Positive rational series for reciprocal powers of Catalan's constant and Dirichlet beta values
Let $\beta(s)=\sum_{k=0}^{\infty}(-1)^k(2k+1)^{-s}$ and let $G=\beta(2)$ be Catalan's constant. We develop two families of positive series for reciprocal powers $\beta(s)^{-r}$. 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 $\pi$ arising from odd values of the Dirichlet beta function.