On Dirichlet's lambda function

Abstract

Letλ(s)=∑n=0∞1(2n+1)s,β(s)=∑n=0∞(−1)n(2n+1)s, andη(s)=∑n=1∞(−1)n−1ns be the Dirichlet lambda function, its alternating form, and the Dirichlet eta function, respectively. According to a recent historical book by Varadarajan ([25, p. 70]), these three functions were investigated by Euler under the notations N(s), L(s), and M(s), respectively. In this paper, we shall present some additional properties for them. That is, we obtain a number of infinite families of linear recurrence relations for λ(s) at positive even integer arguments λ(2m), convolution identities for special values of λ(s) at even arguments and special values of β(s) at odd arguments, and a power series expansion for the alternating Hurwitz zeta function J(s,a), which involves a known one for η(s).