DOI QR코드

DOI QR Code

ON THE INFINITE PRODUCTS DERIVED FROM THETA SERIES II

  • Kim, Dae-Yeoul ;
  • Koo, Ja-Kyung
  • Published : 2008.09.30

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

algebraic number;theta series;Rogers-Ramanujan identities

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. 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
  8. M. Waldschmidt, Nombres Transcendants, Lecture Notes in Mathematics, Vol. 402. Springer-Verlag, Berlin-New York, 1974
  9. 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
  10. C. L. Siegel, Transcendental Numbers, Annals of Mathematics Studies, no. 16. Princeton University Press, Princeton, N. J., 1949