In this paper, we investigate the restricted bipartition function [c.sub.N] (n) for n = 7, 11, and 5l, for any integer l [greater than or equal to] 1, and prove some congruence properties modulo 2, 3, and 5 by using Ramanujan's theta-function
and the Jacobi theta-function [[upsilon].sub.3](*, [theta]) is given by
We use the Jacobi theta-function [[upsilon].sub.1](*,[theta]), which for an arbitrary parameter [theta] [member of] C, [??]([theta]) > 0 and variable z [member of] C is given by
Each modular equation is equivalent to a certain theta-function identity, but a theta-function identity may not have an equivalent modular equation.
In Chapter 16 of his second notebook (, ), Ramanujan develops a theory of theta-functions. His theta-function is defined by
Even when the letter is spelled out, as at the beginning of a sentence, the compound should remain hyphenated: Theta-function.
Theta-functions, for instance, are often identified by [Theta] rather than by [Theta].
In Chapter 16 of his second notebook (15), Ramanujan develops the theory of theta-function
and his theta-function
is defined by
In his last letter to Hardy, dated January 12, 1920, he wrote: "I discovered very interesting functions recently which I can call "Mock" theta-functions
. Unlike the "False" theta-functions
(studied partially by Prof.