On Rogers–Ramanujan functions, binary quadratic forms and eta-quotients

Berkovich, Alexander; Yesilyurt, Hamza

doi:10.1090/S0002-9939-2013-11816-2

On Rogers–Ramanujan functions, binary quadratic forms and eta-quotients
HTML articles powered by AMS MathViewer

by Alexander Berkovich and Hamza Yesilyurt PDF

Proc. Amer. Math. Soc. 142 (2014), 777-793 Request permission

Abstract:

In a handwritten manuscript published with his lost notebook, Ramanujan stated without proofs forty identities for the Rogers–Ramanujan functions. We observe that the function that appears in Ramanujan’s identities can be obtained from a Hecke action on a certain family of eta products. We establish further Hecke-type relations for these functions involving binary quadratic forms. Our observations enable us to find new identities for the Rogers–Ramanujan functions and also to use such identities in return to find identities involving binary quadratic forms.

References

Alexander Berkovich and Hamza Yesilyurt, Ramanujan’s identities and representation of integers by certain binary and quaternary quadratic forms, Ramanujan J. 20 (2009), no. 3, 375–408. MR 2574780, DOI 10.1007/s11139-009-9215-8
Bruce C. Berndt, Ramanujan’s notebooks. Part III, Springer-Verlag, New York, 1991. MR 1117903, DOI 10.1007/978-1-4612-0965-2
Bruce C. Berndt and Hamza Yesilyurt, New identities for the Rogers-Ramanujan functions, Acta Arith. 120 (2005), no. 4, 395–413. MR 2190072, DOI 10.4064/aa120-4-5
Bruce C. Berndt, Geumlan Choi, Youn-Seo Choi, Heekyoung Hahn, Boon Pin Yeap, Ae Ja Yee, Hamza Yesilyurt, and Jinhee Yi, Ramanujan’s forty identities for the Rogers-Ramanujan functions, Mem. Amer. Math. Soc. 188 (2007), no. 880, vi+96. MR 2323810, DOI 10.1090/memo/0880
Anthony J. F. Biagioli, A proof of some identities of Ramanujan using modular forms, Glasgow Math. J. 31 (1989), no. 3, 271–295. MR 1021804, DOI 10.1017/S0017089500007850
D. M. Bressoud, Some identities involving Rogers-Ramanujan-type functions, J. London Math. Soc. (2) 16 (1977), no. 1, 9–18. MR 447158, DOI 10.1112/jlms/s2-16.1.9
Kathrin Bringmann and Holly Swisher, On a conjecture of Koike on identities between Thompson series and Rogers-Ramanujan functions, Proc. Amer. Math. Soc. 135 (2007), no. 8, 2317–2326. MR 2302552, DOI 10.1090/S0002-9939-07-08735-7
Erich Hecke, Mathematische Werke, Vandenhoeck & Ruprecht, Göttingen, 1970 (German). Mit einer Vorbemerkung von B. Schoenberg, einer Anmerkung von Carl Ludwig Siegel, und einer Todesanzeige von Jakob Nielsen; Zweite durchgesehene Auflage. MR 0371577
Masao Koike, Thompson series and Ramanujan’s identities, Galois theory and modular forms, Dev. Math., vol. 11, Kluwer Acad. Publ., Boston, MA, 2004, pp. 367–373. MR 2059774, DOI 10.1007/978-1-4613-0249-0_{2}0
Srinivasa Ramanujan, The lost notebook and other unpublished papers, Springer-Verlag, Berlin; Narosa Publishing House, New Delhi, 1988. With an introduction by George E. Andrews. MR 947735
L. J. Rogers, Second Memoir on the Expansion of certain Infinite Products, Proc. Lond. Math. Soc. 25 (1893/94), 318–343. MR 1576348, DOI 10.1112/plms/s1-25.1.318
L. J. Rogers, On a Type of Modular Relation, Proc. London Math. Soc. (2) 19 (1921), no. 5, 387–397. MR 1576607, DOI 10.1112/plms/s2-19.1.387
G. N. Watson, Proof of certain identities in combinatory analysis, J. Indian Math. Soc. 20 (1933), 57–69.
Hamza Yesilyurt, Elementary proofs of some identities of Ramanujan for the Rogers-Ramanujan functions, J. Math. Anal. Appl. 388 (2012), no. 1, 420–434. MR 2869757, DOI 10.1016/j.jmaa.2011.11.004
Hamza Yesilyurt, A generalization of a modular identity of Rogers, J. Number Theory 129 (2009), no. 6, 1256–1271. MR 2521474, DOI 10.1016/j.jnt.2009.01.007