Approximations of the generalized-Euler-constant function and the generalized Somos’ quadratic recurrence constant
Abstract In this paper, we provide an estimate for approximating the generalized-Euler-constant function γ(z)=∑k=1∞zk−1(1k−lnk+1k) $\gamma (z)=\sum_{k=1}^{\infty }z ^{k-1} (\frac{1}{k}-\ln \frac{k+1}{k} )$ by its partial sum γN−1(z) $\gamma _{N-1}(z)$ when 0<z<1 $0< z<1$. We obtain an as...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2019-07-01
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| Series: | Journal of Inequalities and Applications |
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| Online Access: | http://link.springer.com/article/10.1186/s13660-019-2153-0 |
| _version_ | 1819093443037626368 |
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| author | Aimin Xu |
| author_facet | Aimin Xu |
| author_sort | Aimin Xu |
| collection | DOAJ |
| description | Abstract In this paper, we provide an estimate for approximating the generalized-Euler-constant function γ(z)=∑k=1∞zk−1(1k−lnk+1k) $\gamma (z)=\sum_{k=1}^{\infty }z ^{k-1} (\frac{1}{k}-\ln \frac{k+1}{k} )$ by its partial sum γN−1(z) $\gamma _{N-1}(z)$ when 0<z<1 $0< z<1$. We obtain an asymptotic expansion for the generalized-Euler-constant function and show that the coefficients of the asymptotic expansion are explicitly expressed by the Eulerian fractions. Also, we find a recurrence relation for those coefficients. Using its relation with the generalized-Euler-constant function, we establish two inequalities for the generalized Somos’ quadratic recurrence constant. Moreover, two asymptotic expansions for the natural logarithm of the generalized Somos quadratic recurrence constant are presented. |
| first_indexed | 2024-12-21T23:11:35Z |
| format | Article |
| id | doaj.art-7a0f9a8635d54403972ce417b98a52bd |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-21T23:11:35Z |
| publishDate | 2019-07-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-7a0f9a8635d54403972ce417b98a52bd2022-12-21T18:47:01ZengSpringerOpenJournal of Inequalities and Applications1029-242X2019-07-012019111810.1186/s13660-019-2153-0Approximations of the generalized-Euler-constant function and the generalized Somos’ quadratic recurrence constantAimin Xu0Institute of Mathematics, Zhejiang Wanli UniversityAbstract In this paper, we provide an estimate for approximating the generalized-Euler-constant function γ(z)=∑k=1∞zk−1(1k−lnk+1k) $\gamma (z)=\sum_{k=1}^{\infty }z ^{k-1} (\frac{1}{k}-\ln \frac{k+1}{k} )$ by its partial sum γN−1(z) $\gamma _{N-1}(z)$ when 0<z<1 $0< z<1$. We obtain an asymptotic expansion for the generalized-Euler-constant function and show that the coefficients of the asymptotic expansion are explicitly expressed by the Eulerian fractions. Also, we find a recurrence relation for those coefficients. Using its relation with the generalized-Euler-constant function, we establish two inequalities for the generalized Somos’ quadratic recurrence constant. Moreover, two asymptotic expansions for the natural logarithm of the generalized Somos quadratic recurrence constant are presented.http://link.springer.com/article/10.1186/s13660-019-2153-0Asymptotic expansionEulerian fractionGeneralized-Euler-constant functionInequalitySomos’ quadratic recurrence constant |
| spellingShingle | Aimin Xu Approximations of the generalized-Euler-constant function and the generalized Somos’ quadratic recurrence constant Journal of Inequalities and Applications Asymptotic expansion Eulerian fraction Generalized-Euler-constant function Inequality Somos’ quadratic recurrence constant |
| title | Approximations of the generalized-Euler-constant function and the generalized Somos’ quadratic recurrence constant |
| title_full | Approximations of the generalized-Euler-constant function and the generalized Somos’ quadratic recurrence constant |
| title_fullStr | Approximations of the generalized-Euler-constant function and the generalized Somos’ quadratic recurrence constant |
| title_full_unstemmed | Approximations of the generalized-Euler-constant function and the generalized Somos’ quadratic recurrence constant |
| title_short | Approximations of the generalized-Euler-constant function and the generalized Somos’ quadratic recurrence constant |
| title_sort | approximations of the generalized euler constant function and the generalized somos quadratic recurrence constant |
| topic | Asymptotic expansion Eulerian fraction Generalized-Euler-constant function Inequality Somos’ quadratic recurrence constant |
| url | http://link.springer.com/article/10.1186/s13660-019-2153-0 |
| work_keys_str_mv | AT aiminxu approximationsofthegeneralizedeulerconstantfunctionandthegeneralizedsomosquadraticrecurrenceconstant |