ALGEBRAIC INDEPENDENCE OF CERTAIN NUMBERS RELATED TO MODULAR FUNCTIONS

In previous papers the authors established a method how to decide on the algebraic independence of a set {y(1) ,..., y(n)} when these numbers are connected with a set {x(1) ,..., x(n)} of algebraic independent parameters by a system f(i)(x(1) ,..., x(n), y(1) ,..., y(n)) = 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 pi, e(pi root 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 is an element of Q with 0 < |q| < 1 and r = 1, 2, 3, 5, 7, 1 0, and the values given by reciprocal sums of polynomials.