AI 中文总结
该研究构建了两类Dirichlet beta函数倒数幂的正项级数，推导了系数公式与收敛性质，得到卡塔兰常数及π的倒数幂的正有理级数表达式。
AI 中文摘要
设$β(s)=\u2062∑_{k=0}^{∞}(-1)^k(2k+1)^{-s}$，且$G=β(2)$为卡塔兰常数。我们构建了两类用于表示倒数幂$β(s)^{-r}$的正项级数。第一类由经典欧拉变换得到，在$1/2$处取值，对所有实数$s>0$均成立。第二类变换适用于$s≥2$的情况，可得到在$1/3$处取值的收敛更快的级数。我们推导了基系数的有限和公式与积分公式，以及正项递推关系和任意倒数幂的合成公式。当$s$为整数时，所有系数均为有理数。我们还给出了显式余项估计，并确定了这两类级数的确切根收敛速度。作为应用，我们得到了卡塔兰常数的所有倒数幂的正有理级数，以及由Dirichlet beta函数的奇数值导出的$π$的倒数幂的正有理级数。
英文摘要
Let $β(s)=\sum_{k=0}^{\infty}(-1)^k(2k+1)^{-s}$ and let $G=β(2)$ be Catalan's constant. We develop two families of positive series for reciprocal powers $β(s)^{-r}$. The first is obtained from the classical Euler transformation and is evaluated at $1/2$; it is valid for real $s>0$. A second transformation, valid for $s\geq2$, gives a faster series evaluated at $1/3$. We derive finite-sum and integral formulas for the base coefficients, together with a positive recurrence and a composition formula for arbitrary reciprocal powers. When $s$ is an integer, all coefficients are rational. We also give explicit remainder estimates and determine the exact root-convergence rates of the two families. As applications, we obtain positive rational series for every reciprocal power of Catalan's constant and for reciprocal powers of $π$ arising from odd values of the Dirichlet beta function.
Comments19 pages
