Jacobsthal-Lucas series and their applications
In this paper we study the properties of positive series such that its terms are reciprocals of the elements of Jacobsthal-Lucas sequence. In particular, we consider the properties of the set of incomplete sums as well as their applications. We prove that the set of incomplete sums of this series is...
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irk-123456789-1566452019-06-19T01:26:44Z Jacobsthal-Lucas series and their applications Pratsiovytyi, M. Karvatsky, D. In this paper we study the properties of positive series such that its terms are reciprocals of the elements of Jacobsthal-Lucas sequence. In particular, we consider the properties of the set of incomplete sums as well as their applications. We prove that the set of incomplete sums of this series is a nowhere dense set of positive Lebesgue measure. Also we study singular random variables of Cantor type related to Jacobsthal-Lucas sequence. 2017 Article Jacobsthal-Lucas series and their applications / M. Pratsiovytyi, D. Karvatsky // Algebra and Discrete Mathematics. — 2017. — Vol. 24, № 1. — С. 169-180. — Бібліогр.: 8 назв. — англ. 1726-3255 2010 MSC:11B83, 11B39, 60G50. http://dspace.nbuv.gov.ua/handle/123456789/156645 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this paper we study the properties of positive series such that its terms are reciprocals of the elements of Jacobsthal-Lucas sequence. In particular, we consider the properties of the set of incomplete sums as well as their applications. We prove that the set of incomplete sums of this series is a nowhere dense set of positive Lebesgue measure. Also we study singular random variables of Cantor type related to Jacobsthal-Lucas sequence. |
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Pratsiovytyi, M. Karvatsky, D. |
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Pratsiovytyi, M. Karvatsky, D. Jacobsthal-Lucas series and their applications Algebra and Discrete Mathematics |
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Pratsiovytyi, M. Karvatsky, D. |
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Jacobsthal-Lucas series and their applications |
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Jacobsthal-Lucas series and their applications |
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Jacobsthal-Lucas series and their applications |
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Jacobsthal-Lucas series and their applications |
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Jacobsthal-Lucas series and their applications |
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jacobsthal-lucas series and their applications |
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Jacobsthal-Lucas series and their applications / M. Pratsiovytyi, D. Karvatsky // Algebra and Discrete Mathematics. — 2017. — Vol. 24, № 1. — С. 169-180. — Бібліогр.: 8 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT pratsiovytyim jacobsthallucasseriesandtheirapplications AT karvatskyd jacobsthallucasseriesandtheirapplications |
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2025-07-14T09:01:45Z |
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1837612353937473536 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 24 (2017). Number 1, pp. 169–180
c© Journal “Algebra and Discrete Mathematics”
Jacobsthal-Lucas series and their applications
Mykola Pratsiovytyi and Dmitriy Karvatsky
Communicated by A. P. Petravchuk
Abstract. In this paper we study the properties of posi-
tive series such that its terms are reciprocals of the elements of
Jacobsthal-Lucas sequence (Jn+2 = 2Jn+1 + Jn, J1 = 2, J2 = 1).
In particular, we consider the properties of the set of incomplete
sums as well as their applications. We prove that the set of incom-
plete sums of this series is a nowhere dense set of positive Lebesgue
measure. Also we study singular random variables of Cantor type
related to Jacobsthal-Lucas sequence.
Introduction
Today mathematicians heavily research structural, topological, metric
and fractal properties of the set of incomplete sums (subsums) of absolutely
convergent series. Despite essential progress for some series, the problem
is quite difficult in general case. In this context scientists focus on series
such that their terms are elements of some sequences with some condition
of homogeneity (depend on finite numbers of parameters and defined by
a formula for general term or some recurrence relation).
In this research article we investigate series
∞∑
n=1
1
2n−1 + (−1)n−1
.
2010 MSC: 11B83, 11B39, 60G50.
Key words and phrases: Jacobsthal-Lucas sequence, the set of incomplete sums,
singular random variable, Hausdorff-Besicovitch dimension.
170 Jacobsthal-Lucas series
It seems that this series is a “simple perturbation” of classic binary series
and changes of properties of the set of subsums are insignificant. However
this is not true.
1. Jacobsthal-Lucas sequence
Definition 1. The sequence of real numbers (un) ≡ (un)∞
n=1 having the
property
un+2 = pun+1 + sun, (1)
where u1, u2, p, s are fixed real numbers, is called a generalized Fibonacci
sequence.
Let p and s be fixed real numbers (p2 + s2 6= 0), and let Fp,s be a set
of all sequences satisfying condition (1). Consider linear operations on
this set defined by formulas
(an) ⊕ (bn) = (an + bn) and λ(an) = (λan), n ∈ N, λ ∈ R.
Then Fp,s forms a two-dimensional vector space with respect to these
operations. It is easy to introduce different mathematical structures in
this space [4].
If p = 1, s = 2, then generalized Fibonacci sequence is called Jacobsthal
sequence. Thus, Jacobsthal sequences form a two-dimensional vector space
containing geometric progression with common ratios 2 and −1. This
family does not contain infinitesimal sequences except for zero sequence.
The general term of Jacobsthal sequence is equal to
un =
(u2 + u1)2n−1 + (2u1 − u2)(−1)n−1
3
.
These sequences can be used for representing real numbers [2], mo-
deling of objects with complicated local structure (sets, functions, random
variables, etc. [3]). One particular case (for u2 = p = 1, u1 = s = 2) of
generalized Fibonacci sequence is Jacobsthal-Lucas sequence defined as
follows
(Jn) = (2, 1, 5, 7, 17, 31, 65, . . . Jn, . . .) , Jn = Jn−1 + 2Jn−2, n > 3.
(2)
Lemma 1. The general term of Jacobsthal-Lucas sequence is determined
by formula
Jn = 2n−1 + (−1)n−1. (3)
M. Pratsiovytyi, D. Karvatsky 171
Proof. Finding a general term of the sequence (2) is equivalent to solving
homogeneous difference equation of order 2
y (x + 2) − y (x + 1) − 2y (x) = 0. (4)
Characteristic equation of (4) has the form λ2 − λ − 2 = 0. Numbers 2
and −1 are solutions of characteristic equation. Functions y1(x) = 2x and
y2(x) = (−1)x are solutions of equation (4). Hence, general solution can
be written as a function y (x) = c12x−1 + c2(−1)x−1. Taking into account
initial conditions
{
c1y1 (1) + c2y2 (1) = 2,
c1y1 (2) + c2y2 (2) = 1,
we can find constants c1
and c2. It is easy to see that c1 = c2 = 1.
So, general term of sequence (2) has form (3).
Theorem 1. Jacobsthal-Lucas sequence has the following properties:
1.
k∑
n=1
Jn =
{
2k if k is odd,
2k − 1 if k is even;
2.
k+1
2∑
n=1
J2n−1 =
Jk+2 − 2
3
+
k + 1
2
; 3.
k
2∑
n=1
J2n =
2(Jk+1 − 2)
3
−
k
2
;
4.
k∑
n=1
J2
n =
J2k+1 + 2Jk+1 + 2
3
+ k if k is odd,
J2k+1 − 2Jk+1 + 2
3
+ k if k is even;
5.
k∑
n=1
(−1)n−1Jn =
−Jk+1 + 2
3
+ k if k is odd,
Jk+1 + 2
3
+ k if k is even.
2. Jacobsthal-Lucas series
Consider the series of the reciprocals of the Jacobsthal-Lucas numbers
r =
∞∑
n=1
un =
∞∑
n=1
1
Jn
=
1
2
+
1
1
+
1
5
+
1
7
+ . . . +
1
Jn
+ . . . . (5)
This is convergent positive series, and its terms form a monotonic
decreasing sequence, starting with the second term. Furthermore, it is
known [8] that infinite sum
∞∑
n=1
tn
Aαn+Bβn is an irrational number if α, β
are positive integers and A · B 6= 0, |α| > |t|, |A · B · t2| < |α|. Hence, the
sum of series (5) is also an irrational number.
172 Jacobsthal-Lucas series
Using equality (3), we have the formula for general term of the sequence
of reciprocal Jacobsthal-Lucas numbers:
un =
1
2n−1 + (−1)n−1
. (6)
Lemma 2. For series (5), the following system of inequalities holds:
{
un > rn if n is even,
un < rn if n is odd.
(7)
Proof. Using (6), for even numbers n, we have un =
1
2n−1 − 1
. Since
1
2k + 1
+
1
2k+1 − 1
<
1
2k
+
1
2k+1
for any k ∈ N , we obtain
rn =
1
2n + 1
+
1
2n+1 − 1
+
1
2n+2 + 1
+· · · <
1
2n
+
1
2n+1
+
1
2n+2
+· · · =
1
2n−1
.
So, rn <
1
2n−1
<
1
2n−1 − 1
= un. Hence un > rn for even numbers n.
Similarly, using (6), for odd numbers n, we have un =
1
2n−1 + 1
. Since
1
2k − 1
+
1
2k+1 + 1
>
1
2k
+
1
2k+1
for any k ∈ N , we obtain
rn =
1
2n − 1
+
1
2n+1 + 1
+
1
2n+2 − 1
+· · · >
1
2n
+
1
2n+1
+
1
2n+2
+· · · =
1
2n−1
.
So, rn >
1
2n−1
>
1
2n−1 + 1
= un. Hence un < rn for odd numbers n.
Lemma 3. For remainders of series (5), the following system of inequa-
lities holds:
1
2n + 1
+
1
2n
< rn <
1
2n−1
if n is even,
1
2n−1
< rn <
1
2n − 1
+
1
2n
if n is odd.
(8)
M. Pratsiovytyi, D. Karvatsky 173
Proof. For even number n, we have
rn =
1
2n + 1
+
1
2n+1 − 1
+ · · · +
1
2n+k + (−1)n+k
+ . . .
<
1
2n
+
1
2n+1
+ · · · +
1
2n+k
+ · · · =
1
2n−1
.
On the other hand,
rn >
1
2n + 1
+
[
1
2n+1
+
1
2n+2
+ · · · +
1
2n+k+1
+ . . .
]
=
1
2n + 1
+
1
2n
.
Hence, for even number n, the following inequalities hold:
1
2n + 1
+
1
2n
< rn <
1
2n−1
.
For odd number n, we have
rn =
1
2n − 1
+
1
2n+1 + 1
+ · · · +
1
2n+k + (−1)n+k
+ . . .
>
1
2n
+
1
2n+1
+ · · · +
1
2n+k
+ · · · =
1
2n−1
.
On the other hand,
rn <
1
2n − 1
+
[
1
2n+1
+
1
2n+2
+ · · · +
1
2n+k+1
+ . . .
]
=
1
2n − 1
+
1
2n
.
Hence, for odd number n, the following inequalities hold:
1
2n−1
< rn <
1
2n − 1
+
1
2n
.
Lemma 4. For any positive integer k, the following inequalities hold:
u2k+2+u2k+3+. . .+u4k < u2k+1 <u2k+2+u2k+3+. . .+u4k+(u4k+1+u4k+2).
To prove this lemma, we estimate the difference of the right-hand and
left-hand side of each inequality and take into account
1
2k + 1
+
1
2k+1 − 1
<
1
2k
+
1
2k+1
<
1
2k − 1
+
1
2k+1 + 1
.
174 Jacobsthal-Lucas series
3. The set of incomplete sums of the series
Definition 2. Let M ∈ 2N that is M ⊆ N . Then number
x = x (M) =
∑
n∈M
un =
∞∑
n=1
εnun, (9)
where εn =
{
1 if n ∈ M,
0 if n /∈ M,
is called the incomplete sum of series
∑
un.
By ∆′ we denote the set of all incomplete sums of series (5).
The set of incomplete sums of convergent positive series such that ine-
quality un 6 rn (un > rn) holds only finitely many times was investigated
in paper [7]. Moreover, it is well known [6] that the set of all subsums
of any convergent positive series always belongs to one of the following
three types: a finite union of closed intervals, a Cantor type set or an
M-Cantorval. However, the question about type and properties of the set
of subsums of series (5) is still open because inequalities un < rn and
un > rn hold infinitely many times for this series.
Definition 3. The set ∆′
c1...ck
of all incomplete sums
k∑
n=1
cnun +
∞∑
n=k+1
εnun, where εn ∈ {0, 1} ,
of series (5) is called the cylinder of rank k with base c1 . . . ck(ci ∈ {0, 1}).
Definition 4. The closed interval ∆c1c2...ck
=
[
inf ∆′
c1...ck
, sup ∆′
c1...ck
]
is
called the cylindrical interval of rank k with base c1 . . . ck(ci ∈ {0, 1}).
It is possible that ∆′
c1...ck
and ∆c1...ck
coincide or not, depending on
properties of series and sequence (c1 . . . ck). However ∆′
c1...ck
⊂ ∆c1...ck
in
any case.
Lemma 5. The cylindrical sets have the following properties:
1. ∆c1c2...ck
=
[ k∑
i=1
ciui,
k∑
i=1
ciui + rk
]
;
2. |∆c1c2...ck
| = rk → 0 as k → ∞;
3. ∆c1c2...ck
⊂∆c1c2...ck0
⋃
∆c1c2...ck1, ∆′
c1c2...ck
= ∆′
c1c2...ck0
⋃
∆′
c1c2...ck1;
4. inf ∆c1c2...ck
= inf ∆c1c2...ck0 < inf ∆c1c2...ck1,
sup ∆c1c2...ck
= sup ∆c1c2...ck1 > sup ∆c1c2...ck0;
M. Pratsiovytyi, D. Karvatsky 175
5.
∞⋂
k=1
∆c1c2...ck
=
∞⋂
k=1
∆′
c1c2...ck
≡ ∆c1c2...ck... = x ⊂ [0, r] ;
6.
|∆c1c2...ckc|
|∆c1c2...ck
|
=
rk+1
rk+1 + uk+1
=
1
δk+1 + 1
, where δk+1 =
uk+1
rk+1
;
7. ∆c1c2...ck
= ∆s1s2...sk
if and only if ci = si, i = 1, k;
8. Ok+1
c1...ck
(1, 0) = ∆c1c2...ck1
⋂
∆c1c2...ck0
=
[
k∑
n=1
cnun + un+1,
k∑
n=1
cnun + rn+1
]
if k is even,
∅ if k is odd;
9. For any even number k,
∆c1c2...ck1
⋂
∆c1c2...ck0 = ∆c1c2...ck10
⋂
∆c1c2...ck01;
10. Gk+1
c1...ck
(1, 0) = ∆c1c2...ck
\
(
∆c1c2...ck1
⋃
∆c1c2...ck0
)
=
( k∑
n=1
cnun+rn+1,
k∑
n=1
cnun + un+1
)
if k is odd,
∅ if k is even;
11. For any positive integer k > 2,
Gk+2
c1...ck
(01, 00)
⋂
Gk+2
c1...ck
(10, 11) = ∅;
12. For any positive integer k > 2,
(
Gk+2
c1...ck
(01, 00)
⋃
Gk+2
c1...ck
(10, 11)
)⋂
Ok+1
c1...ck
(1, 0) = ∅.
Lemma 6. For any odd number n, the following relations hold:
On
c1...cn−1
(0, 1) = ∆c1...cn−10 1 . . . 1︸ ︷︷ ︸
m
⋂
∆c1...cn−11 0 . . . 0︸ ︷︷ ︸
m
if m < n − 2,
On
c1...cn−1
(0, 1) 6= ∆c1...cn−10 1 . . . 1︸ ︷︷ ︸
m
⋂
∆c1...cn−11 0 . . . 0︸ ︷︷ ︸
m
if m > n.
Lemma 7. For any nonempty On+1
c1...cn
(0, 1) there exist a positive integer
m and sequence α1, . . . , αm, β1, . . . , βm, where αi ∈ {0, 1}, βi ∈ {0, 1},
i = 1, m, such that
Gn+m+1
c1...cn1α1...αm
(0, 1)
⋂
Gn+m+1
c1...cn0β1...βm
(0, 1) ∈ On+1
c1...cn
(0, 1).
176 Jacobsthal-Lucas series
Proof. Let us show that there exist a positive integer m and sequence
α1, . . . , αm, β1, . . . , βm, where αi ∈ {0, 1}, βi ∈ {0, 1}, i = 1, m, such that
min Gn+m+1
c1...cn0β1...βm
< min Gn+m+1
c1...cn1α1...αm
< max Gn+m+1
c1...cn0β1...βm
,
0 < un+1 +
m∑
i=1
(αi − βi)un+1+i < un+m+2 − rn+m+2.
Taking into account un+1 < rn+1, we can find the number m̃ > n + 1
such that un+1−un+2−un+3−· · ·−un+m̃ < 0. Since un+m+2−rn+m+2 > 0,
we see that there exist numbers α1, . . . , αm, β1, . . . , βm, where αi ∈ {0, 1}
and βi ∈ {0, 1}, i = 1, m, such that
0 < un+1 +
m∑
i=1
(αi − βi)un+1+i < un+m+2 − rn+m+2.
Corollary 1 (Lemmas 7, 8). Any intersection of cylinders of the same
rank is not completely contained in the set of subsums of series (5).
Theorem 2. The set of incomplete sums of series (5) is a perfect nowhere
dense set of positive Lebesgue measure.
Proof. It is easy to prove that the set of incomplete sums of any positive
series is a perfect set (i.e., closed set without isolated points).
By G we denote the sum of all gaps in the form Gn+1
c1...cn
(0, 1) between
cylinders of even ranks. Then Lebesgue measure of ∆′ is greater than or
equal to some number: L(∆′) >
∞∑
n=1
un − G.
Using properties 10, 11, 12 of cylindrical sets, we have
∞∑
n=1
un − G =
∞∑
n=1
un−3[u1−r2]−8[u4−r4]− . . . −2 · 4n−1[u2n−r2n] − . . .
= 4u3 − 4u4 + 12u5 − 20u6 + 44u7 − . . .
+ un+2[22 − 23 + 24 − . . . + (−2)n+1] + . . .
=
∞∑
n=1
(
u2n+1 ·
4 + 22n+1
3
+ u2n+2 ·
4 − 22n+2
3
)
=
∞∑
n=1
An.
Now we show that An > 0 for all n ∈ N . Taking into account equa-
lity (6), we see that
An =
4 + 22n+1
3 (22n + 1)
+
4 − 22n+2
3 (22n+1 − 1)
=
22n+1
(22n + 1)(22n+1 − 1)
> 0.
M. Pratsiovytyi, D. Karvatsky 177
Using approximate calculation, we can conclude that Lebesgue measure
of ∆′ is greater than some positive number:
L(∆′) >
100∑
n=1
22n+1
(22n + 1)(22n+1 − 1)
≈ 0, 3099984859 . . . > 0.
Finally, we show that the set of incomplete sums is nowhere dense.
Suppose that there exists some closed interval [a, b] ⊂ ∆′. It is obvious
that we can find numbers c1, c2, . . . , ck such that
a <
k∑
n=1
cnun < b, max{uk, rk} < b −
k∑
n=1
cnun.
So, there exists some cylindrical interval ∆c1...ck
⊂ [a, b].
If k is odd, then from the properties of cylindrical sets it follows that
Gk+1
c1...ck
(0, 1) ⊂ ∆c1...ck
. So there exists some gap Gk+1
c1...ck
(0, 1) such that it
is a subset of [a, b] but is not contained in the set of subsums.
If k is even, then from the properties of cylindrical sets it follows that
Ok+1
c1...ck
(0, 1) ⊂ ∆c1...ck
⊂ [a, b]. According to Lemmas 7 and 8, intersection
Ok+1
c1...ck
(0, 1) cannot be contained in the set of subsums. This contradiction
proves the theorem.
4. Distribution of random incomplete sum
Let us consider random variable ξ =
∞∑
k=1
ξk
Jk
, where (ξk) is a sequence of
independent random variables taking the values 0 and 1 with probabilities
P{ξk = 0} = p0k > 0, P{ξk = 1} = p1k > 0 respectively, p0k + p1k = 1,
and (Jk) is Jacobsthal-Lucas sequence.
Properties of distribution of ξ depend both on infinite stochastic
matrix ||pik|| and series (5). According to the Jessen-Wintner theorem
[1], the random variable ξ has a pure distribution (pure discrete, pure
absolutely continuous or pure singular). Criterion for the discreteness of
ξ follows from the P. Lévy theorem (see [5]): the random variable ξ has a
discrete distribution if and only if M =
∏
∞
k=1 max{p0k, p1k} > 0.
It is easy to prove that the spectrum Sξ of the distribution of random
variable ξ is the set Sξ = {x : x = ∆c1c2...ck..., pckk > 0, ∀k ∈ N} and Sξ
is a subset of the set of incomplete sums of series (5). We understand
topological, metric and fractal properties of distribution of random variable
ξ as topological, metric and fractal properties of its spectrum.
178 Jacobsthal-Lucas series
Theorem 3. If
{
p0(2m)p1(2m) = 0,
p0(2m−1)p1(2m−1) 6= 0,
or
{
p0(2m)p1(2m) 6= 0,
p0(2m−1)p1(2m−1) = 0,
then random variable ξ has a singular distribution of Cantor type. The
Hausdorff-Besicovitch dimension of the spectrum of ξ is equal to 0, 5.
Proof. We prove the theorem if ξ satisfies condition p0(2m) · p1(2m) = 0,
p0(2m−1) · p1(2m−1) 6= 0. The proof is similar in second case.
The spectrum of ξ coincides with the set of incomplete sums of series
∞∑
n=1
tn =
1
J1
+
1
J3
+ · · · +
1
J2n−1
+ · · · =
∞∑
n=1
u2n−1. (10)
It is easy to see that tk > rk(t) =
∞∑
n=k+1
tn for any positive integer k.
Then the set of subsums of series (10) is a perfect nowhere dense set [7].
By (3), we have tn = 1
4n−1+1
.
The Lebesgue measure of the set of incomplete sums of series (10)
∆′(t) can be computed by formula λ(∆′(t)) = limn→∞ 2nrn(t). Since
rn(t) =
∞∑
k=n+1
tk <
1
4n
+
1
4n+1
+ · · · =
1
3 · 4n−1
<
1
4n−1
, we see that
λ(∆′(t)) < lim
n→∞
2n
4n−1
= 0.
So, we can conclude that the Lebesgue measure of the set of incomplete
sums of series (10) is equal to zero. However, there is still open question
about Hausdorff-Besicovitch dimension of spectrum of ξ. It is well known
that
Hα(E) = lim
k→∞
2krα
k (t) = lim
k→∞
(2r
α
k
k (t))k =
0 if 2r
α
k
k (t) < 1,
1 if 2r
α
k
k (t) = 1,
∞ if 2r
α
k
k (t) > 1.
Let us consider the case 2r
α
k
k (t) = 1. We have α(k) =
−k ln 2
ln rk(t)
. Since
α depends on k in the last equality, we see that
α = lim
k→∞
α(k) = lim
k→∞
−k ln 2
ln rk(t)
.
M. Pratsiovytyi, D. Karvatsky 179
For the remainders of series (10), the following inequalities hold:
1
4k+1
< rk(t) <
1
4k−1
.
Hence, we have ln
(
1
4k+1
)
< ln rk(t) < ln
(
1
4k−1
)
,
ln 2
ln
(
1
4k−1
) <
ln 2
ln rk(t)
<
ln 2
ln
(
1
4k+1
) ,
−k · ln 2
−(k − 1) · ln 4
<
−k · ln 2
ln rk(t)
<
−k · ln 2
−(k + 1) · ln 4
.
We can find the limit of the sequence on the left and right side of the
above inequality respectively:
lim
k→∞
−k · ln 2
−(k − 1) · ln 4
=
1
2
and lim
k→∞
−k · ln 2
−(k + 1) · ln 4
=
1
2
.
From the squeeze theorem it follows that
lim
k→∞
α(k) = lim
k→∞
−k · ln 2
ln rk(t)
=
1
2
.
So, the Hausdorff-Besicovitch dimension of the set of incomplete sums
of series (10) is equal to 0, 5.
Theorem 4. Let pairs (ξ2k−1ξ2k) of consecutive independent random
variables take values from the set {(0, 0), (1, 1)} with probabilities p0k > 0,
p1k > 0 respectively,
∞∏
k=1
max{p0k, p1k} = 0, and stochastic matrix ‖pik‖
has finite numbers of zeroes. Then random variable ξ has a singular
probability distribution of Cantor type; and the Hausdorff-Besicovitch
dimension of the spectrum of ξ is equal to 0, 5.
References
[1] B. Jessen, A. Wintner, Distribution functions and the Riemann zeta function,
Trans. Amer. Math. Soc., Vol. 38 (1), 1935, pp. 48-88.
[2] D. M. Karvatsky, Representation of real numbers by infinitesimal generalized
Fibonacci sequences, Trans. Natl. Pedagog. Mykhailo Drahomanov Univ. Ser. 1.
Phys. Math., no. 15, 2013, pp. 56-73 (in Ukrainian).
[3] D. M. Karvatsky, Properties of distribution of random incomplete sum of conver-
gent positive series whose terms are elements of generalized Fibonacci sequence,
Bukovinian Mathematical Journal, Vol. 3 (1), 2015, pp. 52-59 (in Ukrainian).
180 Jacobsthal-Lucas series
[4] D. M. Karvatsky, N. M. Vasylenko, Mathematical structures in the spaces of
generalized Fibonacci sequences, Trans. Natl. Pedagog. Mykhailo Drahomanov
Univ. Ser. 1. Phys. Math., no. 13 (1), 2012, pp. 118-127 (in Ukrainian).
[5] P. Lévy, Sur les séries dont les termes sont des variables éventuelles indépendantes,
Studia Math., Vol. 3 (1), 1931, pp. 119-155.
[6] J. E. Nymann, R. A. Sáenz, On a paper of Guthrie and Nymann on subsums of
infinite series, Colloq. Math., Vol. 83 (1), 2000, pp. 1-4.
[7] M. V. Pratsiovytyi, Fractal approach to investigation of singular probability
distributions, Natl. Pedagog. Mykhailo Drahomanov Univ. Publ., Kyiv, 1998
(in Ukrainian).
[8] M. Prévost, On the irrationality of
∑
tn
Aαn+Bβn
, J. Number Theory, Vol. 73 (2),
1998, pp. 139-161.
Contact information
M. Pratsiovytyi,
D. Karvatsky
National Dragomanov Pedagogical University,
Ukraine, Kyiv, vul. Pirogova 9.
E-Mail(s): Prats4444@gmail.com,
D.Karvatsky@gmail.com
Web-page(s): npu.edu.ua
Received by the editors: 12.09.2016
and in final form 29.03.2017.
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