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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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

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

First available in Project Euclid: 25 September 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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