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The generalized-Euler-constant function $gamma(z)$ and a generalization of Somos's quadratic recurrence constant | | Jonathan Sondow
; Petros Hadjicostas
; | | Date: |
17 Oct 2006 | | Subject: | Classical Analysis and ODEs; Number Theory | | Abstract: | We define the generalized-Euler-constant function $gamma(z)=sum_{n=1}^{infty} z^{n-1} (frac{1}{n}-log frac{n+1}{n})$ when $ z leq 1$. Its values include both Euler’s constant $gamma=gamma(1)$ and the "alternating Euler constant" $logfrac{4}{pi}=gamma(-1)$. We extend Euler’s two zeta-function series for $gamma$ to polylogarithm series for $gamma(z)$. Integrals for $gamma(z)$ provide its analytic continuation to $C-[1,infty)$. We prove several other formulas for $gamma(z)$, including two functional equations; one is an inversion relation between $gamma(z)$ and $gamma(1/z)$. We generalize Somos’s quadratic recurrence constant and sequence to cubic and other degrees, give asymptotic estimates, and show relations to $gamma(z)$ and to an infinite nested radical due to Ramanujan. We calculate $gamma(z)$ and $gamma’(z)$ at roots of unity; in particular, $gamma’(-1)$ 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 Jonqui`{e}re’s relation for the polylogarithm. | | Source: | arXiv, math/0610499 | | Services: | Forum | Review | PDF | Favorites | |
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