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

Abstract

Let f ( x ) ∈ Z [ x ] . Set f 0 ( x ) = x and, for n ⩾ 1 , define f n ( x ) = f ( f n − 1 ( x ) ) . We describe several infinite families of polynomials for which the infinite product ∏ n = 0 ∞ ( 1 + 1 f n ( x ) ) has a specializable continued fraction expansion of the form S ∞ = [ 1 ; a 1 ( x ) , a 2 ( x ) , a 3 ( x ) , … ] , where a i ( x ) ∈ Z [ x ] for i ⩾ 1 . 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 k ⩾ 2 , that there are classes of polynomials f ( x , k ) for which the regular continued fraction expansion of the product ∏ n = 0 k ( 1 + 1 f n ( x , k ) ) is specializable but the regular continued fraction expansion of ∏ n = 0 k + 1 ( 1 + 1 f n ( x , k ) ) is not specializable.