ARITHMETIC OF INFINITE PRODUCTS AND ROGERS-RAMANUJAN CONTINUED FRACTIONS

Title & Authors
ARITHMETIC OF INFINITE PRODUCTS AND ROGERS-RAMANUJAN CONTINUED FRACTIONS
Kim, Dae-Yeoul; Koo, Ja-Kyung; Simsek, Yilmaz;

Abstract
Let k be an imaginary quadratic field, h the complex upper half plane, and let $\small{\tau{\in}h{\cap}k}$, \$q
Keywords
transcendental number;algebraic number;theta series;Rogers-Ramanujan continued fraction;
Language
English
Cited by
References
1.
G. E. Andrews, An introduction to Ramanujan's lost notebook, Amer. Math. Monthly 86 (1979), 89-108

2.
G. E. Andrews, Ramanujan's lost notebook. III. The Rogers-Ramanujan continued fraction, Adv. Math. 41 (1981), 186-208

3.
W. N. Bailey, On the simplification of some identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (3) (1951), 217-221

4.
B. C. Berndt, Ramanujan's Notebooks II, Springer-Verlag, 1989

5.
B. C. Berndt, Ramanujan's Notebooks III, Springer-Verlag, 1991

6.
B. C. Berndt, Ramanujan's Notebooks IV, Springer-Verlag, 1993

7.
B. C. Berndt, Ramanujan's Notebooks V, Springer-Verlag, 1997

8.
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

9.
B. C. Berndt and H. H. Chan, Some values for the Rogers-Ramanujan continued fraction, Canad. J. Math. 47 (1995), 897-914

10.
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

11.
P. B. Borwein and P. Zhou, On the irrationality of a certain q series, Proc. Amer. Math. Soc. 127 (1999), 1605-1613

12.
S.-S. Huang, Ramanujan's evaluations of the Rogers-Ramanujan type continued fractions at primitive roots of unity, Acta Arith. 80 (1997), 49-60

13.
N. Ishida, Generators and equations for modular function fields of principal congruence subgroups, Acta Arith. 85 (1998), 197-207

14.
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

15.
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

16.
D. Kim and J. K. Koo, Algebraic integer as values of elliptic functions, Acta Arith. 100 (2001), 105-116

17.
D. Kim and J. K. Koo, On the infinite products derived from theta series I, J. Korean Math. Soc. 44 (2007), 55-107

18.
D. Kubert and S. Lang, Units in the modular function fields, Math. Ann. 218 (1975), 175-189

19.
S. Lang, Elliptic Functions, Addison-Wesley, 1973

20.
S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957

21.
S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, 1988

22.
L. J. Rogers, On two theorems of combinatory analysis and some allied identities, Proc. London Math. Soc. (1) 16 (1917), 315-336

23.
L. J. Rogers, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. (1) 25 (1894), 318-343

24.
A. V. Sills, Finite Rogers-Ramanujan type identities, Electron. J. Combin. 10 (2003), 1-122

25.
J. Silverman, The Arithmetic of Elliptic Curves, Springer -Verlag, New York, 1986

26.
L. J. Slater, A new proof of Roger's transformations of series, Proc. London Math. Soc. sereis 2 53 (1951), 461-475

27.
L. J. Slater, Further identities of the Rogers-Ramanujan type, Proc. London Math. Soc. series 2 54 (1952), 147-167

28.
S. Son, Cubic identities of theta functions, The Ramanujan Journal 2 (1998), 303-316

29.
S. Son, Some theta function identities related to Rogers- Ramanujan continued fraction, Proc. Amer. Math. Soc. 126 (1998), 2895-2902

30.
E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Press, 1978