Productly linearly independent sequences
Jaroslav Hančl1 , Katarı́na Korčeková and Lukáš Novotný2
Department of Mathematics, University of Ostrava, 30. dubna 22, 701 03 Ostrava 1
e-mail: [email protected], [email protected], [email protected]
Abstract
We introduce the two new concepts, productly linearly independent sequences and
productly irrational sequences. Then we prove a criterion for which certain infinite
sequences of rational numbers are productly linearly independent. As a consequence
we obtain a criterion for the irrationality of infinite products and a criterion for a
sequence to be productly irrational.
Key words: Linear independence, infinite product.
1
Introduction
Using some techniques from Erdős [2] we prove
Theorem 1 Let K be a non-negative integer and let {an }∞
n=1 be a non-decreasing sequence
of positive integers such that
1
1
(2K+3)n
a
=
lim
sup
a
= ∞.
n
n
n→∞ 2n
n→∞
Q
Q∞
Q∞
an +1
an +1
Then the numbers 1, ∞
n=1 (1 + a2n +1 ),
n=1 (1 + na2n +1 ), · · · , and
n=1 (1 +
early independent over the rational numbers.
lim
an +1
)
nK a2n +1
are lin-
We also prove a criterion for infinite products to be irrational.
Theorem 2 Let {an }∞
n=1 be a non-decreasing sequence of positive integers such that
1
log an
= lim sup an3n = ∞.
n→∞ log n
n→∞
lim
Then the number
Q∞
n=1 (1
+
an +1
)
a2n +1
is irrational.
1
2n
We do not know if there exists a sequence {an }∞
n=1 of positive integers with lim inf an > 1
n→∞
Q
Q∞
1
1
and such that the numbers ∞
n=1 (1 + an ) and
n=1 (1 + an +1 ) are Q-linearly dependent.
1
But
that Q
if limn→∞ an2n = ∞ and an ∈ Z+ for all n ∈ Z+ then the numbers
Q∞ we know
1
1
and ∞
) are Q-linearly independent. This is a consequence of
n=1 (1 + an )Q
n=1 (1 +
an +1
Q
Q
∞
∞
1
1
the fact that n=1 (1 + an )( n=1 (1 + an1+1 ))−1 = ∞
n=1 (1 + an (an +2) ) and the theorem of
1
Hančl and Kolouch
[6] which shows that if limn→∞ an2n = ∞ and an ∈ Z+ for all n ∈ Z+
Q∞
then the number n=1 (1 + a1n ) is irrational.
Q
P∞
1
1
3
A simple calculation shows that ∞
(1
+
)
=
2
and
n=1
n=1 n(n+2) = 4 . So the
n(n+2)
Q∞
P
∞
1
1
numbers n=1 (1 + n(n+2)
) and n=1 n(n+2)
are rational, hence Q-linearly dependent. On
1
2
Supported by the grants no. ME09017 and MSM 6198898701
Supported by the grant 01798/2011/RRC of the Moravian-Silesian region.
Q
P∞ 1
1
the other side we do not know if the numbers ∞
n=1 (1 + n3 ) and
n=1 n3 are Q-linearly
independent.
Q
Let K be a positive integer greater then 1. It is not hard to prove that ∞
n=1 (1 +
Q∞
1
1
K2
) = K 2 −1 , but we do not know if the number n=1 (1 + (K 2n +1)an ) is irrational for all
K 2n
P∞
1
sequences {an }∞
n=1 of positive integers. So it is natural to ask if the number
n=1 (K 2n +1)an
is irrational for all sequences {an }∞
n=1 of positive integers. The authors do not know the
answer to this, and the case K = 2 is an open problem of Erdős [3].
P
[nα ]
Hančl and Tijdeman [8] proved that the numbers 1, e, and ∞
n=1 n! are Q-linearly
independentα for all α ∈ R+ , α ∈
/ Z. On the other side we do not know if the numbers
Q
∞
[n ]
) are Q-linearly independent for all α ∈ R+ . Moreover, we do not know if the
n=1 (1 +
n!
Q∞
number n=1 (1 + n!1 ) is irrational.
In 1975 Erdős [2] defined a so-called irrational sequence in the following way. We
say that the sequence {an }∞
n=1 of positive
P∞real 1numbers is irrational if for every sequence
∞
of
positive
integers
the
series
{cn }∞
n=1
n=1 an cn is an irrational number. If {an }n=1 is
not an irrational sequence, then we say it is a rational sequence. Erdős also proved a
theorem which gives a criterion for a sequence to be irrational in the same paper. Hančl
[5] extended Erdős’ definition to linear independence. For more information see [5] and [7].
Other criteria for linear independence can be found in [1] and [4].
Our main theorem is Theorem 3 which deals with Q-linear independence. As a consequence of Theorem 3 we obtain a criterion for irrationality in Theorem 4.
2
Main results
In the spirit of Erdős we define the so-called productly linearly independent sequences in
the following way.
real
If for
Definition 1 Let {ai,n }∞
n=1 for i = 1, . . . , K be sequences
Qnumbers.
Q∞of positive
∞
1
∞
every sequence {cn }n=1 of positive integers the numbers n=1 1+ a1,n cn , n=1 1+ a2,n1 cn ,
Q
1
∞
..., ∞
n=1 1 + aK,n cn , and 1 are Q-linearly independent, then the sequences {ai,n }n=1
i = 1, . . . , K are said to be productly linearly independent.
We have a criterion for productly linearly independent sequences.
Theorem 3 Let K be a positive integer and ε, a, and b be real numbers such that 0 < ε,
1
∞
0 ≤ a, 0 ≤ b, and 1 − a − b > 1+ε
. Suppose that {ai,n }∞
n=1 and {bi,n }n=1 , i = 1, . . . , K, are
∞
sequences of positive integers with {a1,n }n=1 non-decreasing and such that
1
Vn
= ∞,
lim sup a1,n
n→∞
where V =
K+(K−1)a
1−b−a
+ 1 , and
ai,n bj,n
= 0,
n→∞ bi,n aj,n
lim
i, j = 1, . . . , K,
i > j.
Assume that for every sufficiently large n
a1,n ≥ n1+ε ,
b+
1
log1+ε log a1,n
bi,n ≤ a1,n
,
i = 1, . . . , K,
and
1−(a+
a1,n
1
)
log1+ε log a1,n
1+a+
≤ ai,n ≤ a1,n
1
log1+ε log a1,n
,
i = 1, . . . , K.
a
i,n ∞
Then the sequences { bi,n
}n=1 i = 1, . . . , K, are productly linearly independent.
Example 1 The sequences
∞
6·9n
n
+5
n9n + 3 n=1
and
n
n3·9 + 7
n8n + 5
∞
n=1
are productly linearly independent.
In a similar way we can define the so-called productly irrational sequences.
of positive real numbers. If for every sequence
Definition 2 Let {an }∞
n=1 be a sequenceP
1
∞
{cn }n=1 of positive integers the number ∞
n=1 (1 + an cn ) is irrational, then the sequence
∞
{an }n=1 is said to be productly irrational.
For productly irrational sequences we have the following criterion.
1
Theorem 4 Let ε and b be real numbers such that 0 ≤ b < 1 − 1+ε
and 0 < ε. Suppose
∞
∞
that {an }n=1 and {bn }n=1 are sequences of positive integers with {an }∞
n=1 non-decreasing
1−b n
b+ 1+ε1
( )
and such that lim sup an 2−b = ∞. Assume that an ≥ n1+ε and bn ≤ an log log an hold
n→∞
for every sufficiently large n. Then the sequence { abnn }∞
n=1 is productly irrational and the
Q∞
bn
number n=1 (1 + an ) is irrational.
Example 2 The sequences
2·3n
∞
n
+ n2
n3n + n n=1
and
n
33·4 + n!
32·4n + n3
∞
n=1
are productly irrational.
3
Proofs
Proof: (of Theorem 1) Theorem 1 is an immediate consequence of Theorem 3 when we set
K := K +1, aj,n := nj−1 a2n +1 and bj,n := nj−1 a2n +1 for all n ∈ Z+ and j ∈ {1, · · · , K +1},
a := 0, and b := 21 .
Proof: (of Theorem 2) Theorem 2 is an immediate consequence of Theorem 4 when we
set an := a2n + 1 and bn := an + 1 for all n ∈ Z+ , and b := 12 .
Proof: (of Theorem 4) Theorem 4 is an immediate consequence of Theorem 3 when we
set K := 1, a1,n := an , and b1,n := bn for all n ∈ Z+ , a := 0, and b := b.
Proof: (of Example 1) Example 1 is an immediate consequence of Theorem 3 when we
n
n
n
n
set K := 2, a1,n := n6·9 + 5, b1,n := n9 + 3, a2,n := n3·9 + 7, and b2,n := n8 + 5 for all
n ∈ Z+ , a := 12 , and b := 16 .
Proof: (of Example 2) Example 2 is an immediate consequence of Theorem 4 when we
n
n
set an := n2·3 + n2 and bn := n3 + n for all n ∈ Z+ , and b := 12 for the first sequence, and
n
n
an := 33·4 + n! and bn := 32·4 + n3 for all n ∈ Z+ , and b := 32 for the second sequence.
Proof: (of Theorem 3) We can see that other Theorems are consequences of Theorem 3,
but the proof is difficult and long. So we introduced
here only idea of this proof. To prove
Q∞
Q∞ bK,n
b1,n
that the numbers n=1 1 + a1,n , . . . , n=1 1 + aK,n , and the number 1 are linearly
independent over the rational numbers we will prove that for every K-tuple of integers
β1 , β2 , . . . , βK (not all equal to zero) the sum
β=
K
X
i=1
βi
∞ Y
1+
n=1
is an irrational number. We will suppose that β =
(1) we obtain that for every N ∈ Z+ the number
bi,n ai,n
p
q
(1)
is a rational number. From this and
N
−1
−1 Y
K
K
NY
Y
X
bi,n 1+
ai,n p − q
βi
β(N ) = ai,n
n=1
n=1 i=1
i=1
is an integer. We will show that 0 < β(N ) < 1 for all sufficiently large N and it will be a
contradiction with rationality of β.
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