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

Jonathan Sondow; Petros Hadjicostas

doi:10.1016/j.jmaa.2006.09.081

paper

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

arXiv:math/0610499 · doi:10.1016/j.jmaa.2006.09.081

Abstract

We define the generalized-Euler-constant function $γ (z) = \sum_{n = 1}^{\infty} z^{n - 1} (\frac{1}{n} - lo g \frac{n + 1}{n})$ when $∣ z ∣ \leq 1$ . Its values include both Euler's constant $γ = γ (1)$ and the "alternating Euler constant" $lo g \frac{4}{π} = γ (- 1)$ . We extend Euler's two zeta-function series for $γ$ to polylogarithm series for $γ (z)$ . Integrals for $γ (z)$ provide its analytic continuation to $\C-[1,\infty)$. We prove several other formulas for $γ (z)$ , including two functional equations; one is an inversion relation between $γ (z)$ and $γ (1/ z)$ . We generalize Somos's quadratic recurrence constant and sequence to cubic and other degrees, give asymptotic estimates, and show relations to $γ (z)$ and to an infinite nested radical due to Ramanujan. We calculate $γ (z)$ and $γ^{'} (z)$ at roots of unity; in particular, $γ^{'} (- 1)$ 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 Jonquière's relation for the polylogarithm.

26 pages, 2 figures, to appear in J. Math. Anal. Appl

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

Abstract

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