号依赖系数的展开，并应用于新毕达哥拉斯均值及最近提出的双参数族。
英文摘要
This paper studies asymptotic expansions of the Gaussian and Archimedean compounds of two arbitrary bivariate means that need not be symmetric. Recursive algorithms for the coefficients of such expansions were previously obtained only in the symmetric case. Here, this restriction is removed and recursions are derived for the general nonsymmetric setting. All coefficient calculations are carried out in the algebra of formal power series. This introduces an additional case in the recursion, as well as singular configurations in which the formal invariance equation does not determine the coefficients recursively. The formal problem is kept separate from the existence and convergence of the iterative procedures. We also connect the first formal coefficients with Farhi's metric, recall its known global convergence criterion, and derive the corresponding local rates. The algorithms are illustrated using weighted power means and several known identities. Finally, the method is extended to expansions with sign-dependent coefficients and is applied to the neo-Pythagorean means and to a recently introduced two-parameter family.
