Symmetry and specializability in the continued fraction expansions of some infinite products Original Research Article

Let image. Set f0(x)=x and, for ngreater-or-equal, slanted1, define fn(x)=f(fn−1(x)). We describe several infinite families of polynomials for which the infinite productimage has a specializable continued fraction expansion of the formS∞=[1;a1(x),a2(x),a3(x),…], where image for igreater-or-equal, slanted1. When the infinite product and the continued fraction are specialized by letting x take integral values, we get infinite classes of real numbers whose regular continued fraction expansion is predictable. We also show that, under some simple conditions, all the real numbers produced by this specialization are transcendental. We also show, for any integer kgreater-or-equal, slanted2, that there are classes of polynomials f(x,k) for which the regular continued fraction expansion of the productimage is specializable but the regular continued fraction expansion ofimage is not specializable.