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

Title & Authors
ON EVALUATIONS OF THE MODULAR j-INVARIANT BY MODULAR EQUATIONS OF DEGREE 2
Paek, Dae Hyun; Yi, Jinhee;

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

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.

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.