Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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REMARKS FOR BASIC APPELL SERIES

  • Seo, Gyeong-Sig (Department of Mathematics and Institute of Pure and Applied Mathematics Chonbuk National University) ;
  • Park, Joong-Soo (Department of Mathematics Education Woosuk University)
  • Received : 2009.09.28
  • Accepted : 2009.11.06
  • Published : 2009.12.25
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Abstract

Let k be an imaginary quadratic field, ℌ the complex upper half plane, and let ${\tau}{\in}k{\cap}$ℌ, q = exp(${\pi}i{\tau}$). And let n, t be positive integers with $1{\leq}t{\leq}n-1$. Then $q^{{\frac{n}{12}}-{\frac{t}{2}}+{\frac{t^2}{2n}}}{\prod}^{\infty}_{m=1}(1-q^{nm-t})(1-q^{nm-(n-t)})$ is an algebraic number [10]. As a generalization of this result, we find several infinite series and products giving algebraic numbers using Ramanujan's $_{1{\psi}1}$ summation. These are also related to Rogers-Ramanujan continued fractions.

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References

  1. G. E. Andrews, Summations and transformations for basic Appell series, J. London Math. Soc. (2) 4 (1972), 618-622. https://doi.org/10.1112/jlms/s2-4.4.618
  2. B. C. Berndt, Ramanujan's notebooks. Part II, Springer-Verlag, 1991.
  3. N. Ishida, Generators and equations for modular function fields of principal congruence subgroups, Acta Arith. 85 (1998), no. 3, 197-207. https://doi.org/10.4064/aa-85-3-197-207
  4. N. Ishida and N. Ishii, Generator, and defining equation of the modular function field of the group ${\Gamma}_{1}$(N), Acta Arith. 101 (2002), no. 4, 303-320. https://doi.org/10.4064/aa101-4-1
  5. N. Ishida and N. Ishii, The equations for modular function fields of principal congruence subgroup, of prime level, Manuscripta Math. 90 (1996), no. 3, 271-285. https://doi.org/10.1007/BF02568306
  6. N. Ishida and N. Ishii, The equations for the modular curve $X_1$(N) derived from the equation for the modular curve X(N), Tokyo J. math. 22 (1999), no. 1, 167-175. https://doi.org/10.3836/tjm/1270041620
  7. F. J. Jackson, On basic double hypergeometric functions, Quart. J. Math., 13 (1942), 69-82. https://doi.org/10.1093/qmath/os-13.1.69
  8. F. J. Jackson, q-identities of Auluck, Carlitz, and Rogers, Quart. J. Math., 15 (1944), 49-61. https://doi.org/10.1093/qmath/os-15.1.49
  9. D. Kim and J. K. Koo, Algebraic integers as values of elliptic functions, Acta Arith. 100 (2001), 105-116. https://doi.org/10.4064/aa100-2-1
  10. 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
  11. D. Kubert and S. Lang, Units in the modular function field, Math. Ann, 218 (1975), 175-189. https://doi.org/10.1007/BF01370818
  12. S. Lang, Elliptic Functions, Addison-Wesley, 1973.
  13. S. Ramanujan, Proof of certain identities in combinatory analysis, Proc. Cambridge Philos. Soc. 19 (1919), 214-216.
  14. L. J. Rogers, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. 25(1984), 318-343. https://doi.org/10.1112/plms/s1-25.1.318