Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:29 https://doi.org/10.1007/s12044-018-0407-2
Transcendence of some power series for Liouville number arguments
FATMA ÇALISKAN ¸
Department of Mathematics, Faculty of Science, Istanbul University, Vezneciler, Fatih, 34134 Istanbul, Turkey E-mail: [email protected]
MS received 10 June 2016; revised 25 December 2016; accepted 21 May 2018; published online 12 June 2018
Abstract. In this paper, we prove that some power series with rational coefficients take either values of rational numbers or transcendental numbers for the arguments from the set of Liouville numbers under certain conditions in the field of complex numbers. We then apply this result to an algebraic number field. In addition, we establish the p-adic analogues of these results and show that these results have analogues in the field of p-adic numbers.
Keywords. Mahler and Koksma classification; Liouville number; algebraic number field; p-adic number.
2010 Mathematics Subject Classification. 11J17, 11J61, 11J81.
1. Introduction In 1932, Mahler introduced a classification of complex numbers and divided them into four disjointed classes called the A-numbers, S-numbers, T -numbers and U-numbers [12]. Two years later, Mahler also came up with an analogous classification of p-adic numbers which is the p-adic completion of rational numbers with respect to the p-adic valuation for a given prime number p, and the numbers in this field were again divided into four disjointed classes called the A-numbers, S-numbers, T -numbers and U-numbers [13]. In 1939, Koksma proposed another classification based on approximation by algebraic numbers which was similar to the one developed by Mahler in 1932 [9]. This classification was also appropriate for use with p-adic numbers. Several mathematicians have studied the transcendence of values of infinite products and series. For example, in 1971, Baron and Braune [2] investigated the transcendence of Fourier series with rational coefficients at algebraic points. In 1973, this result was extended to the trigonometric series by Bundschuh [4]. Later, in 2002, Zhu [24] generalized this result using a condition dependent on the coefficients. On the other hand, in 1986, Oryan [14] considered the values of certain power series with rational coefficients for the arguments from the set of Liouville numbers and showed that they can be either rational
© Indian Academy of Sciences 1 29 Page 2 of 16 Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:29 numbers or Liouville numbers and then generalized this result taking coefficients from an algebraic number field. Other examples for the transcendence of values of infinite products and series can also be given [1,3,6,22]. Many questions in the approximation theory which have been studied in the field of complex numbers can also be transposed to the field of p-adic numbers [5,15,20,21]. In the present paper, we deal with some power series under certain conditions for obtain- ing new transcendental numbers. We show that the power series with rational coefficients for the arguments from the set of Liouville numbers take values of either rational numbers or transcendental numbers under certain conditions in the field of complex numbers. Later, we give a generalized form of this result for some power series with algebraic coefficients and then prove that some power series with algebraic coefficients from a certain algebraic number field K take values of either algebraic numbers in K or transcendental numbers for the arguments from the set of Liouville numbers under certain conditions in the field of complex numbers. Later, in section 4, we establish p-adic analogues of the results from section 3. In particular, the results in section 3 contains the theorems of [16] and [8]asa special case. Besides, the results in section 3 and section 4 are partially generalizations of the theorems of [14] and [15].
2. Preliminaries In 1932, Mahler [12] developed a classification for complex numbers as follows: Let n P(x) = an x +···+a0 represent a polynomial with integer coefficients. The number H(P) = max{|an|,...,|a0|} is called the height of P.Letξ be a complex number. We can then define the quantity wn(H,ξ)= min{|P(ξ)|:P(x) ∈ Z[x], H(P) ≤ H, deg(P) ≤ n, P(ξ) = 0}, where n and H are natural numbers. It is obvious that 0 It is clear that 0 ≤ wn(ξ) ≤+∞and 0 ≤ w(ξ) ≤+∞for n ≥ 1. If wn(ξ) =+∞for some integers n, then μ(ξ) is defined as the smallest such integer. In addition, if wn(ξ) < +∞ for every n, then μ(ξ) =+∞.Bothμ(ξ) and w(ξ) cannot be simultaneously finite. Therefore, the following four possibilities exist for the complex number ξ: (1) an A-number if w(ξ) = 0 and μ(ξ) =+∞, (2) an S-number if 0 1 ∗ log w∗( ,ξ) w (ξ) w∗(ξ) = H n H w∗(ξ) = n . n lim sup and lim sup H→+∞ log H n→+∞ n ≤ w∗(ξ) ≤+∞ ≤ w∗(ξ) ≤+∞ ≥ The inequalities 0 n and 0 hold true for n 1. If w∗(ξ) =+∞ μ∗(ξ) n for some integers n, then is defined as the smallest such integer, w∗(ξ) < +∞ μ∗(ξ) =+∞ μ∗(ξ) w∗(ξ) and if n for every n, then .Both and cannot be simultaneously finite. Therefore, four possibilities exist for a complex number ξ: (1) an A∗-number if w∗(ξ) = 0 and μ∗(ξ) =+∞, (2) an S∗-number if 0 The inequalities 0 ≤ wn(ξ) ≤+∞and 0 ≤ w(ξ) ≤+∞hold true for n ≥ 1. If wn(ξ) =+∞for some integers n, then μ(ξ) is defined as the smallest such integer, and if wn(ξ) < +∞ for every n, then μ(ξ) =+∞.Bothμ(ξ) and w(ξ) cannot be simultaneously finite. Therefore, there are four possibilities for the p-adic number ξ: (1) an A-number if w(ξ) = 0 and μ(ξ) =+∞, (2) an S-number if 0 α is defined as the degree of P(z).Forthep-adic number ξ and natural numbers n,we w∗( ,ξ) = {|ξ − α| : (α) ≤ , (α) ≤ ,ξ= α}, define the quantity n H min p H H deg n and then we can establish that 1 ∗ log w∗( ,ξ) w (ξ) w∗(ξ) = H n H w∗(ξ) = n . n lim sup and lim sup H→+∞ log H n→+∞ n ≤ w∗(ξ) ≤+∞ ≤ w∗(ξ) ≤+∞ ≥ The inequalities 0 n and 0 hold true for n 1. If w∗(ξ) =+∞ μ∗(ξ) n for some integers n, then is defined as the smallest such integer, w∗(ξ) < +∞ μ∗(ξ) =+∞ μ∗(ξ) w∗(ξ) and if n for every n, then .Both and cannot be simultaneously finite. Therefore, only four possibilities for the p-adic number ξ are possible: (1) an A∗-number if w∗(ξ) = 0 and μ∗(ξ) =+∞, (2) an S∗-number if 0 3. Some power series in the field of complex numbers We consider the power series ∞ n f (x) = cn x (1) n=0 with non-zero rational coefficients c = bn (a and b are integers; a > 1 for sufficiently n an n n n large n) satisfying the following two conditions: log a + σ := lim inf n 1 > 1(2) →+∞ n log an Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:29 Page 5 of 16 29 and log |b | θ := lim sup n < 1. (3) n→+∞ log an From (2), we know that there is a sufficiently small ε1(> 0) such that σ1 = σ − ε1 > 1. Since an > 1 for sufficiently large n, then from (2), we get log an+1 >σ1 log an (n ≥ N1) (4) for a sufficiently large N1 = N1(ε1). Hence, we have − >σn N1 log an 1 log aN1 (5) for n ≥ N1. Therefore, we can infer that log an log an lim an =+∞, lim =+∞ and lim =+∞. (6) n→+∞ n→+∞ n n→+∞ n2 From (3), there exists a sufficiently small ε2(> 0) such that θ+ε | | < 2 bn an (7) for sufficiently large n, where θ + ε2 < 1. Therefore, from (6) and (7) it follows that the radius of convergence of the power series f (x) is infinity. From (4), we obtain (n−m) 1 σ 1 am < an (n > m ≥ N1), (8) and since 0 < 1 < 1, we have σ1 σ 1 σ − 1 1 an ≤ An ≤ C0 an (n > N1), (9) where C0 is a suitable positive number and An =[a0, a1,...,an].From(9), we then get log An σ−ε 1 ≤ lim sup →+∞ ≤ 1 . If we choose a sufficiently small ε (> 0), then we n log an σ−ε1−1 1 have σ 1 ≤ u ≤ , (10) σ − 1 where u := lim sup log An . Furthermore, since 1 − θ − ε > 0, from (5) it follows that log an 2 1−θ−ε2 ν a + ν 1 0 < n 1 |α| < (ν = 1, 2,...) (11) an+ν+1 2 for sufficiently large n. 29 Page 6 of 16 Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:29 In the next two theorems, we consider the Liouville number α for which the following properties hold: There exist rational numbers pn/qn (qn > 1) and a sequence w(n) such that equations pn 1 α − < lim w(n) =+∞ (12) nw(n) n→+∞ qn qn and δ δ 1 ≤ n ≤ 2 an qn an (13) hold for sufficiently large n, where δ1 and δ2 are real numbers such that e <δ1 ≤ δ2 for a real number e. Theorem 1. If e = u and σ(1 − θ) > 4δ2 for the power series (1), then f (α) is either a rational number or a transcendental number. ( ) = n ν ( = , ,...). Proof. We can consider the polynomials fn x ν=0 cν x n 1 2 From (7) and (11), we then get ∞ 1 ν | f (α) − fn(α)|≤ |α| 1−θ−ε2 ν= + aν n 1 |α|n+1 |α|n+1 ≤ + 1 + 1 +··· = 2 −θ−ε 1 −θ−ε (14) 1 2 2 22 1 2 an+1 an+1 for sufficiently large n. In addition, from (6), there is a sufficiently small ε3(> 0) such that ε |α|n+1 < 3 4 an+1 for sufficiently large n. Therefore, from (4) and (14), we have 1 | f (α) − fn(α)|≤ (15) (σ−ε1)(1−θ−ε2−ε3) 2an for sufficiently large n. Since u = lim sup log An and u <δ ≤ δ ,using(15), we obtain log an 1 2 | (α) − (α)|≤ 1 f fn (σ−ε )( −θ−ε −ε ) (16) 1 1 2 3 ( n) 2δ 2 Anqn 2 for sufficiently large n.From(12), we get pn < |α|+1 for sufficiently large n. Hence, qn from this information, (6), (7) and the equation u = lim sup log An , it follows that log an w(n) p 1 qn 2 f (α) − f n ≤ q−nw(n) n n n n u+ε4 qn 2 an w(n) n 2 ≤ 1 −nw(n) qn = 1 qn w(n) (17) 2 An n 2 2 Anqn Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:29 Page 7 of 16 29 for sufficiently large n, where ε4(> 0) is chosen such that 0 < u + ε4 <δ1. Hence it follows from limn→+∞ w(n) =+∞,(16) and (17), (α) − pn ≤ 1 f fn (σ−ε )( −θ−ε −ε ) (18) 1 1 2 3 qn n 2δ Anqn 2 for sufficiently large n. If we choose sufficiently small ε1,ε2 and ε3, then a suitable positive (σ−ε )( −θ−ε −ε ) σ( −θ) number ε exists such that 1 1 2 3 > 1 − ε. Since σ(1 − θ) > 4δ ,itis 2δ2 2δ2 2 then possible to choose a positive number ε such that σ(1 − θ) − ε ≥ 2 + ε. (19) 2δ2 pn hn We can take fn = n , where hn, An ∈ Z. Therefore, from (18) and (19), we get qn Anqn (α) − hn < 1 f +ε (20) A qn n 2 n n Anqn for sufficiently large n, where ε is a suitable positive number dependent on ε1,ε2 and ε3. hn If the sequence n is constant, then f (α) is a rational number. Otherwise, f (α) is a Anqn transcendental number according to Roth’s theorem [17]. Let K be an algebraic number field of degree m. We now consider the power series ∞ η f (x) = n xn (21) an n=0 with non-zero algebraic coefficients, where ηn is an algebraic integer in K and an is a rational integer (an > 1 for sufficiently large n) satisfying the following two conditions: log a + σ := lim inf n 1 > 1(22) →+∞ n log an and log H(η ) θ := lim sup n < 1, (23) n→+∞ log an where H(ηn) represents the height of the algebraic number ηn. From (22), the equations (4), (5), (6), (8), (9), (11) for the sequence {an} hold. In addition, there is a sufficiently small ε2(> 0) such that θ +ε2 < 1. Furthermore, from (23) we know that θ+ε (η )< 2 H n an (24) 29 Page 8 of 16 Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:29 for sufficiently large n. Therefore, from Lemma 2, we have for sufficiently large n, θ+ε |η | < 2 . n 2an (25) Hence, the radius of convergence of the power series f (x) is infinity. Two lemmas are needed to prove the next theorem. Lemma 2[19]. Let ξ be an algebraic number of height h. Then |ξ|≤h + 1. Lemma 3[7]. Let α1,...,αk (k ≥ 1) be algebraic numbers in an algebraic number field Kofdegreeg, and let F(y, x1,...,xk) be a polynomial with integral coefficients so that its degree in y is at least 1.Ifξ is an algebraic number such that F(ξ, α1,...,αk) = 0, then the degree of ξ is ≤ dg, and 2dg+(l1+···+lk )g g l1g lk g H(ξ) ≤ 3 H H(α1) ...H(αk) , where H(ξ) is the height of ξ, H(αi ) is the height of αi (i = 1,...,k), H is the maximum of the absolute values of the coefficients of the polynomial F, li is the degree of F in xi (i = 1,...,k) and d is the degree of F in y. Theorem 4. If e = 1 and 2(σ(1 + θ) + (σ − 1)δ2)m <σ(1 − θ)(σ − 1) for the power series (21), then f (α) is either an algebraic number in K or a transcendental number. ( ) = n ην ν ( = , ,...) Proof. We need to consider the polynomials fn x ν=0 aν x n 1 2 .Let γ := f ( pn ), where γ (n = 1, 2,...)is an algebraic number in K ; therefore deg(γ ) ≤ n n qn n n m. Now we must obtain an upper bound for the height of γn, and to do this, we can use Lemma 3 with the polynomial n ν ( , , ,..., ) = n − n 1 pn , F y x0 x1 xn Anqn y Anqn xν aν q ν=0 n where F(γn,η0,η1,...,ηn) = 0. Here, n 1 p ν = n, n n ≤ n n, H max Anqn Anqn Anqn d0 (26) ν=0 aν qn where | pn |≤d for a sufficiently large integer d . Now from Lemma 3,we qn 0 0 2m+(n+1)m m m m know that H(γn) ≤ 3 H H(η0) ...H(ηn) . Using (24) and (26), we can ( + ) (θ+ε ) (γ ) ≤ n 3 m( n n)m m( ... ) 2 m, obtain H n 3 Anqn d0 C1 a0a1 an for sufficiently large n,in which C is a sufficiently large positive integer. From (6) and (9), we get H(γ ) ≤ 1 σ n 1 ( σ − +ε3)(1+θ+ε2)m nm nm 1 1 C2 qn an for sufficiently large n, in which C2 is a sufficiently large positive integer and ε3 is a sufficiently small positive constant. Finally, using this informa- tion along with (6) and (13), we have σ−ε 1 σ−ε − +ε3 (1+θ+ε2)+δ2+ε4 m 1 1 H(γn) ≤ an (27) Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:29 Page 9 of 16 29 for sufficiently large n, in which ε4 is a sufficiently small positive constant. Next, from (12), we know that (α) − pn ≤ 2 −nw(n)(|α|+ )n−1 {|η |} fn fn n qn 1 max ν qn 1≤ν≤n for sufficiently large n. Thus, using (6), (13), (25) and (27)wehave 2(|α|+ )n−1 1 pn 2n 1 2 fn(α) − fn ≤ ≤ δ1w(n)−θ−ε2 δ1w(n)−(1+θ+ε2) qn an an 1 < 2 δ w( )−( +θ+ε ) (28) 1 n 1 2 σ−ε 1 +ε (θ+ε + )+δ +ε σ −ε −1 3 2 1 2 4 m H(γn) 1 for sufficiently large n.From(4), (6), (11) and (25), + 2|α|n 1 1 1 | f (α) − f (α)|≤ −θ−ε 1 + + + ... n 1 2 2 22 an+1 n+1 1 4|α| 2 ≤ −θ−ε < (σ−ε )( −θ−ε −ε ) 1 2 1 1 2 5 an+1 an for sufficiently large n. Finally we can use this and (27) to get 1 | (α) − (α)|≤ 2 f fn (σ−ε )( −θ−ε −ε ) (29) 1 1 2 5 σ−ε 1 +ε ( +θ+ε )+δ +ε σ−ε −1 3 1 2 2 4 m H(γn) 1 for sufficiently large n. Hence, from (28), (29) and limn→+∞ w(n) =+∞, it follows that (α) − pn ≤ 1 f fn (σ−ε )( −θ−ε −ε ) (30) 1 1 2 5 qn σ−ε 1 +ε ( +θ+ε )+δ +ε σ−ε −1 3 1 2 2 4 m H(γn) 1 for sufficiently large n. If we choose sufficiently small ε1,ε2,ε3,ε4 and ε5, then there is a suitable positive number ε such that (σ − ε )( − θ − ε − ε ) σ( − θ) 1 1 2 5 > 1 − ε. σ−ε σ 1 + ε (1 + θ + ε ) + δ + ε m (1 + θ)+ δ m σ−ε1−1 3 2 2 4 σ−1 2 σ( −θ)(σ− ) Since m < 1 1 , it is possible to choose a positive number ε so that 2(σ(1+θ)+(σ−1)δ2) σ(1 − θ)(σ − 1) − ε ≥ 2 + ε. (31) (σ(1 + θ)+ (σ − 1)δ2)m Since f ( pn ) = γ ∈ K in which deg(γ ) ≤ m (n = 1, 2,...),from(30) and (31), we n qn n n know that | (α) − γ | < 1 f n 2+ε (32) H(γn) 29 Page 10 of 16 Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:29 for sufficiently large n, where ε is a suitable positive number dependent on ε1,ε2,ε3,ε4 and ε5. If the sequence {γn} is constant, then f (α) is an algebraic number in K . Otherwise, f (α) is a transcendental number according to Roth’s theorem [17]. 4. Some power series in the field of p-adic numbers In the p-adic field Qp, we consider the power series ∞ n f (x) = cn x (33) n=0 with non-zero rational coefficients c = bn (a and b are integers; b = 0,(a , b ) = n an n n n n n 1; an > 1 for sufficiently large n) that satisfies the following conditions: u + σ := lim inf n 1 > 1(34) →+∞ n un and logp An Bn λ := lim sup < +∞, (35) n→+∞ un −u where |cn|p ≤ p n ,λ≥ 0, Bn = max0≤i≤n{|bi |}, An =[a0, a1,...,an] and un > 0 for sufficiently large n. From (34), there exists a sufficiently small ε1 > 0 such that σ1 = σ − ε1 > 1. Since an > 1 for sufficiently large n, from (34), we get un+1 >σ1un (n ≥ N1) (36) for sufficiently large natural number N1 = N1(ε1). Then − >σn N1 un 1 u N1 (37) for n ≥ N1. Therefore, we have un lim un =+∞ and lim =+∞. (38) n→+∞ n→+∞ n Hence, the radius of convergence of the power series f (x) is infinity. In this section, we consider the p-adic Liouville number α for which the following holds: There exist rational numbers pn/qn (qn > 1) and a sequence w(n) such that equations pn 1 α − < lim w(n) =+∞ (39) nw(n) n→+∞ qn p Hn Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:29 Page 11 of 16 29 and δ δ un 1 ≤ n ≤ un 2 p Hn p (40) hold for sufficiently large n, where δ1 and δ2 are real numbers such that 0 <δ1 ≤ δ2 and Hn = max{|pn|, |qn|}. Theorem 5. If λ + δ2 > 1 and σ>4(λ + δ2) for the power series (33), then f (α) is either a rational number or a p-adic transcendental number. ( ) = n ν ( = , ,...). Proof. We consider the polynomials fn x ν=0 cν x n 1 2 There is an h integer number h such that |α|p = p .From(38), we can deduce that there is a sufficiently small positive number ε2 such that nh 1 − > 1 − ε2 (41) un −u for sufficiently large n. Since |cn|p ≤ p n , we know from (41) that ν −(1−ε2)uν |cνα |p ≤ p (ν = n + 1, n + 2,...) (42) for sufficiently large n. In addition, from (36) and (42), we know that −(1−ε2)(σ−ε1)un | f (α) − fn(α)|p ≤ p (43) for sufficiently large n. We can determine f ( pn ) := Pn and we have η = n qn Qn n (| |, | |) ≤ ( + ) n. max Pn Qn n 1 An Bn Hn Using (35), we can then deduce that there is a u (λ+ε ) sufficiently small positive number ε3 such that An Bn ≤ p n 3 for sufficiently large n. From this information and (40), it follows that un(λ+ε3+δ2) ηn ≤ (n + 1)p (44) pn for sufficiently large n. Furthermore, from (39), we know that ≤|α|p + 1for qn p −uν sufficiently large n. Since {un} is a monotone increasing sequence, it follows that p < C (ν = 0, 1,...), where C is a suitable positive number. Therefore, we have pn pn − − δ w( ) (α) − ≤ α − (|α| + )n 1 ≤ n un 1 n fn fn p 1 C C1 p qn p qn p for sufficiently large n, where C1 represents a suitable positive constant. Therefore, we know from (38) that there is a sufficiently small positive number ε4 such that p n −un(δ1w(n)−ε4) fn(α) − fn ≤ p (45) qn p 29 Page 12 of 16 Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:29 for sufficiently large n. Therefore, from (38), (43), (45) and limn→+∞ w(n) =+∞,we know that pn −(1−ε )(σ−ε )u −u (δ w(n)−ε ) f (α) − fn ≤ max{p 2 1 n , p n 1 4 } qn p −(1−ε )(σ−ε ) −( −ε )(σ−ε )u u 2 1 = p 1 2 1 n ≤ ((n + 1)p n ) 2 for sufficiently large n. In addition, from (44) and λ + δ2 > 1, we get (α) − pn ≤ 1 f fn ( −ε )(σ−ε ) (46) 1 2 1 qn (λ+ε +δ ) p 2 3 2 ηn for sufficiently large n.Ifε1,ε2 and ε3 are sufficiently small, then we can obtain a suitable ( −ε )(σ−ε ) σ positive number ε such that 1 2 1 > − ε. Since σ>4(λ + δ ),itispossible 2(λ+ε3+δ2) 2(λ+δ2) 2 to choose a positive number ε so that σ − ε ≥ 2 + ε. (47) 2(λ + δ2) Therefore, it follows from (46) and (47) that for sufficiently large n, (α) − Pn < 1 , f 2+ε (48) Qn p ηn where ε is a suitable positive number that is dependent on ε1,ε2 and ε3. If the sequence Pn is constant, then f (α) is a rational number. Otherwise, f (α) is a p-adic transcen- Qn dental number according to Lang’s theorem [10]. Let K be a p-adic algebraic number field of degree m.Inthep-adic field Qp, we consider the power series ∞ η f (x) = n xn (49) a n=0 n with the p-adic algebraic coefficients, in which ηn is a p-adic algebraic integer in K and an is a rational integer (an > 1 for sufficiently large n) that satisfies the following conditions: u + σ := lim inf n 1 > 1(50) →+∞ n un and (η ) logp An H n λ := lim sup < +∞, (51) n→+∞ un −u where |cn|p ≤ p n ,λ≥ 0, An =[a0, a1,...,an] and un > 0 for sufficiently large n. In addition, H(ηn) is the height of the algebraic numbers ηn. Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:29 Page 13 of 16 29 From (50), for the sequence {un},(36), (37) and (38) hold true. Hence, the radius of convergence of the power series f (x) is infinity. We will employ Theorem 7 after the following lemma. Lemma 6[7]. Let α1,...,αk (k ≥ 1) be p-adic algebraic numbers in a p-adic algebraic number field K of degree g, and let F(y, x1,...,xk) be a polynomial with integral coef- ficients so that its degree in y is at least 1.Ifξ is a p-adic algebraic number such that F(ξ, α1,...,αk) = 0, then the degree of ξ is ≤ dg, and 2dg+(l1+···+lk )g g l1g lk g H(ξ) ≤ 3 H H(α1) ...H(αk) , where H(ξ) is the height of ξ, H(αi ) is the height of αi (i = 1,...,k), H is the maximum of the absolute values of the coefficients of F, li is the degree of F in xi (i = 1,...,k) and d is the degree of F in y. Theorem 7. If 4m(σλ + (σ − 1)δ2)<σ(σ− 1) for the power series (49), then f (α) is either a p-adic algebraic number in K or a p-adic transcendental number. ( ) = n ην ν,( = , ,...) γ := Proof. Consider the polynomials fn x ν=0 aν x n 1 2 .Let n f ( pn )(n = 1, 2,...). Since γ (n = 1, 2,...) is a p-adic algebraic number in n qn n K, deg(γn) ≤ m. We can then determine an upper boundary for the height of γn using Lemma 6 with the polynomial n ν ( , , ,..., ) = n − n 1 pn , F y x0 x1 xn Anqn y Anqn xν aν q ν=0 n where F(γn,η0,η1,...,ηn) = 0. By applying Lemma 6, we get 2m+(n+1)m m m m H(γn) ≤ 3 H H(η0) ...H(ηn) , ≤ n where H An Hn . Therefore, we can obtain (γ ) ≤ 4nm( n)m (η )m ... (η )m. H n 3 An Hn H 0 H n (52) Next, from (51), there is a sufficiently small positive number ε2 such that An H(ηn)< (λ+ε ) un 2 ≤ 1 ( ≥ ν ≥ ), p , for sufficiently large n. Then from (36), we get uν σ n−ν un n N1 σ 1 which leads to u +···+u ≤ 1 u (n ≥ N), where N is a suitable natural number. N3 n σ1−1 n Hence, from this information (38), (40) and (52), we obtain 4nm nm (u N +···+un)(λ+ε2)m H(γn) ≤ C03 H p 3 n σ (σ−ε )(λ+ε ) 1 1 2 (λ+ε )mu σ−ε − +δ2+ε3 mun ≤ n nm σ −1 2 n ≤ 1 1 , C1 Hn p 1 p (53) m for sufficiently large n, in which C0 = (H(η0)...H(ηN−1)) and C1 is a sufficiently large − un ≥− + |α| positive number. From (38), we can deduce that 2 un n logp p for sufficiently −u ηn n n large n, and from (54), we can obtain | α | ≤ p 2 for sufficiently large n. Since {u } an p n is a monotone increasing sequence, we then have 29 Page 14 of 16 Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:29 − un+1 | f (α) − fn(α)|p ≤ p 2 (54) for sufficiently large n. Therefore, from (36), (53) and (54), we can obtain 1 | f (α) − f (α)| ≤ σ−ε (55) n p 1 (σ−ε )(λ+ε ) 1 2 +δ +ε 2m σ−ε −1 2 3 H(γn) 1 −uν for sufficiently large n.From(38), it follows that p < C2 (ν = 0, 1,...), where C2 is a suitable positive number. Therefore, from (39) and (40), we have ν pn ην ν pn fn(α) − fn ≤ max | |p α − qn p 1≤ν≤n aν qn p pn n−1 n −u δ w(n) ≤ C2 α − (|α|p + 1) ≤ C p n 1 qn p 3 for sufficiently large n, in which C3 is a suitable positive constant. In addition, from (38), we know that there exists a sufficiently small positive number ε4 such that p n −un(δ1w(n)−ε4) fn(α) − fn ≤ p (56) qn p for sufficiently large n. Furthermore, from (53) and (56), we can then obtain (α) − pn ≤ 1 fn fn δ w( )−ε (57) 1 n 4 qn p (σ−ε )(λ+ε ) 1 2 +δ +ε m σ−ε −1 2 3 H(γn) 1 for sufficiently large n. Thus, since limn→+∞ w(n) =+∞and δ1 > 0, from (55) and (57), we have pn 1 f (α) − f ≤ σ−ε (58) n 1 qn (σ−ε )(λ+ε ) p 1 2 +δ +ε 2m σ−ε −1 2 3 H(γn) 1 for sufficiently large n. If we choose sufficiently small ε1,ε2 and ε3, then there is a suitable positive number ε such that σ − ε σ 1 > − ε. (σ−ε )(λ+ε ) σλ 2m 1 2 + δ + ε 2m + δ σ−ε1−1 2 3 σ−1 2 Since 4m(σλ + (σ − 1)δ2)<σ(σ− 1), it is possible to choose a positive number ε such that σ(σ − 1) − ε ≥ 2 + ε. (59) 2m(σλ + (σ − 1)δ2) Proc. Indian Acad. Sci. (Math. Sci.) (2018) 128:29 Page 15 of 16 29 Therefore, it follows from (58) and (59) that p 1 f (α) − f n < (60) n (γ )2+ε qn p H n for sufficiently large n, in which ε is a suitable positive number that is dependent on ε1,ε2 and ε3. If the sequence {γn} is constant, then f (α) is a p-adic algebraic number in K . Otherwise, f (α) is a p-adic transcendental number according to Lang’s theorem [10]. Acknowledgements The author would like to thank Prof. Dr Bedriye M Zeren for her valuable comments and suggestions. References [1] Amou M and Väänänen K, Arithmetical properties of certain infinite products, J. Number Theory 153 (2015) 283–303 [2] Baron G and Braune E, Zur Transzendenz von Fourierreihen mit Rationalen Koeffizienten und Algebraischem Argument, Arch. 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