[Paper Review] Curious continued fractions, nonlinear recurrences and transcendental numbers

Andrew N. W. Hone|Kent Academic Repository (University of Kent)|Jun 30, 2015

Algebraic structures and combinatorial models6 references3 citations

TL;DR

This paper proves that the sum of reciprocals of certain rapidly growing integer sequences defined by nonlinear recurrences—specifically those in class (iii) of Hone's classification—has a continued fraction expansion that interlaces the original sequence and its ratios. The authors establish that such sums are transcendental using Roth's theorem on Diophantine approximation, leveraging the double-exponential growth of the sequences to satisfy the required approximation conditions.

ABSTRACT

We consider a family of integer sequences generated by nonlinear recurrences of the second order, which have the curious property that the terms of the sequence, and integer multiples of the ratios of successive terms (which are also integers), appear interlaced in the continued fraction expansion of the sum of the reciprocals of the terms. Using the rapid (double exponential) growth of the terms, for each sequence it is shown that the sum of the reciprocals is a transcendental number.

Motivation & Objective

• To generalize Hanna's empirical observation that the continued fraction of the sum of reciprocals of sequence A112373 interlaces the sequence and its ratios.
• To prove that this interlacing property holds for an infinite family of integer sequences generated by second-order nonlinear recurrences of the form $ x_{n+2}x_n = x_{n+1}^2 F(x_{n+1}) $ with $ F(0) = 1 $.
• To establish the transcendence of the sum of reciprocals of such sequences using Diophantine approximation theory.
• To extend the results of Davison and Shallit on transcendental continued fractions to a broader class of nonlinear recurrence-generated sequences.

Proposed method

• Use the Laurent property to ensure integer sequences from recurrences with $ F(0) = 1 $, starting from $ x_0 = x_1 = 1 $.
• Analyze the growth rate of the sequence $ x_n $ via logarithmic recurrence $ \Lambda_{n+1} - (d+2)\Lambda_n + \Lambda_{n-1} = \log c + \alpha_n $, where $ \Lambda_n = \log x_n $.
• Derive asymptotic behavior $ \Lambda_n \sim C\lambda^n $ with $ \lambda = \frac{d+2 + \sqrt{d(d+4)}}{2} > 2 $, showing double-exponential growth.
• Prove that the continued fraction partial quotients interlace $ x_n $ and $ y_n = x_{n+1}/x_n $, using recurrence relations $ a_{2n} = a_{2n-1}a_{2n-2} $, $ a_{2n+1} = a_{2n-1}(a_{2n} + 1) $.
• Apply Roth's theorem on Diophantine approximation: if a number has infinitely many rational approximations $ p/q $ with $ |\alpha - p/q| < 1/q^{\kappa} $ for $ \kappa > 2 $, then it is transcendental.
• Use the growth rate $ x_{n+1} > x_n^{\lambda - \epsilon} $ with $ \lambda > 2 $ to show that the sum $ \mathcal{S} $ satisfies the condition for transcendence.

Experimental results

Research questions

• RQ1 Does the continued fraction expansion of the sum of reciprocals of sequences generated by $ x_{n+2}x_n = x_{n+1}^2 F(x_{n+1}) $ with $ F(0) = 1 $ interlace the sequence and its ratios?
• RQ2 Can the transcendence of such sums be proven using Diophantine approximation, given the double-exponential growth of the terms?
• RQ3 Is the interlacing property in the continued fraction a general feature of all sequences in class (iii) of Hone's recurrence classification?
• RQ4 How does the asymptotic growth rate of the sequence influence the approximation properties of the sum of its reciprocals?
• RQ5 Can the method used for sequence A112373 be generalized to all such recurrences with $ F \in \mathbb{Z}_{\geq 0}[x] $, $ F(0) = 1 $?

Key findings

• The sum of reciprocals $ \mathcal{S} = \sum_{j=0}^\infty \frac{1}{x_j} $ for the sequence defined by $ x_{n+2}x_n = x_{n+1}^2(x_{n+1} + 1) $ has a continued fraction expansion $ [2; 1, 1, 2, 2, 6, 12, 78, \dots] $ that interlaces the original sequence and its ratios.
• The partial quotients $ a_n $ satisfy $ a_{2n} = a_{2n-1}a_{2n-2} $ and $ a_{2n+1} = a_{2n-1}(a_{2n} + 1) $, confirming the interlacing structure.
• The sequence $ x_n $ grows double-exponentially, with $ \log x_n \sim C\lambda^n $, where $ \lambda = \frac{3 + \sqrt{5}}{2} \approx 2.618 $ for the original case.
• For any such sequence with $ F(0) = 1 $, the sum $ \mathcal{S} $ is transcendental because the growth rate $ x_{n+1} > x_n^{\lambda - \epsilon} $ with $ \lambda > 2 $ implies infinitely many good rational approximations.
• The result generalizes to all recurrences of the form $ x_{n+2}x_n = x_{n+1}^2 F(x_{n+1}) $ with $ F \in \mathbb{Z}_{\geq 0}[x] $, $ F(0) = 1 $, ensuring the same continued fraction and transcendence properties.
• The asymptotic formula $ x_n \sim c^{-1/d} \exp(C\lambda^n) $ is derived, with $ \lambda = \frac{d+2 + \sqrt{d(d+4)}}{2} > 2 $, confirming the necessary growth for transcendence.