HPU2. Nat. Sci. Tech. Vol 02, issue 03 (2023), 13-18
HPU2 Journal of Sciences:
Natural Sciences and Technology
journal homepage: https://sj.hpu2.edu.vn
Article type: Research article
Received date: 13-10-2023 ; Revised date: 24-10-2023 ; Accepted date: 06-12-2023
This is licensed under the CC BY-NC-ND 4.0
Applications of the Z transformation in calculating sum of a series
Huyen-My Le Thi*
University of Medicine and Pharmacy, Thai Nguyen University, Thai Nguyen, Vietnam
Abstract
The Z transformation has many applications in Mathematics, especially in solving difference
equations to help handle discrete data models. This article provides one more application of the Z
transformation, which is the summation of a series. The author through the research method of theory
development from some properties of the Z transformation to obtain the result which is a theorem
about the formula for the sum of a series, with the proof attached. From there, apply the theorem
together with the Z transformation to calculate the sum of some series in specific problems. Thus, the
problem of calculating the sum of the series has one more method of solving, as well as expanding the
application of the Z transformation in the field of Mathematics, especially analysis.
Keywords: The Z transformation, the inverse Z transformation, series, summary of series
1. Introduction
The
Z
transformation is useful tool in handling discrete data models, widely used in the fields of
applied mathematics, digital signal processing, control theory and economics [1,2,3,4,5]. These
discrete models are solved by difference equations similar to the continuous models solved by
differential equations. The
Z
transform plays an important role in solving difference equations in the
same way that the Laplace transform plays an important role in solving differential equations. In this
article, the author mentions another application of the
Z
transformation to calculate series sums. The
author has researched and developed the theory to obtain a way to calculate series sums through the
Z
* Corresponding author, E-mail: lethihuyenmy@tump.edu.vn
https://doi.org/10.56764/hpu2.jos.2023.2.3.13-18
HPU2. Nat. Sci. Tech. 2023, 2(3), 13-18
https://sj.hpu2.edu.vn 14
transformation.
Below, the author will describe in detail how to approach the research problem, starting from
introducing the concept and some related properties of the
Z
transformation, accompanied by
illustrative examples. Then the author will give a way to calculate the series sum using the
Z
transformation by a theorem and some examples.
2. Research methods
The author uses the research method of theory development. First of all, we approach the
definition of the
Z
transformation.
Definition 1. Let
T
be a fixed positive number (can be taken
1=T
). Suppose
()ft
identifies with
0t
and
gets the value at
; 0,1,2,...=nT n
. The
Z
transformation of function
()ft
, or sequence
()f nT
, is a complex variable function
z
determined by formula
0
( ) ( ) ( ) ,
−
=
==
n
n
Z f nT F z f nT z
where
1
=zR
,
is the radius of convergence of the series.
Example 1. Let
()=nT
f nT a
. Then
00

−
==

==



n
T
nT nT n
nn
a
Z a a z z
;.=
−
T
T
zza
za
(1)
Example 2. The
Z
transformation of
1
() !!
=f nT nT
is
0
11
! ! ! !
−
=

=


n
n
Zz
n T n T
11
exp ;
!

=

 z
Tz
(2)
Example 3. With
( ) cos
=f nT nT
we have
0
cos 2

−
−
=
+
=inT inT
n
n
ee
Z nT z
( ) ( )
11
00
1
2


− − −
==

=+



nn
iT iT
nn
e z e z
11
1 1 1
2 1 1

− − −

=+

−−

iT iT
e z e z
2
( cos ) ;1
2 cos 1
−
=
−+
z z T z
z z T
.
HPU2. Nat. Sci. Tech. 2023, 2(3), 13-18
https://sj.hpu2.edu.vn 15
The
Z
transformation has many properties. Here, the author only states some necessary related
properties:
i. Proportional properties
( ) ( ).
−
=
nT T
Z a f nT F a z
(3)
ii. Multiplication
()
( ) ,=− dF z
Z nf nT z dz
()
( ) .=− dF z
Z nTf nT Tz dz
iii. Division
1
( ) ( ) ; 0.
+

=

+

m
m
z
f nT F z
Z z dz m
n m z
(4)
iv. Initial value theorem
Suppose
( ) ( )=Z f nT F z
. Then
(0) lim ( ).
→
=z
f F z
v. Final value theorem
Suppose
( ) ( )=Z f nT F z
. Then
1
lim ( ) lim[( 1) ( )],
→ →
=−
nz
f nT z F z
(5)
where, the limits are assumed to exist.
vi. Inverse
Z
transformation
11
1
( ) ( ) ( ) ,
2
−−
==
n
C
Z F z f nT F z z dz
i
where
C
is the closed circuit surrounding the origin and outside the circle
=zR
.
3. Results and discussion
Theorem 1. Suppose
( ) ( )=Z f nT F z
. Then
(i)
1
0
( ) ( ) ,
1
−
=

=
−

n
k
z
f k Z F z
z
(ii)
1
0
( ) lim ( ) (1).
→
=
==
z
k
f k F z F
Prove.
Set
0
( ) ( )
=
=
n
k
g n f k
. Then
( ) ( ) ( 1).= − −g n f n g n
Take the Z transformation both sides of the above equation, we get
HPU2. Nat. Sci. Tech. 2023, 2(3), 13-18
https://sj.hpu2.edu.vn 16
1
( ) ( ) ( ),
−
=+G z F z z G z
( ) ( ).
1
=
−
z
G z F z
z
Therefore
11
( ) ( ) ( ) .
1
−−

==

−

z
g n Z G z Z F z
z
So
1
0
( ) ( ) .
1
−
=

=
−

n
k
z
f k Z F z
z
According to (5) we have
1
lim ( ) lim ( 1) ( )
→ →
=−
nz
g n z G z
, or
1
0
lim ( ) lim ( 1) ( ) (1).
1
→ →
=

= − =

−

n
nz
k
z
f k z F z F
z
So,
0
( ) (1).
=
=
k
f k F
We will apply the above theorem to some specific problems below.
Problem 1. Use the
Z
transformation to calculate the sum of the series
0
.
!
=
n
n
x
n
According to (3) we have
() 
=

nz
Z x f n F x
.
Set
1
() !
=fn n
. From result (2) we have
1
( ) ( ) exp .

==


F z Z f n z
From there it can be deduced
exp .
!
 
=
 


n
xx
Znz
Using Theorem 1 (ii) we get
0
.
!
=
=
n
x
n
xe
n
Problem 2. Use the
Z
transformation to show that
0
1
( 1) log(1 ).
1
=
+
− = +
+
n
n
n
xx
n
Using (1) we have
1.
+=−
nxz
Zx zx
According to (4) we get
HPU2. Nat. Sci. Tech. 2023, 2(3), 13-18
https://sj.hpu2.edu.vn 17
1
2
1 ( )

+

= =

+ − −


n
zz
x xz dz dz
Z z xz
n z x z z z x
1log log .

−−
= = −


z
z x z x
xz z
x z z
Replaced
x
by
−x
,
1
( 1) log .
1
+
+

−=


+

n
nx z x
Zz
nz
Applying Theorem 1 (ii) gives us the result
0
1
( 1) log(1 ).
1
=
+
− = +
+
n
n
n
xx
n
Problem 3. Calculate the sum of the series
0
sin .
=
n
n
a nx
Set
( ) sin=f n nx
, we have the
Z
transformation
0
( ) sin .
−
=
=n
n
Z f n nx z
02
−
−
=
−
=inx inx
n
n
ee
z
11
00
1( ) ( )
2

− − −
==

=−



ix n ix n
nn
e z e z
11
1 1 1
2 1 1
− − −

=−

−−

ix ix
e z e z
2
sin .
2 cos 1
=−+
zx
z z x
According to (3) we get
22
sin
sin .
2 cos
=−+
naz x
Z a nx a az x z
Therefore, using Theorem 1 (ii) we get
2
0
asin
sin .
2 cos 1
=
=−+
n
n
x
a nx a a x
4. Conclusions
Thus, the above result shows that the sum of many series can be calculated through the
Z
transformation. From there, we see another application of the
Z
transformation in Mathematics, as
well as opening up research suggestions about its other applications.
Declaration of Competing Interest
The author declare no competing interests.