On a class of parameter dependent series generalizing Euler’s constant. (English) Zbl 1424.11174
Summary: Series which depend on a parameter and generalize the constant discovered by Euler are introduced and studied. Convergence results are established. An infinite series expansion is obtained from these generalized formulas which can be used to evaluate the generalized constant. Euler’s constant can be obtained as a special case. Some asymptotic results are formulated and limits of some closely related sequences are given at the end.
MSC:
| 11Y60 | Evaluation of number-theoretic constants |
| 40A25 | Approximation to limiting values (summation of series, etc.) |
References:
| [1] | D. Knuth, Euler’s Constant to 1271 Places, Math. of Computation, 16, 79: 275-281, 1962. · Zbl 0117.10801 |
| [2] | J. C. Lagarias, Euler’s Constant: Euler’s Work and Modern Developments, Bull. Amer. Math. Soc. 50, 4: 527-628, 2013. · Zbl 1282.11002 |
| [3] | A. Sintamarian, A generalization of Euler’s constant, Numer. Algor. 46: 141-151, 2007. · Zbl 1130.11075 |
| [4] | T. Tasaka, Note on the generalized Euler constants, Math. J. Okayama Univ. 36: 29-34, 1994. · Zbl 0840.11032 |
| [5] | E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, 1996. · JFM 45.0433.02 |
| [6] | C. Elsner, On a sequence transformation with integral coefficients for Euler’s constant, Proc. Am. Math. Soc. 123: 1537-1541, 1995. · Zbl 0828.65001 |
| [7] | The Euler Archive: The works of Leonard Euler online, http://www.eulerarchive.org |
| [8] | L. T‘oth, Problem E3432. Amer. Math. Monthly, 99: 684-685, 1992. |
| [9] | L. Yingying, On Euler’s constant-calculating sums by integrals, Amer. Math. Monthly, 109: 845-850, 2002. · Zbl 1027.40004 |
| [10] | D. Lu, Some quicker classes of sequences convergent to Euler’s constant, Applied Math. and Comp. 232: 172-177, 2014. · Zbl 1410.40002 |
| [11] | J. Ser, Questions, 5561,5562, L’Intermediare des Math‘ematiciens, S‘erie 2, 126-127, 1925. · JFM 51.0280.04 |
| [12] | G. G. Polya and G. Szeg¨o, Problems and Theorems in Analysis I, pg. 22, SpringerVerlag, Berlin, 1976. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
