(*continued from yesterday’s post*)

**III. Icosahedral group**

Given the *Rogers-Ramanujan identities* (see also here),

I observed that,

where, as in the previous post, is the j-function, , , and . Since it is known that,

this implies that,

Example. Let , hence . Then,

Furthermore, since Ramanujan established that,

if we define the two functions,

then the counterpart hypergeometric identity is also beautifully simple and given by,

In the next post, we will use one of the hypergeometric formulas to solve the general *quintic*.