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On some power series with algebraic coefficients and Liouville numbers
Journal of Inequalities and Applications volume 2013, Article number: 178 (2013)
In this work, we consider some power series with algebraic coefficients from a certain algebraic number field K of degree m and investigate transcendence of the values of the given series for some Liouville number arguments.
The theory of transcendental numbers has a long history and was originated back to Liouville in his famous paper  in which he produced the first explicit examples of transcendental numbers at a time where their existence was not yet known. 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 theory of transcendental numbers is closely related to the study of Diophantine approximation. Recent advances in Diophantine approximation can be found in the excellent surveys of Moshchevitin  and Waldschimidt .
Mahler  introduced a classification of the set of all transcendental numbers into three disjoint classes, termed S, T and U and this classification has 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 to Mahler’s classification attaches by for the most interest. Mahler described this classification in the following way.
Let be a polynomial with integral coefficients. The height of P is defined by and the degree of P is denoted by . Given an arbitrary complex number ξ, Mahler puts
where n and H are positive integers. Next Mahler puts
The inequalities and hold. From , we get . If for an index , 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 , whence . Therefore, there are the following four possibilities for ξ, ξ is called
In , Koksma introduced an analogous classification of complex numbers. He divided the complex numbers into four classes , , and in the following way.
Let α be an arbitrary algebraic number. If we denote its minimal defining polynomial by , then the height of α is defined by and the degree of α is defined by . Given an arbitrary complex number ξ and positive integers n, H, let α be an algebraic number with degree at most n and height at most H such that takes the smallest positive value; Koksma defines by the following equation:
Next, Koksma puts
The inequalities and hold. If for an index , the is defined as the smallest of them, otherwise . Thus, is uniquely determined. Furthermore, the two quantities and are never finite simultaneously. Therefore, there are the following four possibilities for ξ, ξ is called:
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. 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 , 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. Later in , similar relations are investigated for Liouville number arguments, and it is proved that these series take values in the set of Mahler’s U-numbers. In , Saradha and Tijdeman considered certain convergent sums and showed that they are either rational or transcendental. Later in , Yuan and Li obtained further results for some convergent sums. In , Nyblom employed a variation on the proof used to established Liouville’s theorem concerning the rational approximation of algebraic numbers, to deduce explicit growth conditions for a certain series to converge to a transcendental number. Later, Nyblom  derived a sufficiency condition for a series of positive rational terms to converge to a transcendental number. Further, Duverney  proved a theorem that gives a criterion for the sums of infinite series to be transcendental. The terms of these series consist of the rational numbers and converge regularly and very quickly to zero. In , Hančl introduced the concept of transcendental sequences and proved a criterion for sequences to be transcendental. Later, a new concept of a Liouville sequence was introduced in  by means of the related Liouville series. Some recent results for the transcendence of infinite series can also be found in Borwein and Coons , Hančl and Rucki , Hančl and Štěpnička , Murty and Weatherby , Weatherby .
In the present work, we considered certain power series with algebraic coefficients from a certain algebraic number field K of degree m and showed that under certain conditions these series take values belonging to either the algebraic number field K or in Mahler’s classification of the complex numbers for some Liouville number arguments.
In this paper, means the absolute value of x and the least common multiple of is denoted by .
Definition 1 A real number ξ is called a Liouville number if and only if for every positive integer n there exists integers , () with
The set of all Liouville numbers is identical with the subclass. More information about Liouville numbers may be found in [26–28]. Now, in order to prove our main theorem we need the following lemmas.
Lemma 2 
Let () be algebraic numbers which belong to an algebraic number field K of degree m, and let be a polynomial with rational integral coefficients and with degree at least 1 in y. If η is any algebraic number such that , then and
where H is the height of the polynomial F, d is the degree of F in y and is the degree of F in for .
Lemma 3 
Let α be an algebraic number of degree m, and let be its conjugates. Then , where .
Lemma 4 
Let α be an algebraic number of degree m, then , where .
3 Main result
Theorem 5 Let K be an algebraic number field of degree m, and let
be a power series such that are non-zero algebraic numbers and are rational integers satisfying the following conditions:
Further, let ξ be a Liouville number satisfying the following two properties:
ξ has an approximation with rational numbers () so that the following inequality holds for sufficiently large n(4)
There exist two positive real constants and with and(5)
for sufficiently large n.
Then belongs to either the algebraic number field K or .
Proof It follows from (1) that
for sufficiently large n, where and is to be chosen as . It follows from (6) that the sequence is strictly increasing, thus we have
Furthermore, from (6) we get
for sufficiently large n.
Let . Then by using (7) and (9), we obtain for sufficiently large n
where is to be chosen as . We can easily deduce from (3) and that
for sufficiently large n. Since (11) holds, there is a natural number such that
for every . On the other hand, we get from Lemma 3 that
since are algebraic numbers. From here and (12), we obtain
for every . Now, we shall define the algebraic numbers
for . Since , for . Let us determine an upper bound for the heights of the algebraic numbers . By multiplying both sides of this equality by and putting for , we obtain the equality
Since ξ is a Liouville number, we can assume that for . Then we get a polynomial
with rational integral coefficients such that . Further, this polynomial is of degree 1 in each . Thus, we deduce from Lemma 2 that
where H is the height of the polynomial . By using (4), we obtain for , where is a real constant. From here, we can easily get
It follows from (15) and (16) that
where is a real constant. Moreover, we get
from Lemma 4. Then we deduce from (17) and (18) that
By using (10), (13) and (14), we obtain
where is a real constant. It follows from here, (5) and (10) that
where and are real constants.
Now, we consider the following polynomials:
for . Since are continuous and differentiable for all real numbers, at least one real number exists between ξ and such that for every n
It is obvious that . Since , we obtain for sufficiently large n. Furthermore, from here and (4) and (21), we get
for sufficiently large n.
Define . Then we obtain
for sufficiently large n. It follows from (14) that for sufficiently large n. We get from here and (22), (23)
By using (5), we obtain from here that
From (8) and , it is possible to find a sequence with such that
Therefore, we get from (20), (24) and (25)
for sufficiently large n. Moreover, the following inequality holds:
We get from here and (14)
Thus, we deduce from (9) that
Since (7) holds, then we obtain . Similarly, since , we have
for and, therefore,
On the other hand from (8), we get for sufficiently large n. From here, we obtain
for sufficiently large n. Now if we define , then we have
Using (2), then it follows that there exists a subsequence of such that . Therefore, we get
for sufficiently large . From (8) and , there exists a suitable sequence with such that
and, therefore, from (20) and (27), we obtain
for sufficiently large . On the other hand by using (26), we deduce that
for sufficiently large . Let . It follows from (28) and (29) that
where . It follows from here that . Thus, if the sequence is constant, then is an algebraic number in K. Otherwise . □
In this work, the series with algebraic coefficients are treated and it is shown that under certain conditions the values of these series are either algebraic numbers or U-numbers 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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Dedicated to Professor Hari M Srivastava.
The author would like to express her sincere thanks to the referees for their careful reading and for making some valuable comments, which have essentially improved the presentation of this paper.
The author declares that she has no competing interests.
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Cite this article
Karadeniz Gözeri, G. On some power series with algebraic coefficients and Liouville numbers. J Inequal Appl 2013, 178 (2013). https://doi.org/10.1186/1029-242X-2013-178
- Power Series
- Real Constant
- Algebraic Number
- Infinite Series
- Diophantine Approximation