Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-12-02T20:04:09.987Z Has data issue: false hasContentIssue false
  • English
  • Français

COMMON SOURCE OF NUMEROUS THETA FUNCTION IDENTITIES

Published online by Cambridge University Press:  01 January 2007

CHU WENCHANG
CHU WENCHANG*
Affiliation:
Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, P. R. China
Rights & Permissions [Opens in a new window]

Abstract.

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Motivated by the recent work due to Warnaar (2005), two new and elementary proofs are presented for a very useful q-difference equation on eight shifted factorials of infinite order. As the common source of theta function identities, this q-difference equation is systematically explored to review old and establish new identities on Ramanujan's partition functions. Most of the identities obtained can be interpreted in terms of theorems on classical partitions.

Keywords

Primary 05A30Secondary 14K25
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 49 , Issue 1 , January 2007 , pp. 61 - 79
Copyright
Copyright © Glasgow Mathematical Journal Trust 2007

References

REFERENCES

1.Andrews, G. E., The theory of partitions in Encyclopedia of Mathematics and its Applications (Vol. 2) (Addison-Wesley, Reading, MA, 1976).Google Scholar
2.Bailey, W. N., On the simplification of some identities on the Rogers-Ramanujan type, Proc. London Math. Soc. (3) 1 (1951), 217–221.CrossRefGoogle Scholar
3.Berndt, B. C., Ramanujan's Notebooks (Part III) (Springer-Verlag, 1989).CrossRefGoogle Scholar
4.Berndt, B. C., Choi, G., Choi, Y. S., Hahn, H., Yeap, B. P., Yee, A. J., Yesilyurt, H. and Yi, J., Ramanujan's forty identities for the Rogers-Ramanujan functions, Memoirs of American Mathematical Society, to appear.Google Scholar
5.Berndt, B. C., Chan, S. H., Liu, Z. G. and Yesilyurt, H., A new identity for (q;q)10∞ with an application to Ramanujan's partition congruence modulo 11, Quart. J. Math. Oxford Ser. 2 55 (2004), 13–30.CrossRefGoogle Scholar
6.Birch, B. J., A look back at Ramanujan's notebooks, Math. Proc. Cambridge Philos. Soc. 78 (1975), 73–79.Google Scholar
7.Blecksmith, R., Brillhart, J. and Gerst, I., Parity results for certain partition functions and identities similar to theta function identities, Mathematics of Computation 48 (177) (1987), 29–38.CrossRefGoogle Scholar
8.Borwein, J. M. and Borwein, P. B., π and the AGM (John Wiley & Sons, New York, 1987).Google Scholar
9.Borwein, J. M., Borwein, P. B. and Garvan, F. G., Some cubic modular identities of Ramanujan, Trans. Amer. Math. Soc. 343 (1) (1994), 35–47.CrossRefGoogle Scholar
10.Chu, W., A trigonometric identity with its q-analogue, Amer. Math. Monthly 99 (5) (1992), Problem 10226; ibid 103:2 (1996), Solution.Google Scholar
11.Chu, W., Theta function identities and Ramanujan's congruences on partition function, Quart. J. Math. Oxford Ser. 2 56 (4), (2005), 491–506.CrossRefGoogle Scholar
12.Ewell, J. A., Completion of a Gaussian derivation, Proc. Amer. Math. Soc. 84 (2) (1982), 311–314.Google Scholar
13.Ewell, J. A., Arithmetical consequences of a sextuple product identity, Rocky Mountain J. Math. 25 4 (1995), 1287–1293.Google Scholar
14.Ewell, J. A., A note on a Jacobian identity, Proc. Amer. Math. Soc. 126 2 (1998), 421–423.Google Scholar
15.Farkas, H. M. and Kra, I., Partitions and theta constant identities, in Analysis, geometry, and Number thoery: The mathematics of Leon Ehrenpreis, Contemp. Math. 251 (2000), 197–203.Google Scholar
16.Gasper, G. and Rahman, M., Basic hypergeometric series (second edition) (Cambridge University Press, 2004).Google Scholar
17.Gordon, B., Some identities in combinatorial analysis, Quart. J. Math. Oxford Ser. 2 12 (1961), 285–290.Google Scholar
18.Liu, Z. G., A three-term theta function identity and its applications, Advances in Math. 195 (2005), 1–23.Google Scholar
19.Kongsiriwong, S. and Liu, Z. G., Uniform proofs of q-series-product identities, Results in Math. 44 (2003), 312–339.Google Scholar
20.Robins, S., Arithmetic properties of modular forms, Ph. D Thesis, University of California at Los Angeles (1991).Google Scholar
21.Rogers, I. J., Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. 25 (1894), 318–343.Google Scholar
22.Shen, L. C., On the products of three theta functions, Ramanujan Journal 3 (1999), 343–357.CrossRefGoogle Scholar
23.Ole, S.. Warnaar, A generalization of the Farkas and Kra partition theorem for modulo 7, J Combinatorial Theory (Series A) 110 (2005), 43–52.Google Scholar
24.Watson, G. N., Proof of certain identities in combinatory analysis, J. Indian Math. Society 20 (1933), 57–69.Google Scholar
25.Whittaker, E. T. and Watson, G. N., A course of modern analysis [Fourth Edition] Cambridge University Press, 1952).Google Scholar