The generalized-Euler-constant function [formula omitted] and a generalization of Somos's quadratic recurrence constant

Abstract

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 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.