Automatic Proof of Theta-Function Identities
This is a tutorial for using two new MAPLE packages, thetaids and ramarobinsids. The thetaids package is designed for proving generalized eta-product identities using the valence formula for modular functions. We show how this package can be used to find theta-function identities as well as prove them. As an application, we show how to find and prove Ramanujan’s 40 identities for his so called Rogers–Ramanujan functions G(q) and H(q). In his thesis Robins found similar identities for higher level generalized eta-products. Our ramarobinsids package is for finding and proving identities for generalizations of Ramanujan’s G(q) and H(q) and Robin’s extensions. These generalizations are associated with certain real Dirichlet characters. We find a total of over 150 identities.
- 9.F. Garvan, A \(q\)-product tutorial for a \(q\)-series MAPLE package, Sém. Lothar. Combin. 42 (1999). Art. B42d, 27 pp. (electronic), The Andrews Festschrift (Maratea 1998) MR 1701583 (2000f:33001)Google Scholar
- 14.D.A. Ireland, A Dirichlet character table generator (2013), http://www.di-mgt.com.au/dirichlet-character-generator.html
- 16.J. Lovejoy, R. Osburn, Mixed mock modular \(q\)-series. J. Indian Math. Soc. (N.S.) (2013). Special volume to commemorate the 125th birth anniversary of Srinivasa Ramanujan, 45–61. MR 3157335Google Scholar
- 19.R.A. Rankin, Modular Forms and Functions (Cambridge University Press, Cambridge, 1977). MR 0498390 (58 #16518)Google Scholar
- 20.S. Robins, Arithmetic properties of modular forms, ProQuest LLC, Ann Arbor, MI, 1991. Thesis (Ph.D.)–University of California, Los Angeles. MR 2686433Google Scholar
- 21.S. Robins, Generalized Dedekind \(\eta \)-products, The Rademacher Legacy to Mathematics (University Park, PA, 1992). Contemporary Mathematics, vol. 166 (American Mathematical Society, Providence, 1994), pp. 119–128. MR 1284055 (95k:11061)Google Scholar