ON THE INFINITE PRODUCTS DERIVED FROM THETA SERIES I

Title & Authors
ON THE INFINITE PRODUCTS DERIVED FROM THETA SERIES I
Kim, Dae-Yeoul; Koo, Ja-Kyung;

Abstract
Let k be an imaginary quadratic field, h the complex upper half plane, and let $\small{\tau{\in}h{\cap}k,\;q=e^{{\pi}i\tau}}$. In this article, we obtain algebraic numbers from the 130 identities of Rogers-Ramanujan continued fractions investigated in [28] and [29] by using Berndt's idea ([3]). Using this, we get special transcendental numbers. For example, $\small{\frac{q^{1/8}}{1}+\frac{-q}{1+q}+\frac{-q^2}{1+q^2}+\cdots}$ ([1]) is transcendental.
Keywords
transcendental number;algebraic number;theta series;Rogers-Ramanujan identities;
Language
English
Cited by
1.
REMARKS FOR BASIC APPELL SERIES,Seo, Gyeong-Sig;Park, Joong-Soo;

호남수학학술지, 2009. vol.31. 4, pp.463-478
2.
ARITHMETIC OF INFINITE PRODUCTS AND ROGERS-RAMANUJAN CONTINUED FRACTIONS,Kim, Dae-Yeoul;Koo, Ja-Kyung;Simsek, Yilmaz;

대한수학회논문집, 2007. vol.22. 3, pp.331-351
3.
ON THE INFINITE PRODUCTS DERIVED FROM THETA SERIES II,Kim, Dae-Yeoul;Koo, Ja-Kyung;

대한수학회지, 2008. vol.45. 5, pp.1379-1391
4.
DIVISOR FUNCTIONS AND WEIERSTRASS FUNCTIONS ARISING FROM q-SERIES,Kim, Dae-Yeoul;Kim, Min-Soo;

대한수학회보, 2012. vol.49. 4, pp.693-704
1.
A note on the transcendence of infinite products, Czechoslovak Mathematical Journal, 2012, 62, 3, 613
2.
REMARKS FOR BASIC APPELL SERIES, Honam Mathematical Journal, 2009, 31, 4, 463
3.
DIVISOR FUNCTIONS AND WEIERSTRASS FUNCTIONS ARISING FROM q-SERIES, Bulletin of the Korean Mathematical Society, 2012, 49, 4, 693
References
1.
C. Adiga and T. Kim, On a continued fraction of Ramanujan, Tamsui Oxf. J. Math. Sci. 19 (2003), no. 1, 55-65

2.
K. Barre-Sirieix, G. Diaz, F. Gramain, and G. Philibert, Une preuve de la conjecture de Mahler-Manin, Invent. Math. 124 (1996), no. 1-3, 1-9

3.
B. C. Berndt, Ramanujan's Notebooks III, Springer, 1991

4.
B. C. Berndt, Ramanujan's Notebooks V, Springer, 1998

5.
B. C. Berndt, H. H. Chan, and L.-C. Zhang, Ramanujan's remarkable product of theta- functions, Proc. Edinburgh Math. Soc. (2) 40 (1997), no. 3, 583-612

6.
B. C. Berndt and A. Yee, On the generalized Rogers-Ramanujan continued fraction, Ramanujan J. 7 (2003), no. 1-3, 321-331

7.
D. Bertrand, Series d'Eisenstein et transcendence, Bull. Soc. Math. France 104 (1976), no. 3, 309-321

8.
D. Bertrand, Theta functions and transcendence, Ramanujan J. 1 (1997), no. 4, 339-350

9.
H. H. Chan and Y. L. Ong, On Eisenstein series and $\sum_{m,n}^{\infty}=_{-{\infty}}q^{m^2+mn+2n^2}$, Proc. Amer. Math. Soc. 127 (1999), no. 6, 1735-1744

10.
D. Duverney, Ke. Nishioka, Ku. Nishioka, and I. Shiokawa, Transcendence of Rogers- Ramanujan continued fraction and reciprocal sums of Fibonacci numbers, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), no. 7, 140-142

11.
A. B. Ekin, The rank and the crank in the theory of partitions, Ph. D Thesis, University of Sussex, 1993

12.
L. Euler, Introduction to Analysis of the In¯nite, Springer-Verlag, 1988

13.
N. J. Fine, Basic Hypergeometric Series and Applications, American Mathematical So- ciety, 1988

14.
M. D. Hirschhorn, An identity of Ramanujan, and application, in 'q-series from a contemporary perspective', Contemp. Math. 254 (2000), 229-234

15.
A. Hurwitz, Uber die Entwickelungscoefficienten der lemniscatischen Funktionen., Math. Ann. 51 (1898), no. 2, 196-226

16.
K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer- Verlag, 1990

17.
D. Kim and J. K. Koo, Algebraic integer as values of elliptic functions, Acta Arith. 100 (2001), no. 2, 105-116

18.
D. Kim and J. K. Koo, Algebraic numbers, transcendental numbers and elliptic curves derived from in¯nite products, J. Korean Math. Soc. 40 (2003), no. 6, 977-998

19.
O. Kolberg, Some identities involving the partition function, Math. Scand. 5 (1957), 77-92

20.
S. Lang, Elliptic Functions, Addison-Wesley, 1973

21.
V. A. Lebesgue, Sommation de quelques series, J. Math. Pure. Appl. 5 (1840), 42-71

22.
D. Mumford, Tata Lectures on Theta I, Birkhauser Boston, Inc., Boston, MA, 1983

23.
S. Ramanujan, Collected Papers, Chelsea, 1962

24.
S. Ramanujan, Modular equations and approximations to ${\pi}$ , Quart. J. Math (Oxford) 45 (1914), 350-372

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

26.
M. Rosen, Abel's theorem on the lemniscate, Amer. Math. Monthly 88 (1981), no. 6, 387-395

27.
T. Schneider, Transzendenzeigenschaften elliptischer Funktionen, J. Reine Angew. Math. 14 (1934), 70-74

28.
A. V. Sills, Finite Rogers-Ramanujan type identities, Electron. J. Combin. 10 (2003), 1-122

29.
L. J. Slater, Further identies of the Rogers-Ramanujan type, Proc. London Math. Soc. (2) 54 (1952), 147-167

30.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Press, 1962