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Approximations to the Euler–Mascheroni Constant

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Abstract

In this paper, we establish an asymptotic expansion for the Euler–Mascheroni constant. Based on this expansion, we establish a two-sided inequality and a continued fraction approximation for the Euler–Mascheroni constant.

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References

  1. Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs\(,\) and Mathematical Tables, Applied Mathematics Series 55. Ninth Printing, National Bureau of Standards, Washington, D.C. (1972)

  2. Allasia, G., Giordano, C., Pećarić, J.: Inequalities for the gamma function relating to asymptotic expansions. Math. Inequal. Appl. 5(3), 543–555 (2002)

    MathSciNet  MATH  Google Scholar 

  3. Alzer, H.: On some inequalities for the gamma and psi functions. Math. Comput. 66, 373–389 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, C.-P.: Inequalities for the Lugo and Euler-Mascheroni constants. Appl. Math. Lett. 25(4), 787–792 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, C.-P.: Approximation formulas and inequalities for the Euler-Mascheroni constant, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 115(2), Article 56 (2021). https://doi.org/10.1007/s13398-021-00999-4

  6. Chen, C.-P., Choi, J.: Inequalities and asymptotic expansions for the constants of Landau and Lebesgue. Appl. Math. Comput. 248, 610–624 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, C.-P., Mortici, C.: New sequence converging towards the Euler-Mascheroni constant. Comput. Math. Appl. 64, 391–398 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, C.-P., Srivastava, H.M.: New representations for the Lugo and Euler-Mascheroni constants. Appl. Math. Lett. 24(7), 1239–1244 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, C.-P., Srivastava, H.M.: New representations for the Lugo and Euler-Mascheroni constants, II. Appl. Math. Lett. 25(3), 333–338 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, C.-P., Srivastava, H.M., Li, L., Manyama, S.: Inequalities and monotonicity properties for the psi (or digamma) function and estimates for the Euler-Mascheroni constant. Integral Transforms Spec. Funct. 22, 681–693 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, C.-P., Srivastava, H.M., Wang, Q.: A method to construct continued fraction approximations and its applications, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 115(3), Article 97 (2021)

  12. Chen, C.-P., Wang, Q.: Asymptotic expansions and continued fraction approximations for the harmonic numbers. Appl. Anal. Discrete Math. 13(2), 569–582 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  13. Choi, J.: Some mathematical constants. Appl. Math. Comput. 187, 122–140 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Choi, J., Srivastava, H.M.: Integral representations for the Euler-Mascheroni constant \(\gamma \). Integral Transforms Spec. Funct. 21(9), 675–690 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dence, T.P., Dence, J.B.: A survey of Euler’s constant. Math. Mag. 82, 255–265 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. DeTemple, D.W.: The non-integer property of sums of reciprocals of consecutive integers. Math. Gaz. 75, 193–194 (1991)

    Article  Google Scholar 

  17. DeTemple, D.W.: A quicker convergence to Euler’s constant. Am. Math. Mon. 100, 468–470 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gavrea, I., Ivan, M.: Optimal rate of convergence for sequences of a prescribed form. J. Math. Anal. Appl. 402, 35–43 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Havil, J.: Gamma: exploring Euler’s constant. Princeton University Press, Princeton (2003)

    MATH  Google Scholar 

  20. Lagarias, J.C.: Euler’s constant: Euler’s work and modern developments. Bull. Am. Math. Soc. 50(4), 527–628 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lin, L.: Asymptotic formulas associated with psi function with applications. J. Math. Anal. Appl. 405, 52–56 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mortici, C.: On new sequences converging towards the Euler-Mascheroni constant. Comput. Math. Appl. 59, 2610–2614 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Negoi, T.: A faster convergence to the constant of Euler. Gazeta Matematică, seria A 15, 111–113 (1997). ((in Romanian))

    Google Scholar 

  24. Rippon, P.J.: Convergence with pictures. Am. Math. Mon. 93, 476–478 (1986)

    Article  MathSciNet  Google Scholar 

  25. Sondow, J.: Criteria for irrationality of Euler’s constant. Proc. Am. Math. Soc. 131(11), 3335–3345 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Srivastava, H.M.: A survey of some recent developments on higher transcendental functions of analytic number theory and applied mathematics. Symmetry 13 2294, 1–22 (2021)

    Google Scholar 

  27. Yang, S.: On an open problem of Chen and Mortici concerning the Euler-Mascheroni constant. J. Math. Anal. Appl. 396, 689–693 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Young, R.M.: Euler’s constant. Math. Gaz. 75, 187–190 (1991)

    Article  MATH  Google Scholar 

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Acknowledgements

This work was the Fundamental Research Funds for the Universities of Henan Province (NSFRF210446).

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Authors and Affiliations

  1. School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, 454000, Henan, China

    Xue-Feng Han & Chao-Ping Chen

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  1. Xue-Feng Han
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  2. Chao-Ping Chen
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Correspondence to Chao-Ping Chen.

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Communicated by Saeid Maghsoudi.

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Han, XF., Chen, CP. Approximations to the Euler–Mascheroni Constant. Bull. Iran. Math. Soc. 49, 76 (2023). https://doi.org/10.1007/s41980-023-00820-5

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