# ARITHMETIC OF INFINITE PRODUCTS AND ROGERS-RAMANUJAN CONTINUED FRACTIONS

• Kim, Dae-Yeoul (NATIONAL INSTITUTE FOR MATHEMATICAL SCIENCES) ;
• Koo, Ja-Kyung (KOREA ADVANCED INSTITUTE OF SCIENCE AND TECHNOLOGY DEPARTMENT OF MATHEMATICS) ;
• Simsek, Yilmaz (UNIVERSITY OF ADKENIZ, FACULTY OF SCIENCE DEPARTMENT OF MATHEMATICS)
• Published : 2007.07.31
• 96 7

#### Abstract

Let k be an imaginary quadratic field, h the complex upper half plane, and let $\tau{\in}h{\cap}k$, $q=e^{{\pi}i\tau}$. We find a lot of algebraic properties derived from theta functions, and by using this we explore some new algebraic numbers from Rogers-Ramanujan continued fractions.

#### Keywords

transcendental number;algebraic number;theta series;Rogers-Ramanujan continued fraction

#### References

1. G. E. Andrews, An introduction to Ramanujan's lost notebook, Amer. Math. Monthly 86 (1979), 89-108 https://doi.org/10.2307/2321943
2. G. E. Andrews, Ramanujan's lost notebook. III. The Rogers-Ramanujan continued fraction, Adv. Math. 41 (1981), 186-208 https://doi.org/10.1016/0001-8708(81)90015-3
3. B. C. Berndt, Ramanujan's Notebooks II, Springer-Verlag, 1989
4. B. C. Berndt, Ramanujan's Notebooks III, Springer-Verlag, 1991
5. B. C. Berndt, Ramanujan's Notebooks IV, Springer-Verlag, 1993
6. B. C. Berndt, Ramanujan's Notebooks V, Springer-Verlag, 1997
7. B. C. Berndt, S.-S. Huang, J. Sohn, and S. H. Son, Some theorems on the Rogers Ramanujan continued fraction in Ramanujan's lost notebook, Trans. Amer. Math. Soc. 352 (2000), 2157-2177 https://doi.org/10.1090/S0002-9947-00-02337-0
8. B. C. Berndt and H. H. Chan, Some values for the Rogers-Ramanujan continued fraction, Canad. J. Math. 47 (1995), 897-914 https://doi.org/10.4153/CJM-1995-046-5
9. B. C. Berndt, H. H. Chan, and L.-C. Zhang, Explicit evaluations of the Rogers-Ramanujan continued fraction, J. Reigne Angew. Math. 480 (1996), 141-159
10. S.-S. Huang, Ramanujan's evaluations of the Rogers-Ramanujan type continued fractions at primitive roots of unity, Acta Arith. 80 (1997), 49-60 https://doi.org/10.4064/aa-80-1-49-60
11. N. Ishida, Generators and equations for modular function fields of principal congruence subgroups, Acta Arith. 85 (1998), 197-207 https://doi.org/10.4064/aa-85-3-197-207
12. S.-Y. Kang, Some theorems on the Rogers-Ramanujan continued fraction and associated theta function identities in Ramanujan's lost notebook, The Ramanujan Journal 3 (1999), 91-111 https://doi.org/10.1023/A:1009869426750
13. S.-Y. Kang, Ramanujan's formulas for the explicit evaluation of the Rogers-Ramanujan continued fraction and theta functions, Acta Arith. 90 (1999), 49-68 https://doi.org/10.4064/aa-90-1-49-68
14. D. Kim and J. K. Koo, Algebraic integer as values of elliptic functions, Acta Arith. 100 (2001), 105-116 https://doi.org/10.4064/aa100-2-1
15. D. Kim and J. K. Koo, On the infinite products derived from theta series I, J. Korean Math. Soc. 44 (2007), 55-107 https://doi.org/10.4134/JKMS.2007.44.1.055
16. D. Kubert and S. Lang, Units in the modular function fields, Math. Ann. 218 (1975), 175-189 https://doi.org/10.1007/BF01370818
17. S. Lang, Elliptic Functions, Addison-Wesley, 1973
18. S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957
19. S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, 1988
20. L. J. Rogers, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. (1) 25 (1894), 318-343 https://doi.org/10.1112/plms/s1-25.1.318
21. A. V. Sills, Finite Rogers-Ramanujan type identities, Electron. J. Combin. 10 (2003), 1-122
22. L. J. Slater, A new proof of Roger's transformations of series, Proc. London Math. Soc. sereis 2 53 (1951), 461-475
23. L. J. Slater, Further identities of the Rogers-Ramanujan type, Proc. London Math. Soc. series 2 54 (1952), 147-167 https://doi.org/10.1112/plms/s2-54.2.147
24. S. Son, Cubic identities of theta functions, The Ramanujan Journal 2 (1998), 303-316 https://doi.org/10.1023/A:1009751614537
25. S. Son, Some theta function identities related to Rogers- Ramanujan continued fraction, Proc. Amer. Math. Soc. 126 (1998), 2895-2902 https://doi.org/10.1090/S0002-9939-98-04516-X
26. E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Press, 1978
27. W. N. Bailey, On the simplification of some identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (3) (1951), 217-221 https://doi.org/10.1112/plms/s3-1.1.217
28. P. B. Borwein and P. Zhou, On the irrationality of a certain q series, Proc. Amer. Math. Soc. 127 (1999), 1605-1613 https://doi.org/10.1090/S0002-9939-99-04722-X
29. J. Silverman, The Arithmetic of Elliptic Curves, Springer -Verlag, New York, 1986
30. L. J. Rogers, On two theorems of combinatory analysis and some allied identities, Proc. London Math. Soc. (1) 16 (1917), 315-336