On Dirichlet's lambda functions

Abstract

Let $$\lambda (s)=\sum _{n=0}^{\infty }\frac{1}{(2n+1{)}^s},$$ $$\beta (s)=\sum _{n=0}^{\infty }\frac{(-1{)}^n}{(2n+1{)}^s},$$ 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 $\eta (s)$.

21 pages

References in corpus (1)

• Special values and integral representations for the Hurwitz-type Euler zeta functions