Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON THE INFINITE PRODUCTS DERIVED FROM THETA SERIES II

  • Kim, Dae-Yeoul (National Institute for Mathematical Sciences) ;
  • Koo, Ja-Kyung (Korea Advanced Institute of Science and Technology Department of Mathematical Sciences)
  • Published : 2008.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

Let k be an imaginary quadratic field, ${\eta}$ the complex upper half plane, and let ${\tau}{\in}{\eta}{\cap}k,\;q=e^{{\pi}{i}{\tau}}$. For n, t ${\in}{\mathbb{Z}}^+$ with $1{\leq}t{\leq}n-1$, set n=${\delta}{\cdot}2^{\iota}$(${\delta}$=2, 3, 5, 7, 9, 13, 15) with ${\iota}{\geq}0$ integer. Then we show that $q{\frac}{n}{12}-{\frac}{t}{2}+{\frac}{t^2}{2n}{\prod}_{m=1}^{\infty}(1-q^{nm-t})(1-q^{{nm}-(n-t)})$ are algebraic numbers.

Keywords

References

  1. B. C. Berndt, Ramanujan's Notebooks. Part II, Springer-Verlag, New York, 1989
  2. B. C. Berndt, Ramanujan's Notebooks. Part III, Springer-Verlag, New York, 1991
  3. B. C. Berndt, Ramanujan's Notebooks. Part IV, Springer-Verlag, New York, 1994
  4. D. Kim and J. K. Koo, Algebraic integers as values of elliptic functions, Acta Arith. 100 (2001), no. 2, 105-116 https://doi.org/10.4064/aa100-2-1
  5. D. Kim and J. K. Koo, On the infinite products derived from theta series I, J. Korean Math. Soc. 44 (2007), no. 1, 55-107 https://doi.org/10.4134/JKMS.2007.44.1.055
  6. S. Lang, Elliptic Functions, Addison-Wesley Publishing Co., Inc., Reading, Mass. -London-Amsterdam, 1973
  7. C. L. Siegel, Transcendental Numbers, Annals of Mathematics Studies, no. 16. Princeton University Press, Princeton, N. J., 1949
  8. A. V. Sills, On identities of the Rogers-Ramanujan type, Ramanujan J. 11 (2006), no. 3, 403-429 https://doi.org/10.1007/s11139-006-8483-9
  9. M. Waldschmidt, Nombres Transcendants, Lecture Notes in Mathematics, Vol. 402. Springer-Verlag, Berlin-New York, 1974
  10. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions. Fourth edition. Reprinted Cambridge University Press, New York 1962