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 $\eta(s)$.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2018
- DOI:
- 10.48550/arXiv.1806.07762
- arXiv:
- arXiv:1806.07762
- Bibcode:
- 2018arXiv180607762H
- Keywords:
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- Mathematics - Number Theory;
- Mathematics - Classical Analysis and ODEs;
- 11B68;
- 11S40
- E-Print:
- 21 pages