The generalized-Euler-constant function $\gamma(z)$ and a generalization of Somos's quadratic recurrence constant

Abstract

We define the generalized-Euler-constant function $\gamma(z)=\sum_{n=1}^{\infty} z^{n-1} (\frac{1}{n}-\log \frac{n+1}{n})$ when $|z|\leq 1$. Its values include both Euler's constant $\gamma=\gamma(1)$ and the "alternating Euler constant" $\log\frac{4}{\pi}=\gamma(-1)$. We extend Euler's two zeta-function series for $\gamma$ to polylogarithm series for $\gamma(z)$. Integrals for $\gamma(z)$ provide its analytic continuation to $\C-[1,\infty)$. We prove several other formulas for $\gamma(z)$, including two functional equations; one is an inversion relation between $\gamma(z)$ and $\gamma(1/z)$. We generalize Somos's quadratic recurrence constant and sequence to cubic and other degrees, give asymptotic estimates, and show relations to $\gamma(z)$ and to an infinite nested radical due to Ramanujan. We calculate $\gamma(z)$ and $\gamma'(z)$ at roots of unity; in particular, $\gamma'(-1)$ involves the Glaisher-Kinkelin constant $A$. Several related series, infinite products, and double integrals are evaluated. The methods used involve the Kinkelin-Bendersky hyperfactorial $K$ function, the Weierstrass products for the gamma and Barnes $G$ functions, and Jonqui\`{e}re's relation for the polylogarithm.