The generalized-Euler-constant function γ(z) and a generalization of Somos's quadratic recurrence constant
By Jonathan Sondow, P. Hadjicostas. 2006
51 citations recorded by the source corpus.
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 Jonquiere's relation for the polylogarithm.
Subfields: Algebra and Number Theory, Applied Mathematics
View Paper: https://www.semanticscholar.org/paper/ca037182c96ba46fed1cfb4441a28e9918f7d2e5
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In Algebra and Number Theory
In Applied Mathematics
Citations in the corpus
- Euler's constant: Euler's work and modern developments (2013, methodology): “These include studies of new special functions involving Euler’s constant, such as the generalized Euler-constant function γ(z) proposed by Sondow and Hadjicostas [280].”