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

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dc.creatorSondow, Jonathan
dc.creatorHadjicostas, Petros
dc.date2006-10-16
dc.date.accessioned2026-07-07T08:08:15Z
dc.date.available2026-07-07T08:08:15Z
dc.descriptionWe define the generalized-Euler-constant function $γ(z)=\sum_{n=1}^{\infty} z^{n-1} (\frac{1}{n}-\log \frac{n+1}{n})$ when $|z|\leq 1$. Its values include both Euler's constant $γ=γ(1)$ and the "alternating Euler constant" $\log\frac{4}π=γ(-1)$. 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.
dc.description26 pages, 2 figures, to appear in J. Math. Anal. Appl
dc.identifierhttps://arxiv.org/abs/math/0610499
dc.identifierhttp://arxiv.org/abs/math/0610499
dc.identifierJ. Math. Anal. Appl. 332 (2007) 292-314
dc.identifierdoi:10.1016/j.jmaa.2006.09.081
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/131202
dc.subjectClassical Analysis and ODEs
dc.subjectNumber Theory
dc.subject11M35, 11Y55, 11Y60, 33B15, 33B30, 40A05, 40A20
dc.titleThe generalized-Euler-constant function $γ(z)$ and a generalization of Somos's quadratic recurrence constant
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