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.

- (SOMOS) means that I have not found a reference for it yet, but that in these cases I found the identity and its formulation.
- (ARNDT) means that it is from Joerg Arndt by personal communication.
- (A004016) indicates a reference to Sloane's On-Line Encyclopedia of Integer Sequences at OEIS.
- The table is incomplete and may have errors but is still useful. I keep a high level of quality control so please inform me of any errors.
- Note that I claim no proofs of the identities, in the spirit of Ramanujan. However, using the theory of modular forms, it is possible to prove any of these identities by checking sufficiently many q-series coefficients.

- Naika & Denis & Bairy, "On Some Ramanujan-Selberg Continued Fraction", Indian J. of Math. (2009) v. 51 n. 3 pp. 585-596.
- N. Baruah & J. Bora, "New Proofs of Ramanujan's Modular Equations of Degree 9", Indian J. of Math. (2005) v. 47 n. 1 pp. 99-122.
- N. D. Baruah, "On some of Ramanujan's Schläfli-type ''mixed'' modular equations", J. of Number Theory 100 (2003) pp. 270-294.
- R. Barman & N. Baruah, Theta function identities associated with Ramanujan's modular equations of degree 15, Proc. Indian Acad. Sci., Vol 120, No 3, Jun 2010, pp. 267-284.
- J. M. Borwein, P. B. Borwein, F. G. Garvan, "Some Cubic Modular Identities of Ramanujan", Trans. Amer. Math. Soc. v. 343 n. 1 (1994) pp. 35-47.
- G. Andrews & B. Berndt, Ramanujan's Lost Notebook, Part I, Springer-Verlag.
- Bruce C. Berndt, "Ramanujan's Notebooks", 5 volumes, Springer-Verlag.
- B. C. Berndt, et. al., "Ramanujan's Forty Identities For the Rogers-Ramanujan Functions", Memoirs Amer. Math. Soc., n. 880 (2007).
- John A. Ewell, "A Note on a Jacobian identity", Proc. Amer. Math. Soc., 126 (1998), pp. 421-423.
- Nathan J. Fine, "Basic Hypergeometric Series and Applications", AMS 1988.
- R. W. Gosper, "Experiments and Discoveries in q-Trigonometry".
- B. Gordon & D. Sinor, Multiplicative properties of eta-products, Number theory, Madras 1987, Lecture Notes in Math., 1395, Springer, 1989.
- M. Rogers & W. Zudilin, On the Mahler measure of $1+X+1/X+Y+1/Y$, arXiv:1102.1153
- Michael Trott, The Mathematica Guidebook for Symbolics, Springer 2006.
- G. N. Watson, The Mock Theta Functions (2), Proc. London Math. Soc., (2) 42 (1937), pp. 274-304.
- M. D. Hirschhorn & J. A. Sellers, "Elementary Proofs of parity results for 5-Regular Partitions", Bull. Aust. Math. Soc., 81 (2010), pp. 58-63.

Last updated 10 Apr 2017

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