Abstract
We consider series of the form
where \(x_1=q\) and the integer sequence \((x_n)\) satisfies a certain non-autonomous recurrence of second order, which entails that \(x_n|x_{n+1}\) for \(n\ge 1\). It is shown that the terms of the sequence, and multiples of the ratios of successive terms, appear interlaced in the continued fraction expansion of the sum of the series, which is a transcendental number.
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1 Introduction
In recent work [5], we considered the integer sequence
(sequence A112373 in Sloane’s Online Encyclopedia of Integer Sequences), which is generated from the initial values \(x_0=x_1=1\) by the nonlinear recurrence relation
and proved some observations of Hanna, namely that the sum
has the continued fraction expansion
where \(y_j=x_{j+1}/x_j\in {\mathbb N}\) and we use the notation
for continued fractions. Furthermore, we generalized this result by obtaining the explicit continued fraction expansion for the sum of reciprocals (1.3) in the case of a sequence \((x_n)\) generated by a nonlinear recurrence of the form
with \(F(x)\in {\mathbb Z}_{\ge 0}[x]\) and \(F(0)=1\); so (1.2) corresponds to the particular case \(F(x)=x+1\).
All of the recurrences (1.5) exhibit the Laurent phenomenon [4], and starting from \(x_0=x_1=1\) they generate a sequence of positive integers satisfying \(x_n|x_{n+1}\). The latter fact means that the sum (1.3) is an Engel series (see Theorem 2.3 in Duverney’s book [3], for instance).
The purpose of this note is to present a further generalization of the results in [5], by considering a sum
with the terms \(x_n\) satisfying the recurrence
for \(n\ge 2\), where \((z_n)\) is a sequence of positive integers, \(x_1=q\), and \(x_2\) is specified suitably. Observe that, in contrast to (1.5), the recurrence (1.7) can be viewed as a non-autonomous dynamical system for \(x_n\), because the coefficient \(z_n\) can vary independently (unless it is taken to be \(G(x_n)\), for some function G). The same argument as used in [5], based on Roth’s theorem, shows the transcendence of any number S defined by a sum of the form (1.6) with such a sequence \((x_n)\).
2 The main result
We start with a rational number written in lowest terms as p / q, and suppose that the continued fraction of this number is given as
for some \(k\ge 0\). Note that, in accordance with a comment on p. 230 of [7], there is no loss of generality in assuming that the index of the final coefficient is even. For the convergents we denote numerators and denominators by \(p_n\) and \(q_n\), respectively, and use the correspondence between matrix products and continued fractions, which says that
yielding the determinantal identity
Now for a given sequence \((z_n)\) of positive integers, we define a new sequence \((x_n)\) by
where
It is clear from (2.4) and (2.5) that \((x_n)\) is an increasing sequence of positive integers such that \(x_n|x_{n+1}\) for all \(n\ge 1\); \((y_n)\) also consists of positive integers, and is an increasing sequence as well. The recurrence (1.7) for \(n\ge 2\) follows immediately from (2.4) and (2.5).
Theorem 2.1
The partial sums of (1.6) are given by
for all \(n\ge 1\), where the coefficients appearing after \(a_{2k}\) are
Proof
For \(n=1\), \(S_1\) is just (2.1), and we note that \(q_{2k-1}=y_0-1\) and \(q_{2k}=q=x_1\). Proceeding by induction, we suppose that \(q_{2k+2n-3}=y_{n-1}-1\) and \(q_{2k+2n-2}=x_n\), and calculate the product
By making use of (2.4) and (2.5), this gives \(p_{2k+2n}=(x_ny_{n-1}z_n+1)p_{2k+2n-2}+x_n p_{2k+2n-3}\),
and
which are the required denominators for the \((2k+2n-1)\)th and \((2k+2n)\)th convergents. Thus we have
From (2.3) and (2.4), the bracketed expression above can be rewritten as
giving
which is the required result.\(\square \)
Upon taking the limit \(n\rightarrow \infty \) we obtain the infinite continued fraction expansion for the sum S, which is clearly irrational. To show that S is transcendental, we need the following growth estimate for \(x_n\):
Lemma 2.2
The terms of a sequence defined by (2.4) satisfy
for all \(n\ge 3\).
Proof
Since \((x_n)\) is an increasing sequence, the recurrence relation (1.7) gives
for \(n\ge 2\). Hence \(x_{n-1}<x_n^{1/2}\) for \(n\ge 3\), and putting this back into the first inequality above yields \(x_{n+1}>x_n^3/x_n^{1/2}=x_n^{5/2}\), as required.\(\square \)
The preceding growth estimate for \(x_n\) means that S can be well approximated by rational numbers.
Theorem 2.3
The sum
is a transcendental number.
Proof
This is the same as the proof of Theorem 4 in [5], which we briefly outline here. Let \(P_n=p_{2k+2n-2}\) and \(Q_n=q_{2k+2n-2}\). Approximating the irrational number S by the partial sum \(S_n=P_n/Q_n\), then using Lemma 2.2 and a comparison with a geometric sum, gives the upper bound
for any \(\epsilon >0\), whenever n is sufficiently large. Roth’s theorem [6] (see also chapter VI in [1]) says that, for an arbitrary fixed \({\kappa }>2\), an irrational algebraic number \({\alpha }\) has only finitely many rational approximations P / Q for which \( \left| {\alpha }-\frac{P}{Q}\right| <\frac{1}{Q^{{\kappa }}}; \) so S is transcendental.\(\square \)
For other examples of transcendental numbers whose continued fraction expansion is explicitly known, see [2] and references therein.
3 Examples
The autonomous recurrences (1.5) considered in [5], where the polynomial F has positive integer coefficients and \(F(0)=1\), give an infinite family of examples. In that case, one has \(p=1\) and \(x_1=q=1\), so that \(k=0\), \(y_0=1\) and \(z_n=(F(x_n)-1)/x_n\). More generally, one could take \(z_n=G(x_n)\) for any non-vanishing arithmetical function G.
In general, it is sufficient to take the initial term in (1.6) lying in the range \(0<p/q\le 1\), since going outside this range only alters the value of \(a_0\). As a particular example, we take
so that \(k=1\), and \(q_{1}=3\) which gives \(y_0=2\). Hence \(x_1=7\), \(x_2=112\), and the sequence \((x_n)\) continues with
The sum S is the transcendental number
with continued fraction expansion
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Acknowledgments
This work is supported by Fellowship EP/M004333/1 from the Engineering and Physical Sciences Research Council. The original inspiration came from Paul Hanna’s observations concerning the nonlinear recurrence sequences described in [5], which were communicated via the Seqfan mailing list. The author is grateful to Jeffrey Shallit for helpful correspondence on related matters.
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Communicated by A. Constantin.
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Hone, A.N.W. Continued fractions for some transcendental numbers. Monatsh Math 182, 33–38 (2017). https://doi.org/10.1007/s00605-015-0844-2
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DOI: https://doi.org/10.1007/s00605-015-0844-2