Abstract
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.
A preliminary version of this paper was presented by J. Frye on January 10, 2013 at JMM2013, San Diego. F. Garvan was supported in part by a grant from the Simon’s Foundation (#318714).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
G.E. Andrews, Ramunujan’s “lost” notebook. III. The Rogers–Ramanujan continued fraction. Adv. Math. 41(2), 186–208 (1981). MR 625893 (83m:10034c)
A. Berkovich, H. Yesilyurt, On Rogers–Ramanujan functions, binary quadratic forms and eta-quotients. Proc. Am. Math. Soc. 142(3), 777–793 (2014). MR 3148513
B.C. Berndt, Ramanujan’s Notebooks. Part III (Springer, New York, 1991). MR 1117903
B.C. Berndt, An overview of Ramanujan’s notebooks, Ramanujan: Essays and Surveys. History of Mathematics, vol. 22 (American Mathematical Society, Providence, 2001), pp. 143–164. MR 1862749
B.C. Berndt, G. Choi, Y.-S. Choi, H. Hahn, B.P. Yeap, A.J. Yee, H. Yesilyurt, J. Yi, Ramanujan’s forty identities for the Rogers–Ramanujan functions. Mem. Am. Math. Soc. 188(880), vi+96 (2007). MR 2323810
A.J.F. Biagioli, A proof of some identities of Ramanujan using modular forms. Glasgow Math. J. 31(3), 271–295 (1989). MR 1021804
B.J. Birch, A look back at Ramanujan’s notebooks. Math. Proc. Camb. Philos. Soc. 78, 73–79 (1975). MR 0379372
B. Cho, J.K. Koo, Y.K. Park, Arithmetic of the Ramanujan–Göllnitz–Gordon continued fraction. J. Number Theory 129(4), 922–947 (2009). MR 2499414
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)
H. Göllnitz, Partitionen mit Differenzenbedingungen. J. Reine Angew. Math. 225, 154–190 (1967). MR 0211973
B. Gordon, Some continued fractions of the Rogers–Ramanujan type. Duke Math. J. 32, 741–748 (1965). MR 0184001
S.-S. Huang, On modular relations for the Göllnitz–Gordon functions with applications to partitions. J. Number Theory 68(2), 178–216 (1998). MR 1605895
T. Huber, D. Schultz, Generalized reciprocal identities. Proc. Am. Math. Soc. 144(11), 4627–4639 (2016). MR 3544515
D.A. Ireland, A Dirichlet character table generator (2013), http://www.di-mgt.com.au/dirichlet-character-generator.html
J. Lovejoy, R. Osburn, The Bailey chain and mock theta functions. Adv. Math. 238, 442–458 (2013). MR 3033639
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 3157335
J. Lovejoy, R. Osburn, \(q\)-hypergeometric double sums as mock theta functions. Pac. J. Math. 264(1), 151–162 (2013). MR 3079764
J. Lovejoy, R. Osburn, On two 10th-order mock theta identities. Ramanujan J. 36(1–2), 117–121 (2015). MR 3296714
R.A. Rankin, Modular Forms and Functions (Cambridge University Press, Cambridge, 1977). MR 0498390 (58 #16518)
S. Robins, Arithmetic properties of modular forms, ProQuest LLC, Ann Arbor, MI, 1991. Thesis (Ph.D.)–University of California, Los Angeles. MR 2686433
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)
L. Ye, A symbolic decision procedure for relations arising among Taylor coefficients of classical Jacobi theta functions. J. Symb. Comput. 82, 134–163 (2017). MR 3608235
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Frye, J., Garvan, F. (2019). Automatic Proof of Theta-Function Identities. In: Blümlein, J., Schneider, C., Paule, P. (eds) Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-04480-0_10
Download citation
DOI: https://doi.org/10.1007/978-3-030-04480-0_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-04479-4
Online ISBN: 978-3-030-04480-0
eBook Packages: Computer ScienceComputer Science (R0)