On Dirichlet's lambda functions

Abstract

Let $$\lambda(s)=\sum_{n=0}^\infty\frac1{(2n+1)^s},$$ $$\beta(s)=\sum_{n=0}^\infty\frac{(-1)^{n}}{(2n+1)^s},$$ and $$\eta(s)=\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^s}$$ be the Dirichlet lambda function, its alternating form, and the Dirichlet eta function, respectively. According to a recent historical book by Varadarajan (\cite[p.~70]{Varadarajan}), 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 $\lambda(s)$ at positive even integer arguments $\lambda(2m)$, convolution identities for special values of $\lambda(s)$ at even arguments and special values of $\beta(s)$ at odd arguments, and a power series expansion for the alternating Hurwitz zeta function $J(s,a)$, which involves a known one for $\eta(s)$.