DOI: 10.1515/auom-2015-0029
An. Şt. Univ. Ovidius Constanţa
Vol. 23(2),2015, 107–120
A Criterion for linear independence of infinite
products
Jaroslav Hančl,∗ Ondřej Kolouch† and Lukáš Novotný†
Abstract
Using an idea of Erdős the paper establishes a criterion for the linear
independence of infinite products which consist of rational numbers. A
criterion for irrationality is obtained as a consequence.
1
Introduction
Following Erdős [1] we prove
Theorem 1.1. Let K be a non-negative integer and let {an }∞
n=1 be a nondecreasing sequence of positive integers such that
1
1
n
n
1 < lim inf an(K+2) < lim sup an(K+2) < ∞.
n→∞
n→∞
Q∞
Q∞
Q∞
Then the numbers 1, n=1 (1 + an1+1 ), n=1 (1 + nan1+1 ), · · · , and n=1 (1 +
1
) are linearly independent over the rational numbers.
nK an +1
As a consequence of this theorem we obtain a criterion for infinite products
to be irrational.
Key Words: Linear independence, infinite product, irrationality.
2010 Mathematics Subject Classification: 11J72.
Received: October, 2013.
Revised: December, 2013.
Accepted: December, 2013.
∗ This work was supported by the European Regional Development Fund in the
IT4Innovations Centre of Excellence project (CZ.1.05/1.1.00/02.0070) and by grants no.
P201/12/2351 and MSM 6198898701.
† The authors were supported by grants 01798/2011/RRC and 02508/2013/RRC of the
Moravian-Silesian region and by SGS08/PřF/2014.
107
A CRITERION FOR LINEAR INDEPENDENCE OF INFINITE PRODUCTS
108
Theorem 1.2. Let {an }∞
n=1 be a non-decreasing sequence of positive integers
such that
1
1
1 < lim inf an2n < lim sup an2n < ∞.
n→∞
Q∞
1
an )
n→∞
is irrational.
Q∞
The authors do not know if the number n=1 (1 + 22n a1n +1 ) is irrational
for all sequences {an }∞
n=1 of positive integers
P∞ although we know from another
theorem of Erdős [1] that the number n=1 22n1an is irrational for every sequence {an }∞
n=1 of positive integers. Hančl and Kolouch [4] proved that if
1
Q∞
2n
limn→∞ an = ∞ and an ∈ Z+ then the number n=1 (1 + a1n ) is irrational,
but we do not know if it is transcendental.
Q∞
It is not difficult to prove that n=1 (1 + 221n ) = 43 , but we do not know
Q∞
1
if the number n=1 (1 + (22n +1)a
) is irrational for all sequences {an }∞
n=1 of
n
P∞
1
positive integers. Erdős [2] asked if the number n=1 (22n +1)an is irrational
for all sequences {an }∞
integers.
n=1 of positiveQ
∞
1
1
A simple calculation shows that n=2 (1 − Q
n2 ) = 2 . On the other side the
∞
authors are not able to decide if the number Qn=1 (1 + n12 ) is irrational. In
∞
fact we are not able to prove that the number n=1 (1 + n1k ) is irrational for
any k ∈ Z+ , k 6= 1. P
This is analogous
Q∞ to the problem of the irrationality of
∞
the function ζ(k) = n=1 n1k = n=1 (1 + pk1−1 ) for k ∈ Z+ , k 6= 1 where
n
{pn }∞
n=1 is the increasing sequence of all primes. We know that for even k the
number ζ(k) is transcendental
Q∞and that ζ(3) is an irrational number. But we
do not know if the number n=1 (1 + p1k ) is irrational for any k ∈ Z+ , k 6= 1.
n
Q∞
Q∞
π(n)
1
) and n=1 (1 + n n! ) are
We also do not know if the numbers n=1 (1 + n!
Q-linearly independent where π(n) denotes the number
less than or
Q∞ of primes
1
equal to n. Moreover, we do not know if the number n=1 (1+ n!
) is irrational.
There exists a nice book by Nishioka [6] which contains a review of results
concerning the linear and algebraic independence of infinite products and series which use the strong tools of Mahler’s method. Several general results
concerning the linear independence of infinite series can be found in [3]. For
other results in this theory see [5], for instance.
Our main theorem is Theorem 2.1 concerning Q-linear independence. Its
proof falls into two parts, the second part separated into two main cases, is
based on the location of specific gap and makes use of some suitable tricks.
As a consequence of Theorem 2.1 we obtain a criterion for irrationality in
Theorem 2.2. Šustek [7] used similar method of Erdős to prove the irrationality
measures.
We denote by Z, Z+ , N, and Q the set of all integers, positive integers,
nonnegative integers, and rational numbers, respectively. The functions [x],
Then the number
n=1 (1
+
A CRITERION FOR LINEAR INDEPENDENCE OF INFINITE PRODUCTS
109
and log2 x are the greatest integer less than or equal to x, and the logarithm to the base 2 of the number x, respectively. Notation loga2 log2 x means
(log2 (log2 x))a .
2
Main results
Our first theorem is a basic result which deals with the Q-linear independence
of infinite products of rational numbers.
Theorem 2.1. Let K be a positive integer and let ε be a positive real number.
∞
Assume that {ai,n }∞
n=1 and {bi,n }n=1 (i = 1, . . . , K) are sequences of positive
∞
integers such that {a1,n }n=1 is non-decreasing,
1
1
n
n
(K+1)
(K+1)
< lim sup a1,n
<∞
lim inf a1,n
n→∞
and
lim
n→∞
(1)
n→∞
ai,n bj,n
= 0,
bi,n aj,n
j, i ∈ {1, . . . , K},
for all
i > j.
(2)
Suppose that for every sufficiently large number n
log
1+ε
bi,n < a1,n2
1
log2 a1,n
,
i = 1, . . . , K,
(3)
a1,n ≥ n1+ε ,
(4)
and
−
1
1+ε
log2
log2 a1,n
ai,n a1,n
log
1+ε
< a1,n < ai,n a1,n2
1
log2 a1,n
,
i = 2, . . . , K.
(5)
Q∞
Q∞
b
b
,. . . , and n=1 1+ aK,n
, and the number 1
Then the products n=1 1+ a1,n
1,n
K,n
are Q-linearly independent.
Corollary 2.1. Let K be a positive integer. Assume that {ai,n }∞
n=1 and
∞
{bi,n }∞
n=1 (i = 1, . . . , K) are sequences of positive integers such that {a1,n }n=1
∞
is non-decreasing, {bi,n }n=1 (i = 1, . . . , K) is bounded,
lim inf
n→∞
and
1
1
log2 a1,n ≥ 1
n
1
n
n
(K+1)
(K+1)
lim inf a1,n
< lim sup a1,n
< ∞.
n→∞
n→∞
A CRITERION FOR LINEAR INDEPENDENCE OF INFINITE PRODUCTS
110
Suppose that for every i ∈ {2, . . . , K}
lim
n→∞
ai,n
=0
ai−1,n
and
−1 < lim inf
n→∞
a a log22 n
log22 n
1,n
1,n
≤ lim sup
< 1.
log2
log2
n
ai,n
n
ai,n
n→∞
Q∞
Q∞
b
Then the products n=1 (1 + a1,n
),. . . , and n=1 (1 +
1,n
1 are Q-linearly independent.
bK,n
aK,n ),
and the number
Our second theorem is a consequence of the previous theorem and deals
with the irrationality of infinite products over the rational numbers
Theorem 2.2. Let ε be a positive real number. Suppose that {an }∞
n=1 and
∞
{bn }∞
n=1 are two sequences of positive integers with {an }n=1 non-decreasing
1
1
and such that lim inf n→∞ an2n < lim supn→∞ an2n < ∞. Assume that an ≥
log
−(1+ε)
log a
2 n
hold for every large n. Then the product
n1+ε and bn ≤ an 2
Q
∞
bn
n=1 (1 + an ) is an irrational number.
∞
Corollary 2.2. Assume that {an }∞
n=1 and {bn }n=1 are two sequences of pos∞
itive integers such that {an }n=1 is non-decreasing, lim inf n→∞ n1 log2 an > 1,
1
1
{bn }∞
is bounded and lim inf n→∞ an2n < lim supn→∞ an2n < ∞. Then the
n=1 Q
∞
product n=1 (1 + abnn ) is irrational.
Q∞
Remark 2.1. From the simple calculation we obtain that n=1 (1 + 221n ) = 43 .
Hence the strict inequality between lim inf and lim sup in all Theorems 1.1, 1.2,
2.1, 2.2 and Corollaries 2.1, 2.2 cannot be omited. It also shows implicitely
that straightforward application of Liouville principle does not work.
3
Proofs
Theorem 1.1 is an immediate consequence of Theorem 2.1 when we set K :=
K + 1, aj,n := nK+1−j an + 1, bj,n := 1 for all j ∈ {1, 2, 3, . . . , K + 1} and
n ∈ Z+ . The number ε can be arbitrary because the sequence {a1,n }∞
n=1
converges to infinity as a composition of two exponential functions.
Theorem 1.2 is an immediate consequence of Theorem 1.1 when we set K := 0
and an := an − 1 for all n ∈ Z+ .
Theorem 2.2 is an immediate consequence of Theorem 2.1 when we set K := 1.
Corollary 2.1 is an immediate consequence of Theorem 2.1 when we set ε := 21 .
A CRITERION FOR LINEAR INDEPENDENCE OF INFINITE PRODUCTS
111
Conditions (1)-(4) are fulfilled immediately. We only verify condition (5).
From lim inf n→∞ n1 log2 a1,n ≥ 1 we obtain that for all sufficiently large n we
n
2
have a1,n > 2 . This and the fact that the function x
for a large x imply that
1
1+ 1
log2 2 log2 a1,n
a1,n
n
2
>2
1
1+ 1
log2 2 ( n )
2
1
1+ 1
log2 2 log2 x
is increasing
2n
2
> 2 log2 n .
(6)
Now for all i ∈ {2, . . . , K} and large n we have
−1 < lim inf
n→∞
a a log22 n
log22 n
1,n
1,n
≤ lim sup
< 1.
log2
log2
n
ai,n
n
ai,n
n→∞
Therefore
−
2
2n
log2
2n
2n
2
ai,n < a1,n < ai,n 2 log2 n .
From this and (6) we obtain (5).
Corollary 2.2 is an immediate consequence of Theorem 2.2 when we suppose
in addition that {bn }∞
n=1 is bounded,
Proof. (of Theorem 2.1) Set
1
n
(K+1)
E = lim inf a1,n
n→∞
1
n
(K+1)
and F = lim sup a1,n
.
n→∞
Assume that there is a K-tuple of integers A1 , A2 , . . . , AK (not all equal to
zero), p ∈ Z and q ∈ Z+ such that
K
∞ R
∞ Y
Y
p X
bj,n X
bj,n =
Aj
1+
=
,
Aj
1+
q
aj,n
aj,n
n=1
n=1
j=1
j=1
where R is the largest index such that AR 6= 0. Let N ∈ Z+ . Then we have
R
N
−1
R
∞ R
N
−1
Y
Y
Y
bj,n X
bj,n X
bj,n p X
−
Aj
1+
Aj
Aj
=
1+
−
1+
.
q j=1
aj,n
aj,n
aj,n
n=1
n=1
n=1
j=1
j=1
So the number
α(N ) = q
q
Y
R N
−1
Y
R
N
−1
Y
bj,n p X
Aj
1+
aj,n −
=
q j=1
aj,n
n=1
j=1 n=1
Y
X
NY
∞ −1
R
bj,n bj,n aj,n 1+
− 1 Aj
1+
aj,n
aj,n
n=1
j=1 n=1
j=1
Y
R N
−1
Y
n=N
(7)
A CRITERION FOR LINEAR INDEPENDENCE OF INFINITE PRODUCTS
112
is an integer. To prove our theorem it is enough to prove that there exists
N0 ∈ Z+ \ {1} such that 0 < α(N0 ) < 1. Let us assume that N is sufficiently
large. Set
∞ Y
bj,n .
D=
max |Aj |
1+
aj,n
j∈{1,2,...,K}
n=1
Q∞ b
From this, the facts that N is sufficiently large and n=1 1+ aj,n
< ∞ for all
j,n
Q∞ b
<
j = 1, . . . , K we obtain that D is positive real number and n=N 1 + aj,n
j,n
P∞ bj,n
2 n=N aj,n holds for every j = 1, . . . , K.
1. Now we provethat α(N ) > 0 for all large N . From (7) and the fact
Q∞
P∞ b
b
that n=N 1 + aj,n
< 2 n=N aj,n
holds for every j = 1, . . . , K we obtain
j,n
j,n
that
NY
Y
R
−1
∞ R N
−1
Y
−1 X
Y
bj,n bj,n Aj
aj,n
=
1+
1+
− 1 ≥
α(N ) q
aj,n
aj,n
n=1
j=1
j=1 n=1
n=N
NY
Y
−1
∞ bR,n bR,n |AR |
1+
1+
−1 −
aR,n
aR,n
n=1
n=N
NY
Y
−1
∞ bj,n bj,n 1+
|Aj |
1+
−1 >
aj,n
aj,n
n=1
j=1
R−1
X
n=N
Y
∞ n=N
1+
R−1
∞ X Y
bR,n bj,n −1 −D
−1 >
1+
aR,n
aj,n
j=1
n=N
∞
∞
R−1
X X
1 X bR,n
bj,n
− 2D
=
2
aR,n
aj,n
j=1
n=N
n=N
R−1
∞
X bj,n aR,n X
bR,n 1
− 2D
.
aR,n 2
a b
j=1 j,n R,n
n=N
This and (2) imply that α(N ) > 0 for all large N .
2. Now we prove that α(N ) < 1 for infinitely many large N . To prove this
QR QN −1
we will estimate the product j=1 n=1 aj,n . From the definition of F we
obtain that
n
a1,n < (2F )(K+1)
(8)
A CRITERION FOR LINEAR INDEPENDENCE OF INFINITE PRODUCTS
113
for all sufficiently large n. Inequality (5) implies that
R N
−1
Y
Y
aj,n ≤ D1
j=1 n=1
R N
−1
Y
Y
log
1+ε
a1,n a1,n2
1
log2 a1,n
j=1 n=1
where D1 is a suitable constant which does not depend on N . Recall that
the notation loga2 log2 x means (log2 (log2 x))a . This, (8) and the fact that the
−(1+ε)
function x(log2 log2 x)
is increasing for all sufficiently large x yield
R N
−1
Y
Y
aj,n ≤ D1
j=1 n=1
D2
R N
−1
Y
Y
NY
−1
K
a1,n
2D3
1+ε
1
log2 a1,n
≤
j=1 n=1
NY
−1
a1,n
K NY
−1
n=1
D2
log
a1,n a1,n2
2
D3
n1+ε
(K+1)n
=
n=1
PN −1
1
n=1 n1+ε
(K+1)n ≤ 2N
−(1+ ε )
2 (K+1)N
NY
−1
K
a1,n
, (9)
n=1
n=1
where D2 , and D3 are positive real constants which do not depend on N .
From (3) and (5) we obtain that
R
NY
Y
−1
∞ X
bj,n bj,n Aj
1+
1+
− 1 ≤
aj,n
aj,n
n=1
j=1
n=N
D
∞ R Y
X
j=1
1+
n=N
2
∞
R X
∞
−1
X
X
1+ε
bj,n bj,n
log
log2 a1,n
−1 ≤ 2D
≤ 2DK
a1,n2
≤
aj,n
aj,n
j=1
n=N
∞
X
n=N
1
log
1+ ε
2 log a
2 1,n
a1,n2
−1
.
n=N
From this, (7) and (9) we obtain for all sufficiently large N
NY
X
Y
−1
∞ R
bj,n bj,n α(N ) = q
aj,n Aj
1+
1+
− 1 ≤
aj,n
aj,n
n=1
j=1 n=1
j=1
Y
R N
−1
Y
n=N
ε
q2
N −(1+ 2 ) (K+1)N
NY
−1
n=1
1
n
a1,n
K X
∞
−(1+ ε )
2
log
a1,n2
n=N
(K+1)
Let Tn = a1,n
. Now the proof falls into two cases.
log2 a1,n −1
.
(10)
114
A CRITERION FOR LINEAR INDEPENDENCE OF INFINITE PRODUCTS
2a. First assume that for every sufficiently large n
a1,n ≥ 2n .
(11)
−(1+ ε )
2
log2 x−1
Then (11) and the fact that the function xlog2
sufficiently large x imply
∞
X
−(1+ ε )
2
log
a1,n2
log2 a1,n −1
is decreasing for
=
n=N
log
X
−(1+ ε )
2
a1,n2
log2 a1,n −1
N ≤n≤log2 a1,N
log
−(1+ ε )
2
a1,N2
log
X
+
−(1+ ε )
2
a1,n2
log2 a1,n −1
≤
n>log2 a1,N
log2 a1,N −1
log
X
log2 a1,N +
−(1+ ε )
2
a1,n2
log2 a1,n −1
≤
n>log2 a1,N
−(1+ ε )
2
log
a1,N2
log2 a1,N −1
log2 a1,N +
−(1+ ε )
2
X
2n(log2
n−1)
≤
n>log2 a1,N
log
−(1+ ε )
2
a1,N2
log2 a1,N −1
−(1+ ε )
2
X
log2 a1,N +
2n(log2
log2 a1,N −1)
≤
n>log2 a1,N
log
−(1+ ε )
2
a1,N2
log2 a1,N −1
log
−(1+ ε )
3
log2 a1,N + a1,N2
−(1+ ε )
4
log2
a1,N
log2 a1,N −1
log2 a1,N −1
.
≤
(12)
for sufficiently large N .
For a sufficiently small positive real number δ, from (5) it follows that there
exists a positive integer t0 which is sufficiently large such that for every n ≥ t0
we have max(1, E − δ) < Tn < F + δ. This implies that for every n ≥ t0
n
n
max(1, (E − δ))(K+1) < a1,n < (F + δ)(K+1) .
(13)
Let t1 be the least positive integer greater than (K + 1)t0 +1 such that the
inequality max(1, E − δ) < Tt1 < E + δ holds. Then
max(1, (E − δ))(K+1)
t1
t1
< a1,t1 < (E + δ)(K+1) .
(14)
Let t2 be the least positive integer greater than t1 such that
F − δ < T t2 < F + δ
(15)
A CRITERION FOR LINEAR INDEPENDENCE OF INFINITE PRODUCTS
115
and let t3 be the least positive integer greater than t1 such that t1 < t3 ≤ t2
and
1
Tt3 > 1 + 1+ ε
max (Tj , F − 2δ).
(16)
t3 6 t1 ≤j<t3
Such a number t3 must exist since otherwise using (15) we obtain
1
F − δ < Tt2 ≤ 1 + 1+ ε
max (Tj , F − 2δ) ≤
t2 6 t1 ≤j<t2
1
1
1+
max (Tj , F − 2δ) < . . .
1 + 1+ ε
ε
(t2 − 1)1+ 6 t1 ≤j<t2 −1
t2 6
t2 Y
1
<
1 + 1+ ε (F − 2δ),
j 6
j=t
1
a contradiction for a sufficiently large t0 . From (8),(13), (14) and (16) we
obtain
(K+1)t3
(K+1)t3
1
(K+1)t3
≥
a1,t3 = Tt3
max (Tj , F − 2δ)
> 1 + 1+ ε
6
t
≤j<t
1
3
t3
(K+1)t3
t3 −1
1
+(K+1)t3 −2 +···+1)
1 + 1+ ε
≥
max (Tj , F − 2δ)K((K+1)
6
t
≤j<t
1
3
t3
(K+1)t3 tY
3 −1
K
t1
t1 −1
1
+···+1)
a1,j (F − 2δ)K((K+1) +(K+1)
1 + 1+ ε
≥
6
t3
j=t1 +1
(K+1)t3 tY
j K
t1 3 −1
K Y
1
1
(F − 2δ)(K+1)
≥
1 + 1+ ε
a1,j
Qt0 −1
6
a
1,j
( j=1 a1,j )K
t3
j=1
j=t0
1+
1
1+ 6ε
t3
(K+1)t3 tY
3 −1
a1,j
j=1
K F − 2δ K(K+1)t1
×
E+δ
tY
1 −1
j=t0
1+
1
1+ 6ε
t3
(K+1)t3 tY
3 −1
j=1
a1,j
j
F − 2δ (K+1)
F +δ
1 −1
K tY
(
j=t0
K
D4
Qt0 −1
j=1
(2F )K(K+1)j
≥
j K
(F − 2δ)2
)(K+1)
×
(E + δ)(F + δ)
(3F )−(K+1)
t0 +1
≥
A CRITERION FOR LINEAR INDEPENDENCE OF INFINITE PRODUCTS
1+
(K+1)t3 tY
3 −1
1
1+ 6ε
t3
a1,j
K
(3F )−t3 ,
116
(17)
j=1
where D4 is a positive real constant which does not depend on t0 . Now from
(8), (10), (12), and (17) we obtain
3 −1
tY
−(1+ ε )
2 (K+1)t3
α(t3 ) ≤ q2t3
a1,n
∞
K X
−(1+ ε )
2 (K+1)t3
q2
3 −1
tY
−(1+ ε )
2
log2 a1,n −1
≤
n=t3
n=1
t3
log
a1,n2
a1,n
K
−(1+ ε )
4
log
a1,t32
log2 a1,t3 −1
≤
n=1
3 −1
tY
−(1+ ε )
2 (K+1)t3
t3
q2
a1,n
log
n=1
−(1+ ε )
4
a1,t32
K
1
1+
1+ ε
6
log2 a1,t3
(K+1)t3 Q
t3 −1
j=1
t3
a1,j
(3F )−t3
−(1+ ε )
5 (K+1)t3
2t3
−(1+ ε )
2 (K+1)t3
t3
2
≤
K
1+
=
(K+1)t3
1
(3F )−t3
1+ ε
t3 6
−(1+ ε )
−(1+ ε )
2 (K+1)t3 +t
5 (K+1)t3 −log
2
3
t3
1+
2
1
1+ ε
t3 6
(K+1)t3 +t3 log2 (3F )
<1
for a sufficiently large number t3 .
2b. Now assume that there exist infinitely many n such that
a1,n < 2n .
(18)
−(1+ ε )
2
Then (4) and the fact that the function xlog2
sufficiently large x imply
∞
X
log
−(1+ ε )
2
a1,n2
log2 a1,n −1
log2 x−1
is decreasing for
≤
n=N
−(1+ ε )
2
log
a1,n2
X
N ≤n≤
√
5
log
log2 a1,N −1 √
5
a1,N +
−(1+ ε )
2
log
a1,n2
X
+
n>
a1,N
−(1+ ε )
2
a1,N2
log2 a1,n −1
√
5
a1,N
X
n>
√
5
log2 a1,n −1
a1,N
log
−(1+ ε )
2
a1,n2
log2 a1,n −1
≤
≤
A CRITERION FOR LINEAR INDEPENDENCE OF INFINITE PRODUCTS
log
−(1+ ε )
2
a1,N2
log2 a1,N −1 √
5
n>
log
−(1+ ε )
2
a1,N2
−(1+ ε )
2
X
a1,N +
√
5
n(1+ε)(log2
log2 n1+ε −1)
117
≤
a1,N
log2 a1,N −1 √
5
−
ε
10
a1,N + a1,N
≤
a−A
1,N ,
(19)
1
min(1, ε). Now, set B = 12 (1 + F ) =
for sufficiently large N , where A = 20
1
2 (E + F ). From this and (1) we obtain that there is a sufficiently large k such
that
k
a1,k > B (K+1) .
(20)
Let k0 be the greatest positive integer less than k such that (18) holds. Let
k1 be the least positive integer such that
1
max Tj ,
(21)
Tk1 > 1 + 1+ ε
k1 6 k0 ≤j<k1
and k0 < k1 ≤ k. As in the previous case such a k1 must exist since otherwise
1
1 < B ≤ Tk ≤ 1 + 1+ ε
max Tj <
k 6 k0 ≤j<k
1
1
1 + 1+ ε
1+
max Tj < · · · <
ε
1+
k 6
(k − 1) 6 k0 ≤j<k−1
k
Y
1
<
1 + 1+ ε Tk0 ,
j 6
j=k +1
0
a contradiction for a sufficiently large number k0 . From (21) and the fact that
the sequence {a1,n }∞
n=1 is non-decreasing we obtain
a1,k1 =
1+
(K+1)k1
Tk1
1
>
(K+1)k1
1+ ε
k1 6
1+
1
(K+1)k1
1+ ε
k1 6
( max Tj )(K+1)
k1 −1
( max Tj )K((K+1)
+(K+1)k1 −2 +···+1)
k0 ≤j<k1
1+
1
(K+1)k1 kY
1 −1
1+ 6ε
k1
1+
a1,j
K Y
k0
j=1
1
1+ ε
k1 6
k1
k0 ≤j<k1
≥
≥
−K
≥
a1,j
j=1
(K+1)k1 kY
1 −1
j=1
K
a1,j
2
2−Kk0 .
(22)
A CRITERION FOR LINEAR INDEPENDENCE OF INFINITE PRODUCTS
118
The definition of k1 implies that for every N (k0 < N < k1 )
1 TN ≤ 1 + 1+ ε
max Tj .
N 6 k0 ≤j<N
Thus
TN ≤
Y
N
1+
j=k0 +1
j
Y
∞ 1 T
<
1+
ε
k
0
1+
6
j=k0
1 ε
j 1+ 6
Tk0 = C = C(k0 ), (23)
where C is a constant which depends on k0 and C tends to 1 as k0 tends to
infinity. From (23) we obtain that for every N = k0 , . . . , k1 − 1
N
a1,N ≤ C (K+1) .
This yields
kY
1 −1
K
a1,j
=
Y
k0
j=1
a1,j
K kY
1 −1
j=1
K
a1,j
2
k1
≤ 2Kk0 C (K+1) .
(24)
j=k0
Inequalities (12) and (19), and the definition of k1 and k imply
∞
X
log
−(1+ ε )
2
a1,n2
log2 a1,n −1
=
n=k1
k−1
X
−(1+ ε )
2
log
a1,n2
log2 a1,n −1
+
∞
X
−(1+ ε )
2
log
a1,n2
log2 a1,n −1
≤
n=k
n=k1
log
−(1+ ε )
4
a1,k21
log2 a1,k1 −1
+ a−A
1,k .
(25)
Now from (7), (8), (10), (20), (22), (24), and (25) we obtain
kY
K X
∞
1 −1
−(1+ ε )
−(1+ ε )
2 log a
2 (K+1)k1
log
2 1,n −1
a1,n
a1,n2
≤
α(k1 ) ≤ q2k1
n=1
−(1+ ε )
3 (K+1)k1
2k1
kY
1 −1
a1,n
K n=k1
log
−(1+ ε )
4
a1,k21
log2 a1,k1 −1
+ a−A
=
1,k
n=1
(
Qk1 −1
n=1
−(1+ ε )
3 (K+1)k1
a1,n )K 2k1
log
−(1+ ε )
4
a1,k21
log2 a1,k1
+
a1,k1
(
Qk1 −1
n=1
−(1+ ε )
3 (K+1)k1
a1,n )K 2k1
aA
1,k
≤
A CRITERION FOR LINEAR INDEPENDENCE OF INFINITE PRODUCTS
119
ε
−(1+ )
−(1+ ε )
4 log a
Qk1 −1
3 (K+1)k1 log2
2 1,k1
a1,n )K 2k1
( n=1
a1,k1
+
(K+1)k1 Qk1 −1
2
1
1 + 1+
( j=1 a1,j )K 2−Kk0
ε
6
k1
2
k1
−(1+ ε )
3 (K+1)k1
2Kk0 C (K+1) 2k1
B A(K+1)k
−(1+ ε )
3 (K+1)k1
2k1
1+
1
1+ ε
k1 6
−(1+ ε )
5 (K+1)k1
2k1
(K+1)k1
2
2−Kk0
2
k1
≤
−(1+ ε )
3 (K+1)k1
2Kk0 C (K+1) 2k1
+
B A(K+1)k
< 1,
a contradiction for a sufficiently large k0 .
We would like to thank Professor James E. Carter of the College of
Charleston for his help with the presentation of this paper. The authors also
thank referee for the valuable suggestions.
References
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Infinite Series, J. Math. Sci. 10 (1975), 1-7.
[2] Erdős P, Erdős problem no. 6, 1995 Prague Midsummer Combinatorial
Workshop, KAM Series (95-309), M. Klazar (ed.), (1995), page 5.
[3] Hančl J., A criterion for linear independence of series, Rocky Mountain
J. Math. 34, no. 1, (2004), 173–186.
[4] Hančl J., Kolouch O, Erdős’ method for determining the irrationality of
products, Bull. Aust. Math. Soc. 84, no. 3, (2011), 414-424.
[5] Hančl J., Sobková S, Special linearly unrelated sequences, J. Math. Kyoto
Univ., vol. 46, no. 1, (2006), 31–45.
[6] Nishioka K, Mahler functions and transcendence, Lecture notes in mathematics 1631, Springer, (1996).
[7] Šustek J, New bounds for irrationality measures of some fast converging
series, Mathematics for Applications, vol. 3, no. 2, (2014), 167-175.
Jaroslav HANČL,
Department of Mathematics and Centre of Excellence IT4Innovation, division
of UO, Institute for Research and Applications of Fuzzy Modeling
University of Ostrava
30. dubna 22, 701 03 Ostrava 1, Czech Republic.
Email: hancl@osu.cz
A CRITERION FOR LINEAR INDEPENDENCE OF INFINITE PRODUCTS
Ondřej KOLOUCH
Department of Mathematics
Faculty of Science
University of Ostrava
30. Dubna 22, 701 03 Ostrava 1,Czech republic.
Email: ondrej.kolouch@osu.cz
Lukáš NOVOTNÝ
Department of Mathematics
Faculty of Science
University of Ostrava
30. Dubna 22, 701 03 Ostrava 1,Czech republic.
Email: lukas.novotny@osu.cz
120