Automatic sequences and the Glaisher–Kinkelin constant

Abstract

Let $\left(a\right(n):n\in N_0)$ denote an automatic sequence. Previous research on infinite products involving automatic sequences has mainly dealt with identities for products as in ${\prod }_{n}R{\left(n\right)}^{a\left(n\right)}$ for rational functions $R\left(n\right)$. This inspires the development of techniques for evaluating ${\prod }_{n}f{\left(n\right)}^{a\left(n\right)}$ more generally, for functions $f\left(n\right)$ that are not rational functions. This leads us to apply Euler's product expansion for the Γ-function together with recursive properties of $a\left(n\right)$ to obtain identities as in ${\prod }_{n}f{(n,z)}^{a\left(n\right)}=\Gamma (z+1)$, and this is motivated by how the equivalent series identity ${\sum }_{n}a\left(n\right)\ln f(n,z)=\ln \Gamma (z+1)$ could be applied in relation to the remarkable results due to Gosper on the integration of $\ln \Gamma (z+1)$. We succeed in applying this approach, using Gosper's integration identities, to obtain new infinite products that we evaluate in terms of the Glaisher–Kinkelin constant A and that involve the Thue–Morse sequence, the period-doubling sequence, and the regular paperfolding sequence. A byproduct of our method gives us a way of generalizing a Dirichlet series identity due to Allouche and Sondow, and we also explore applications related to a product evaluation due to Gosper involving A.