Functiones et Approximatio Commentarii Mathematici

Algebraic independence of certain numbers related to modular functions

Carsten Elsner, Shun Shimomura, and Iekata Shiokawa

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Abstract

In previous papers the authors established a method how to decide on the algebraic independence of a set $\{ y_1,\dots ,y_n \}$ when these numbers are connected with a set $\{ x_1,\dots ,x_n \}$ of algebraic independent parameters by a system $f_i(x_1,\dots ,x_n,y_1,\dots ,y_n) =0$ $(i=1,2,\dots ,n)$ of rational functions. Constructing algebraic independent parameters by Nesterenko's theorem, the authors successfully applied their method to reciprocal sums of Fibonacci numbers and determined all the algebraic relations between three $q$-series belonging to one of the sixteen families of $q$-series introduced by Ramanujan. In this paper we first give a short proof of Nesterenko's theorem on the algebraic independence of $\pi$, $e^{\pi\sqrt{d}}$ and a product of Gamma-values $\Gamma (m/n)$ at rational points $m/n$. Then we apply the method mentioned above to various sets of numbers. Our algebraic independence results include among others the coefficients of the series expansion of the Heuman-Lambda function, the values $P(q^r), Q(q^r)$, and $R(q^r)$ of the Ramanujan functions $P,Q$, and $R$, for $q\in \overline{\ACADQ}$ with $0<|q|<1$ and $r=1,2,3,5,7,10$, and the values given by reciprocal sums of polynomials.

Article information

Source
Funct. Approx. Comment. Math., Volume 47, Number 1 (2012), 121-141.

Dates
First available in Project Euclid: 25 September 2012

Permanent link to this document
https://projecteuclid.org/euclid.facm/1348578282

Digital Object Identifier
doi:10.7169/facm/2012.47.1.10

Mathematical Reviews number (MathSciNet)
MR2987116

Zentralblatt MATH identifier
1290.11109

Subjects
Primary: 11J85: Algebraic independence; Gelʹfond's method
Secondary: 11J89: Transcendence theory of elliptic and abelian functions 11J91: Transcendence theory of other special functions 11F03: Modular and automorphic functions

Keywords
algebraic independence Ramanujan functions Nesterenko's theorem complete elliptic integrals Gamma function.

Citation

Elsner, Carsten; Shimomura, Shun; Shiokawa, Iekata. Algebraic independence of certain numbers related to modular functions. Funct. Approx. Comment. Math. 47 (2012), no. 1, 121--141. doi:10.7169/facm/2012.47.1.10. https://projecteuclid.org/euclid.facm/1348578282


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References

  • G.E. Andrews and B.C. Berndt, Ramanujan's Lost Notebook, Part II, Springer, 2009.
  • J.M. Borwein and I.J. Zucker, Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals for the first kind, IMA J. Num. Anal. 12 (1992), 519–526.
  • B.C. Bruce, Ramanujan's Notebooks, Part II, Springer, 1989.
  • B.C. Bruce, Ramanujan's Notebooks, Part V, Springer, 1998.
  • P. Bundschuh, Zwei Bemerkungen über transzendente Zahlen, Monatsh. Math. 88 (1979), 293–304.
  • P.F. Byrd and M.D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, Springer, 1954.
  • Sh. Cooper and H.Y. Lam, Sixteen Eisenstein series, Ramanujan J. 18 (2009), 33–59.
  • Sh. Cooper and P.Ch. Toh, Quintic and septic Eisenstein series, Ramanujan J. 19 (2009), 163–181.
  • Sh. Cooper, On Ramanujan's function (k(q) = r(q)r^2(q^2)), Ramanujan J. 20 (2009), 311–328.
  • C. Elsner, S. Shimomura, and I. Shiokawa, Algebraic relations for reciprocal sums of Fibonacci numbers, Acta Arith. 130 (2007), 37–60.
  • C. Elsner, S. Shimomura, and I. Shiokawa, A remark on Nesterenko's theorem for Ramanujan functions, Ramanujan J. 21 no. 2 (2010), 211–221.
  • C. Elsner, S. Shimomura, I. Shiokawa, and Y. Tachiya, Algebraic independence results for the sixteen families of (q)-series, Ramanujan J. 22, no. 3 (2010), 315–344.
  • C. Elsner, S. Shimomura, and I. Shiokawa, Algebraic independence results for reciprocal sums of Fibonacci numbers, Acta Arith. 148 (2011), 205–223.
  • S. Gun, M.R. Murty, and P. Rath, Transcendence of the log gamma function and some discrete periods, J. Number Theory 129 (2009), 2154-2165.
  • Yu.V. Nesterenko, Modular functions and transcendence questions, Mat. Sb. 187 (1996), 65–96; English transl. Sb. Math. 187 (1996), 1319–1348.
  • Yu.V. Nesterenko and P. Philippon (Eds.), Introduction to Algebraic Independence Theory. Lecture Notes in Math., vol. 1752. Springer, Berlin, 2001.
  • Yu.V. Nesterenko, Algebraic Independence, Tata Inst. Fund. Research, Narosa Publ. House, 2009.
  • S. Ramanujan, On the product (\prod_n=0^n=\infty [1+(\fracxax+nd)^3]), J. Indian Math. Soc. 7 (1915), 209–211.
  • S. Ramanujan, On certain arithmetical functions, Trans. Cambridge Philos. Soc. 22 (1916), 159–184.
  • S. Ramanujan, Notebooks, vol. 2, Tata Institute of Fundamental Research, Bombay, 1957.
  • S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.
  • A. Selberg and S. Chowla, On Epstein's Zeta-function, J. Reine Angew. Math. 227 (1967), 86–110.
  • E.T. Whittaker and G.N. Watson, Modern Analysis, $4$th ed., Cambridge Univ. Press, Cambridge, 1927.
  • I.J. Zucker, The summation of series of hyperbolic functions, SIAM J. Math. Anal. 10 (1979), 192–206.