2026/08/01 by Narendra Bhandari
Mathematics · #math.NT #msc:11Y60 #msc:40G05 #msc:11M06 #msc:33B30
19 pages
arxiv created 2026/08/01 · arxiv updated 2026/08/04
Let β(s)=∑k=0∞(-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≥2, 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.