# SOME EISENSTEIN SERIES IDENTITIES RELATED TO MODULAR EQUATION OF THE FOURTH ORDER

• Published : 2011.01.31

#### Abstract

We find some Eisenstein series related to modulus 4 using a theta function identity of McCullough and Shen and residue theorem for elliptic functions.

#### References

1. G. E. Andrews and B. C. Berndt, Ramanujan's Lost Notebook. Part I, Springer, New York, 2005.
2. G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990.
3. Z.-G. Liu, Some Eisenstein series identities, J. Number Theory 85 (2000), no. 2, 231-252. https://doi.org/10.1006/jnth.2000.2543
4. Z.-G. Liu, Residue theorem and theta function identities, Ramanujan J. 5 (2001), no. 2, 129-151. https://doi.org/10.1023/A:1011427622187
5. Z.-G. Liu, Some theta functions identities associated with the modular equations of degree 5, Integers 1 (2001), A3, 14 pp.
6. Z.-G. Liu, Some Eisenstein series identities related to modular equations of the seventh order, Pacific J. Math. 209 (2003), no. 1, 103-130. https://doi.org/10.2140/pjm.2003.209.103
7. S. McCullough and L.-C. Shen, On the Szego kernel of an annulus, Proc. Amer. Math. Soc. 121 (1994), no. 4, 1111-1121.
8. S. Ramanujan, Collected Papers, Cambridge University Press, Cambridge, 1927.
9. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. An Introduction to the General Theory of Infinite Processes and of Analytic Functions: with an account of the principal transcendental functions, Cambridge University Press, New York 1962.