AI 中文总结
该研究建立了形式幂级数的莫比乌斯协方差与系数对偶性，推广了伯努利级数的相关恒等式，得到拉马努金型求和公式，还应用于生成匹配、多项式系数等的递推关系。
AI 中文摘要
本文证明了形式伯努利级数中首次发现的系数对偶性等价于形式幂级数的一般莫比乌斯协方差律，得到了该对偶性的结构刻画、本征空间解释及加权形式。源于伯努利背景的卡特兰卷积和切比雪夫恒等式可推广至任意莫比乌斯协变族，得到涵盖连续半整数幂的一般拉马努金型求和公式；该框架还可复现经典伯努利与欧拉递推关系，生成带色匹配、广义中心三项式系数的递推族，还能从反射对称阿佩尔序列、戈伦斯坦希尔伯特级数中得到进一步实现。
英文摘要
A coefficient duality first encountered for formal Bernoulli series is shown to be equivalent to a general Möbius covariance law for formal power series. We obtain a structural characterization, an eigenspace interpretation, and a weighted form of this duality. The Catalan convolution and Chebyshev identities from the motivating Bernoulli setting extend to arbitrary Möbius-covariant families and yield a general Ramanujan-type summation formula encompassing consecutive half-integer powers. The framework also recovers classical Bernoulli and Euler recurrences and produces recurrence families for colored matchings and generalized central trinomial coefficients, with further realizations from reflection-symmetric Appell sequences and Gorenstein Hilbert series.
