ALGEBRAIC INDEPENDENCE OF RECIPROCAL SUMS OF POWERS OF CERTAIN FIBONACCI-TYPE NUMBERS

The Fibonacci-type numbers in the title look like R-n = g(1)gamma(n)(1) + g(2)gamma(n)(2) and S-n = h(1)gamma(n)(1) + h(2)gamma(n)(2) for any n is an element of Z, where the g's, h's, and gamma's are given algebraic numbers satisfying certain natural conditions. For fixed k is an element of Z(>0), and for fixed non-zero periodic sequences (a(h)), (b(h)), (c(h)) of algebraic numbers, the algebraic independence of the series Sigma(infinity)(h=0) a(h)/gamma(krh)(1), Sigma(infinity)(h=0)' b(h)/(R-kr(+l)h)(m), Sigma(infinity)(h=0)' c(h)/(S-kr(+l)h)(m) ((l, m, r) is an element of z x z(>0) x z(>1)) is studied. Here the main tool is Mahler's method which reduces the investigation of the algebraic independence of numbers (over Q) to that of functions (over the rational function field) if they satisfy certain types of functional equations.