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On the transcendence of some power series
Advances in Difference Equations volume 2013, Article number: 17 (2013)
In this study, we consider some power series with rational coefficients and investigate transcendence of the values of these series for Liouville number arguments. It is proved that these values are either a Liouville number or a rational number under certain conditions.
AMS Subject Classification:11J81, 11J17.
A real number α is algebraic if it is a zero of a polynomial with integer coefficients. The real numbers which are not algebraic are known as transcendental. The theory of transcendental numbers has a long history and was originated back to Liouville in his famous paper  where he explicitly constructed a number and proved that it is transcendental. Later, Cantor  gave another proof of the existence of transcendental numbers by establishing the denumerability of the set of algebraic numbers. It follows from this that almost all real numbers are transcendental. Further, the development of the theory of transcendental numbers has proved to have a strong influence on some new studies in Diophantine equations; see [3, 4].
A classification of the set of all transcendental numbers into three disjoint classes, termed S, T and U, which was introduced by Mahler , proved to be of considerable value in the general development of the subject. The first classification of this kind was outlined by Maillet in , and others were described by Perna in  and Morduchai-Boltovskoj , but Mahler’s classification receives most of the interest. Mahler described this classification in the following way.
Let be a polynomial with integer coefficients. The height of the polynomial P is defined by
and if the degree of P is denoted by , then , and for a given arbitrary complex number ξ, it can be written as
where n and H are positive integers; see . Next, Mahler puts
The inequalities and hold. From , we get . If index , then the is defined as the smallest of them; otherwise . Thus, is uniquely determined. Furthermore, the two quantities and are never finite simultaneously, for the finiteness of implies that there is an such that , hence .
Therefore, there are the following four possibilities for ξ: it is said to be
In , Wirsing proved that both classifications are equivalent. Namely, A, S, T and U numbers are the same as , , and numbers. The class A is precisely the set of algebraic numbers. The ξ is called a U-number of degree m if . The set of U-numbers of degree m is denoted by . It is obvious that for any , the is a subclass of U, and U is the union of all disjoint sets .
Leveque proved that is not empty for any in . Later, in [13, 14], Oryan considered a class of power series with algebraic coefficients and proved that under certain conditions these series take values in the subclass for algebraic arguments and in the set of Liouville numbers for Liouville number arguments, respectively.
Now, consider the infinite convergent sum , where , and has only simple rational zeros. Then, Saradha and Tijdeman have obtained the sufficient and necessary conditions for the transcendence of T if the degree of is 3; see . Similarly, Ping and Yuan gave sufficient and necessary conditions for the transcendence of T if the degree of is 4 and is reduced; see .
We also note that a transcendental function is an analytic function having a single value or many values, and to calculate the values, we need a limiting process. However, an analytic function is transcendental if and only if its Riemann surface is non-compact; see .
In the present work, we consider certain power series with rational coefficients and show that these series take values of either the set of Liouville numbers or rational numbers under certain conditions. Thus, we give a new result for obtaining -numbers.
In this paper, means the absolute value of x and the least common multiple of is denoted by .
Definition 2.1 A real number ξ is called Liouville number if and only if for every positive integer n, there exist integers , () with
Now, in order to prove our main theorem, we need the following lemma which was proved in .
Lemma 2.2 Let ξ be a real number. If there exist a sequence of real numbers tending to infinity and a sequence of rational numbers satisfying
then ξ is a Liouville number.
3 Main theorem
Theorem 3.1 Let
be a power series with non-zero rational coefficients (, ) which satisfies the following conditions:
Further, let ξ be a Liouville number and satisfy the following two properties:
The ξ has an approximation with rational numbers () so that the following inequality holds for sufficiently large n:(3.4)
There exist two positive real numbers and with and(3.5)
for sufficiently large n.
Then is either a Liouville number or a rational number.
Proof It follows from (3.1) that
for sufficiently large n, where and is chosen as . Then it follows from (3.6) that the sequence is strictly increasing, thus we have
Now, by using equation (3.6), we get
Let . Then on using (3.9), we obtain
where is chosen as .
Now, if we consider the following polynomials:
then we have
and from (3.4) it follows that
Further, on using equations (3.4), (3.11) and (3.12), we get
Then we obtain
for sufficiently large n.
On the other hand, we can easily deduce from (3.3) and that
and since the sequence is strictly increasing, then it follows that
On using , and from (3.7), (3.8) and (3.16), we get
That is, on using (3.5), (3.10) and (3.14),
for sufficiently large n. Then on using (3.13), we have
for sufficiently large n.
Moreover, the following inequality holds:
It follows from (3.15) that
for sufficiently large n. We get from here and (3.18)
Thus, we can deduce from (3.9) that
Since (3.7) holds, we then obtain
Similarly, since , we have
On the other hand, from (3.8) we get
for sufficiently large n. From here we obtain
for sufficiently large n. Now, if further we define
then we have
Using (3.2) then it follows that there exists a subsequence of such that
for sufficiently large . On using (3.5), (3.7) and (3.10), we deduce that there exists a suitable sequence with . Then from (3.20) for we obtain
for sufficiently large . On the other hand, using (3.17) we get
for sufficiently large .
It follows from (3.21) and (3.22) that
where . Moreover,
are rational numbers with integers. It follows from (3.23) that
Thus, if the sequence is constant, then is a rational number. Otherwise, using Lemma 2.2 we get from (3.23) that is a Liouville number. □
Corollary 3.2 If () and in Theorem 3.1, then is a Liouville number.
Proof Since , it is possible to choose a subsequence of so that the terms are positive and strictly increasing or decreasing. Let us assume that is strictly increasing. From it follows
From (3.24) we deduce that the sequence is not constant. Thus, is a Liouville number. In the case of strictly decreasing , the proof follows similarly. □
In this paper, the series with rational coefficients are treated and it is shown that under certain conditions these series take values belonging to either the set of Liouville numbers or the rational number field for Liouville number arguments.
The similar results can be proved for the power series which are defined in the p-adic field and in the field of formal Laurent series.
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The authors express their sincere thanks to the referees for the careful and noteworthy reading of the manuscript and very helpful suggestions that improved the manuscript substantially. The first author acknowledges that this work was partially supported by Scientific Research Projects Coordination Unit of Istanbul University under project number 3414.
The authors declare that they do not have any competing interest.
All the authors contributed equally.
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Karadeniz Gözeri, G., Pekin, A. & Kılıçman, A. On the transcendence of some power series. Adv Differ Equ 2013, 17 (2013). https://doi.org/10.1186/1687-1847-2013-17
- power series
- Liouville numbers
- Mahler’s classification