Explanation of using the table of identities

Suppose you have an identity involving functions that can be expressed in terms of Ramanujan's theta function f(-q) (the page Commonly used functions in modular equations has a list of these with definitions of each in terms of f(-q)). For example, consider the identity

a^3(q) = b^3(q) + c^3(q) where a(), b(), c() are cubic AGM theta functions.

Using the definitions given there you can rewrite the identity as

( (f^3(-q) + 9 q f^3(-q^9))/f(-q^3) )^3 =
( f^3(-q)/f(-q^3) )^3 + 27 q ( f^3(-q^3)/f(-q) )^3 .

The arguments in f(-q), f(-q^3), f(-q^9), namely, -q, -q^3, -q^9, all have exponents dividing 9, which is thus equal to the "level" of the identity. When the exponents have GCD g, replace q^g with q lowering the level of the identity. Next replace f(-q) with u1, f(-q^2) with u2, f(-q^3) with u3, and so on, and put everything under a common denominator on one side of the identity and simplify to get

0 = 1*u1^9*u9^3 +27*q^2*u1^3*u9^9 +9*q*u1^6*u9^6 -1*u3^12 .

This identity has four terms and the level is 9. It equates a homogeneous polynomial in u1, u3, u9 of total degree 12 to zero. Define the rank of u1 to be 1, of u3 to be 3, and so on, and let the rank be additive so the rank of u1^9*u9^3 is the sum of the ranks of its factors yielding 36. The rank of all terms must be the same modulo 24. Where the ranks are different, factors of q must appear as a consequence as in the second and third terms. I have written PARI/GP code to automatically perform the steps outlined above. This example processed using my code is

> gp -q
? read("formula.gp");

formula.gp dated 20 Mar 2012 loading
? doit(" a3^3 = b3^3 + c3^3 ");
/* equivalent to (SOMOS) :
   a3^3 = b3^3 + c3^3
q9_12_36 = +1*u1^9*u9^3 +27*q^2*u1^3*u9^9 +9*q*u1^6*u9^6 -1*u3^12 ;

This identity is labeled 'q9_12_36' where 'q' is the prefix for four terms. The prefix is 't' for three terms and 'x' for more than four terms. After the first letter the three numbers are the level, total degree, and the rank separated by '_'. My preferred TeX notation would be 'q_{9,12,36}' but I use 'q9_12_36' in the PARI/GP code. If more than one identity has the same parameters, a letter is appended to the label to distinguish them. Irreducible identities only are included in the table, and only a few with more than four terms. References, if any, appear as comments before the identity in the table.

After determining the level of an identity, you can search for it in the list of Dedekind eta function product identities by level or else the entire collection of Dedekind eta function product identities in PARI/GP code.

Reference Bibliography

Last updated 10 Apr 2017
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