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.
Citation
Carsten Elsner. Shun Shimomura. Iekata Shiokawa. "Algebraic independence of certain numbers related to modular functions." Funct. Approx. Comment. Math. 47 (1) 121 - 141, September 2012. https://doi.org/10.7169/facm/2012.47.1.10
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