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The Ramanujan Journal

, Volume 49, Issue 3, pp 653–668 | Cite as

Precise estimates for the solution of Ramanujan’s generalized modular equation

  • Miao-Kun Wang
  • Yu-Ming ChuEmail author
  • Wen Zhang
Article

Abstract

In the article, we present several monotonicity theorems and inequalities for the modular equation functions \(m_{a}(r)\) and \(\mu _{a}(r),\) and find the infinite-series formulas for \(m_{1/3}(r)\) and \(m_{1/4}(r)\) which depend only on r. As applications, we find several precise explicit estimates for the solution of Ramanujan’s generalized modular equation.

Keywords

Gaussian hypergeometric function Ramanujan’s generalized modular equation Infinite-series formula Inequalities 

Mathematics Subject Classification

11F03 33C05 

Notes

Acknowledgements

The authors express their sincere thanks to the referee(s) for careful reading of the manuscript and very helpful suggestions that improved the current manuscript substantially.

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Copyright information

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Authors and Affiliations

  1. 1.Department of MathematicsHuzhou UniversityHuzhouChina
  2. 2.College of ScienceHunan City UniversityYiyangChina
  3. 3.Friedman Brain InstituteIcahn School of Medicine at Mount SinaiNew YorkUSA

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