On Dirichlet's lambda functions

TL;DR

This paper explores new properties of Dirichlet lambda, beta, and eta functions, including recurrence relations, convolution identities, and power series expansions, enriching their mathematical understanding.

Contribution

It introduces additional properties such as recurrence relations, convolution identities, and power series expansions for these classical functions.

Findings

Derived infinite families of recurrence relations for λ(2m).

Established convolution identities for special values of λ(s) and β(s).

Developed a power series expansion for the alternating Hurwitz zeta function.

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 $\eta(s)$.