Period Lengths of Continued Fractions Involving Fibonacci Numbers

For given nonsquare positive integers C ≡ 5(mod 8), we investigate families {Dk(X)}k∈N of integral polynomials of the form Dk(X) = AkX +2BkX+C where (Bk/2)−(Ak/2)C = 4, and show that the period length of the simple continued fraction expansion of (1+ √ Dk(X))/2 is a multiple of k, and independent ofX. For each member of the families involved, we show how to easily determine the fundamental unit of the underlying quadratic order Z[(1+ √ Dk(X))/2]. We also demonstrate how the simple continued fraction expansion of (1+ √ Dk(X))/2 is related to that of (1 + √ C)/2. As applications, we present infinite families of continued fractions related to the Fibonacci numbers. This continues work in [11]-[12] and corrects errors in [15] (see Theorem 3.1). In 1949, Nyberg [14] found the first example of a parametric family in which a fundamental unit can be easily produced even though the period length of the continued fraction gets arbitrarily large. Since this discovery, there have been a number of generalizations: Dan Shanks [16]-[17] (see also Yamamoto [22]) in 1969 1971, Hendy [5] in 1974, Bernstein [2]-[3] in 1976, Williams [20] in 1985, Levesque and Rhin [7] in 1986, Azuhata [1] in 1987, Levesque [6] in 1988, Halter-Koch [4] in 1989, Mollin and Williams [13] in 1992, Williams [19] in 1995, and van der Poorten and Williams in [15] in 1999. In [12], we found infinite families of quadratic Pellian polynomials Dk(X) such that the continued fraction expansions √ Dk(X) have unbounded period length for arbitrary k, while for fixed k and arbitrary X > 0, have constant period length. In this paper, we continue the investigation for simple continued fraction expansions of (1 + √ Dk(x))/2, where Dk(X) are quadratic polynomials of Eisenstein type (see Section 3 below). We are also able to find explicit fundamental units of the order Z[(1 + √ Dk(X))/2]. A consequence of the result is an infinite family of continued fraction expansions whose fundamental units and discriminant are related to Fibonacci numbers. Ostensibly the first such finding of these types of continued fractions was given in [21], and the findings herein appear to be the second, albeit distinctly different from those given in [21]. The relatively “small” fundamental units for the underlying quadratic order which we explicitly determine means that we have “large” class numbers hD(X) for Z[(1 + √ D(X))/2]. The reason behind this fact is Siegel’s class number result [18] which tells us that for positive discriminants ∆, lim ∆→∞ log(h∆R)/ log( √ ∆) = 1