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

We define the generalized-Euler-constant function γ (z) = ∞n=1 zn−1( 1 n − log n+1 n ) when |z| 1. Its values include both Euler’s constant γ = γ (1) and the “alternating Euler constant” log 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,∞).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. © 2006 Elsevier Inc. All rights reserved