On some power series with algebraic coefficients and Liouville numbers
© Karadeniz Gözeri; licensee Springer 2013
Received: 14 December 2012
Accepted: 2 April 2013
Published: 16 April 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.
In , Koksma introduced an analogous classification of complex numbers. He divided the complex numbers into four classes , , and in the following way.
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 .
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 
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
- 1.ξ has an approximation with rational numbers () so that the following inequality holds for sufficiently large n(4)
- 2.There exist two positive real constants and with and(5)
for sufficiently large n.
Then belongs to either the algebraic number field K or .
for sufficiently large n.
where and are real constants.
for sufficiently large n.
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.
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.
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