Document Type

Article

Publication Date

2007

Abstract

Let f(x) ∈ Z[x]. Set f0(x) = x and, for n ≥ 1, define fn(x) = f(fn−1(x)). We describe several infinite families of polynomials for which the infinite product Y∞ n=0 ( 1 + 1 fn(x) ) has a specializable continued fraction expansion of the form S∞ = [1; a1(x), a2(x), a3(x), . . . ], where ai(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 Yk n=0 (1 + 1 fn(x, k) ) is specializable but the regular continued fraction expansion of kY +1 n=0 ( 1 + 1 fn(x, k) ) is not specializable.

Publication Title

Journal of Number Theory

ISSN

0022-314X

Publisher

Elsevier

Volume

127

Issue

2

First Page

184

Last Page

219

Comments

Preprint version is available here.

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