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 z n−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 GlaisherKinkelin 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.