In their recent paper, Farkas and Kra proved five three-term identities for the coefficient of certain finite products. In this paper we will show that each of the five identities is just one special case of an infinite family of identities. We will briefly introduce modular forms, and show that the coefficients of these products are closely related to the coefficients of certain modular form. Consequently in our proof we will use the properties of modular forms, specifically the Hecke operators.

Author Bio

This was a summer research project sponsored by theDivision of Natural Science and Mathematics of Colgate University, which startedlast May and was completed July, 2001. I just finished my freshman year ofcollege at the time I started my research. My advisor for this project was Professor Scott Ahlgren, who is now a professor at UIUC. I plan to do moreresearch with faculty members of Colgate in the near future; my ultimate goal isto pursue a doctorate degree in mathematics.