Algebraic independence of certain numbers related to modular functions

Carsten Elsner, Shun Shimomura, Iekata Shiokawa

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In previous papers the authors established a method how to decide on the algebraic independence of a set {y1, . . . , yn} when these numbers are connected with a set {x1, . . . , xn} of algebraic independent parameters by a system fi(x1, . . . , xn, y1, . . . , yn) = 0 (i = 1, 2, . . . , 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 π, eπ√d and a product of Gamma-values Γ(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(qr),Q(qr), and R(qr) of the Ramanujan functions P,Q, and R, for q ∈ Q with 0 < |q| < 1 and r = 1, 2, 3, 5, 7, 10, and the values given by reciprocal sums of polynomials.

Original languageEnglish
Pages (from-to)121-141
Number of pages21
JournalFunctiones et Approximatio, Commentarii Mathematici
Issue number1
Publication statusPublished - 2012


  • Algebraic independence
  • Complete elliptic integrals
  • Gamma function
  • Nesterenko's theorem
  • Ramanujan functions

ASJC Scopus subject areas

  • General Mathematics


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