Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON EVALUATIONS OF THE MODULAR j-INVARIANT BY MODULAR EQUATIONS OF DEGREE 2

  • Paek, Dae Hyun (Department of Mathematics Education Busan National University of Education) ;
  • Yi, Jinhee (Department of Mathematics and Computer Science Korea Science Academy of KAIST)
  • Received : 2014.01.11
  • Published : 2015.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

We derive modular equations of degree 2 to establish explicit relations for the parameterizations for the theta functions ${\varphi}$ and ${\psi}$. We then find specific values of the parameterizations to evaluate some new values of the modular j-invariant in terms of $J_n$.

Keywords

References

  1. B. C. Berndt, Ramanujan's Notebooks. Part V, Springer-Verlag, New York, 1998.
  2. B. C. Berndt, Ramanujan's Notebooks. Part III, Springer-Verlag, New York, 1991.
  3. B. C. Berndt, Number Theory in the Spirit of Ramanujan, American Mathematical Society, 2006.
  4. B. C. Berndt and H. H. Chan, Ramanujan and the modular j-invariant, Canad. Math. Bull. 42 (1999), no. 4, 427-440. https://doi.org/10.4153/CMB-1999-050-1
  5. D. A. Cox, Primes of the Form $x^2$ + $ny^2$, Wiley, New York, 1989.
  6. J. Yi, The Construction and Applications of Modular Equations, Ph.D. Thesis, University of Illinois at Urbana-Champaign, 2001.
  7. J. Yi, Theta-function identities and the explicit formulas for theta-function and their applications, J. Math. Anal. Appl. 292 (2004), no. 2, 381-400. https://doi.org/10.1016/j.jmaa.2003.12.009
  8. J. Yi, Y. Lee, and D. H. Paek, The explicit formulas and evaluations of Ramanujan's theta-function ${\Psi}$, J. Math. Anal. Appl. 321 (2006), no. 1, 157-181. https://doi.org/10.1016/j.jmaa.2005.07.062