Two series which generalize Dirichlet’s lambda and Riemann’s zeta functions at positive integer arguments

The series (cid:80) ∞ k =0 G N ( k ) (2 k +1) r and (cid:80) ∞ k =1 H N ( k ) k r are considered, where G N ( k ) and H N ( k ) are the Borwein-Chamberland sums appeared in the expansions of integer powers of the arcsine reported in the paper [D. Borwein, M. Chamberland, Int. J. Math. Math. Sci. 2007 (2007) #1981]. For 3 ≤ r ∈ N , representations for these series in terms of zeta values are derived, extending a theorem proved in the paper [J. Ewell, Canad. Math. Bull. 34 (1991) 60–66]. Several corollaries (especially for the case r = 3 ) are obtained, extending some known representations, including Euler’s famous rapidly converging series for ζ (3) . The technique can be applied to the case r = 2 and it yields generalizations of the formulas (cid:80) ∞ k =0 1 (2 k +1