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 $\tau{\in}h{\cap}k,\;q Keywords transcendental number;algebraic number;theta series;Rogers-Ramanujan identities; Language English Cited by 1. ARITHMETIC OF INFINITE PRODUCTS AND ROGERS-RAMANUJAN CONTINUED FRACTIONS,;;; 대한수학회논문집, 2007. vol.22. 3, pp.331-351 2. ON THE INFINITE PRODUCTS DERIVED FROM THETA SERIES II,;; 대한수학회지, 2008. vol.45. 5, pp.1379-1391 3. REMARKS FOR BASIC APPELL SERIES,;; 호남수학학술지, 2009. vol.31. 4, pp.463-478 4. DIVISOR FUNCTIONS AND WEIERSTRASS FUNCTIONS ARISING FROM q-SERIES,;; 대한수학회보, 2012. vol.49. 4, pp.693-704 1. REMARKS FOR BASIC APPELL SERIES, Honam Mathematical Journal, 2009, 31, 4, 463 2. DIVISOR FUNCTIONS AND WEIERSTRASS FUNCTIONS ARISING FROM q-SERIES, Bulletin of the Korean Mathematical Society, 2012, 49, 4, 693 3. A note on the transcendence of infinite products, Czechoslovak Mathematical Journal, 2012, 62, 3, 613 References 1. C. Adiga and T. Kim, On a continued fraction of Ramanujan, Tamsui Oxf. J. Math. Sci. 19 (2003), no. 1, 55-65 2. K. Barre-Sirieix, G. Diaz, F. Gramain, and G. Philibert, Une preuve de la conjecture de Mahler-Manin, Invent. Math. 124 (1996), no. 1-3, 1-9 3. B. C. Berndt, Ramanujan's Notebooks III, Springer, 1991 4. B. C. Berndt, Ramanujan's Notebooks V, Springer, 1998 5. B. C. Berndt, H. H. Chan, and L.-C. Zhang, Ramanujan's remarkable product of theta- functions, Proc. Edinburgh Math. Soc. (2) 40 (1997), no. 3, 583-612 6. B. C. Berndt and A. Yee, On the generalized Rogers-Ramanujan continued fraction, Ramanujan J. 7 (2003), no. 1-3, 321-331 7. D. Bertrand, Series d'Eisenstein et transcendence, Bull. Soc. Math. France 104 (1976), no. 3, 309-321 8. D. Bertrand, Theta functions and transcendence, Ramanujan J. 1 (1997), no. 4, 339-350 9. H. H. Chan and Y. L. Ong, On Eisenstein series and$\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

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