C-Type Ops Transformation


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 7(3) (2022), Pages 401-410 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: April 05, 2022 Reviewed: June 29, 2022 Accepted: July 07, 2022 
DOI: http://dx.doi.org/10.18860/ca.v7i3.15749      

C-Type Ops Transformation 

Ahmad Lazwardi*, Iin Ariyanti, Soraya Djamilah 

Universitas Muhammadiyah Banjarmasin, Indonesia 

 

Email: lazwardiahmad@gmail.com  

ABSTRACT 

The scope of the ops transformation is limited to the McLaurin series only. While there are still 
many cases in mathematical modeling which are modeled in the form of a more general series. 
The purpose of this study is to generalize the ops transformation into a more general form that 
can be used for any Taylor series. This study uses a literature study method, namely by reviewing 
the ops transformation and then observing aspects that can be generalized. Next is to construct a 
more general definition of ops transformation which is referred to as c-type ops transformation. 
At the end of this research, the ops transformation will be applied to solve ordinary differential 
equations with variable coefficients very briefly. 

Keywords: C-Type ops transformation; ops transformation; power series  

INTRODUCTION 
Taylor series is accessible to all students and it is a useful mathematical tool to 

nonlinear equations [1]. Power series is an essensial method for solving many problems 
in mathematics such as algebra and differential equations [2].  Many cases also appear in 
algebra which is involving power series such as solving polynomial homotopies 
equations [3]. As we know on algebraic geometry topics we talk about rings, ring 
extensions and ideals which recently appears as series forms. For example, is on 
commutative ring topics. If R be a commutative ring with identity. Let R[x] and R[[x]] be 
the collection of polynomials and, respectively, of power series with coefficients in R. 
Their multiplications are from a class of sequences 𝜆 = {𝜆𝑛} of positive integers which is 
related one-one correspondence to its power series [4].  

Another branch of mathematics which also involved by power series recently is 
differential equations[5]. The problem of finding formal power series solutions of 
differential equations has a long history and it has been extensively studied in literature. 
Using power series method however, is a more systematic way and standard basic 
method for approximating the solutions of such differential equations analytically and 
thus studying the method is of greater importance [6]. Ordinary Power series has also 
appear as solutions for fractional differential equations as told by Angstmann and Henry 
in their publication namely Generalized series expansions involving integer powers and 
fractional powers in the independent variable have recently been shown to provide 
solutions to certain linear fractional order differential equations [7]. This incredible 
discovery is related to I. Area and J. Nieto who discover the solution for fractional 
logistics equation which is appeared as power series form [8].  

http://dx.doi.org/10.18860/ca.v7i3.15749


C-Type Ops Transformation 

Ahmad Lazwardi 402 

This research discovers new theory of ordinary power series expecially the 
alternating method to analyze ordinary power series by considering ordinary power 
series as a transformation which called ops transformation. This transformation will 
simplify the counting process of some equations involving sigma notation. Reducing the 
use of sigma algebra, alternating it by algebra of linear transformation. 
 The aim of this research is to generalize the concept of ops transformation for 
power series of form (1). We will name it as c-type ops transformation. The letter c is 
indicated the center of the power series. 
 
METHODS  
We will proceed this reseach through the following procedures. First we will define the 
more general form of ops transforamtion. After that we will find some basic results of 
our new definition. We will use such results to extend the theory of ops transformation. 
Next we will make sure that our previous definition of ops transformation to become 
special case for our new definition. Next we will find some theorems regarding our new 
transformation properties. We also will make sure that our new definition is able to be 
applied much wider than our previous one.  
 
RESULTS AND DISCUSSION  
First we shall define the c-Type of ops transformation.  Recall that power series centered 
at c is defined to be the real valued function of the form [9] 
 

.)(
0







n

n

n
cxa  

The series has a value depending on what value of x we choose. Some value of x will 
result the series tend to infinity. Some other will result the series converges[10]. The set 
of x which result (1) converges is called convergence interval. The term “interval” makes 
sense because such set always forms an interval. The half-length of such interval is 
called convergence radius. 
 
Some smooth function f(x) at point c is able to be approximated by power series which 
on some value of x0 lies on its convergence interval centered at c, the result will be same, 
i.e f(x0) = ∑ 𝑎𝑛(𝑥0 − 𝑐)

𝑛∞
𝑛=0 . Such functions are called “real analytic” functions. The 

method of resulting such series was given by Taylor which is called Taylor series as 
below [11] 
 

.)(
!

)(

0

)(







n

n
n

cx
n

cf
 

Special case of Taylor series is when the value of c = 0, the series is called McLaurin 
series. Lazwardi (2021) has already able to reformulate such series into more simple 
form called ops transformation.  The ops transformation is defined as below 
 

.)})(({
0







n

n

nn
xaxaOps  

Therefore, for some McLaurin series of the form 

.
!

)0(

0

)(




n

n
n

x
n

f
 

(1) 

(2) 

(4) 



C-Type Ops Transformation 

Ahmad Lazwardi 403 

its enough to write the series simply as 𝑂𝑝𝑠{
𝑓(𝑛)(0)

𝑛!
} .  Simplification of the form will 

make calculations and manipulations easier [12]. There are some properties regarding 
ops transformation as following: 
 
Theorem 1. (shifting-entry) For each {𝑎𝑛} sequence, we have  

𝑂𝑝𝑠{0,𝑎0,𝑎1,…} = 𝑥𝑂𝑝𝑠{𝑎𝑛}. 
  
Theorem 2. For each {𝑎𝑛}  sequence, we have  

𝑂𝑝𝑠({𝑎𝑛})− 𝑎0 = 𝑥𝑂𝑝𝑠({𝑎𝑛+1}). 
  
Beside two above theorems, ops transformation inherits linearity properties as well as 
sigma notations.  
  
Theorem 3. For each {𝑎𝑛},{𝑏𝑛} sequences and any real numbers 𝛼,𝛽 , we have 

𝑂𝑝𝑠(𝛼{𝑎𝑛} +𝛽{𝑏𝑛}) = 𝛼𝑂𝑝𝑠{𝑎𝑛} + 𝛽𝑂𝑝𝑠{𝑏𝑛}. 
 
The last theorem notices us that we can view ops transformation as linear 
transformation which mapping from the space of all real sequences to real numbers on 
its convergence radius. We shall use this necessary fact to simplify several calculations. 
 
Besides that Lazwardi was able to prove the formula regarding product of two ops 
transformations as following. 
 
Theorem 4. For each {𝑎𝑛},{𝑏𝑛} sequences, we have 

.}{}{
0 








 




n

k

knknn
baOpsbOpsaOps  

This is just similiar with the product two power series but wihout involving double 
sigma notation. Another important result of previous research is we can use the fact that 
the power series is always able to differentiate n-times, to construct the rule of 
differentiation for ops transformation.  
Talking about differentiation of ops transformation meas we have to state the symbol 
for its derivative. We use 𝐷𝑥𝑂𝑝𝑠{𝑎𝑛} to notate the derivative of ops transformation on its 
radius convergence.  Therefore we have the following theorems. 
 
Theorem 5. For each {𝑎𝑛} sequence, we have  

𝐷𝑥𝑂𝑝𝑠({𝑎𝑛}) = 𝑂𝑝𝑠({(𝑛 + 1)𝑎𝑛+1}). 
 
Here is some nice modification formula 
Theorem 6. For each {𝑎𝑛} sequence, we have 

𝑥𝐷𝑥𝑂𝑝𝑠({𝑎𝑛}) = 𝑂𝑝𝑠({𝑛𝑎𝑛}). 
 
If we pay more attention to the (1). There are some difference between (1) and (3) i.e 
the value of c will be varied and able to consider it as a variable. Therefore we have at 
least  4 variable involved in calculations of (1) which is more complicated than (3) 
expecially special type of ordinary power series which called Taylor series. As told by 
Salwa in her research that many infinite series form are recently appear in sequence 
spaces expecially on 𝛽 − 𝑑𝑢𝑎𝑙  sequence spaces which is defined as infinite series form 
[13] One of popular application from Taylor series is the iterative method of the 

(5) 

(6) 

(7) 

(8) 

(9) 

(10) 



C-Type Ops Transformation 

Ahmad Lazwardi 404 

differential transform methor has already been used for a while, by the ‘‘Traditional’’ 
Taylor series method users which have even better developed the method. 
 
Suppose that we have power series of form (1) centered at c. We define 
Definition 1. Let c be a real number and suppose power series ∑𝑎𝑛(𝑥 − 𝑐)

𝑛  has 
positive convergence radius near c. Define the c-type ops transformation as following 
 

𝑂𝑝𝑠𝑐{𝑎𝑛}(𝑥) = ∑𝑎𝑛

∞

𝑛=0

(𝑥 − 𝑐)𝑛. 

 
It looks similiar to the previous form with additional superscript c. Note that the 
additional superscript c on Ops roles as index depending on value c on the right side.  For 
some reason, we shall keep c to become upper index because we shall use lower index 
with another use on the next research. 
 
Recall that one of the most suitable form which is similiar to our last definition is Taylor 
series of analytic function on c. If 𝑓(𝑥) is an analytic function near c, then we can write f  
as Taylor series on some neighborhood c as below 
 

𝑓(𝑥) = ∑
𝑓(𝑛)(𝑐)

𝑛!

∞

𝑛=0

(𝑥 − 𝑐)𝑛. 

 
Hence, Taylor series of f can be written as c-type ops transformation as 
 

𝑓(𝑥) = 𝑂𝑝𝑠𝑐 {
𝑓(𝑛)(𝑐)

𝑛!
}(𝑥). 

 
For some reason, we just write 
 

𝑓 = 𝑂𝑝𝑠𝑐 {
𝑓(𝑛)(𝑐)

𝑛!
}. 

 
Please pay more attention here. Upper index c is viewed as variable (not necessary 
fixed). We can write 

𝑂𝑝𝑠𝑎 {
𝑓(𝑛)(𝑐)

𝑛!
} = ∑

𝑓(𝑛)(𝑐)

𝑛!

∞

𝑛=0

(𝑥 − 𝑎)𝑛. 

i.e when we change the value of upper index c by a, the value of 
𝑓(𝑛)(𝑐)

𝑛!
 doesn’t change but 

the center of power series on the right side changes to a. 
 
Its clear that ops transformation is a special case of c-type of ops transformation by 
taking value c = 0 [14], i.e 
 

𝑂𝑝𝑠0{𝑎𝑛} = 𝑂𝑝𝑠{𝑎𝑛}. 
 
For more brief information. We shall discuss some more examples as following. 
 

(11) 

(12) 

(13) 

(14) 

(15) 



C-Type Ops Transformation 

Ahmad Lazwardi 405 

Example 1. 𝑂𝑝𝑠𝑐{1} =
1

1−(𝑥−𝑐)
. 

 
Proof: Observe that 𝑂𝑝𝑠𝑐{1} = ∑(𝑥 − 𝑐)𝑛. Suppose that 𝑦 = 𝑥 − 𝑐 then the right side of 

equation become ∑𝑦𝑛 =
1

1−𝑦
 for |𝑦| < 1. Hence we have for |𝑥 − 𝑐| < 1 or 𝑐 − 1 < 𝑥 <

𝑐 + 1 we will get the series ∑(𝑥 − 𝑐)𝑛 will converge and we have 𝑂𝑝𝑠𝑐{1} =
1

1−(𝑥−𝑐)
 . 

 
Here is another example 
 

Example 2. 𝑂𝑝𝑠𝑐 {
1

𝑛!
} = 𝑒𝑥−𝑐. 

 
Now we shall analyze more properties of c-type ops transformation. First we success to 
preserve the “shifting index” properties as well as previous form [15] 
 
Theorem 7. (shifting-entry) For each {𝑎𝑛} sequence, we have  

𝑂𝑝𝑠𝑐{0,𝑎0,𝑎1,…} = (𝑥 − 𝑐)𝑂𝑝𝑠
𝑐{𝑎𝑛}. 

 
Proof: Observe that  

𝑂𝑝𝑠𝑐{0,𝑎0,𝑎1,…} = 0 + ∑𝑎𝑛−1

∞

𝑛=1

(𝑥 − 𝑐)𝑛 

 

= ∑𝑎𝑛(𝑥 − 𝑐)
𝑛+1

∞

𝑛=0

 

  

= (𝑥 − 𝑐)∑𝑎𝑛

∞

𝑛=0

(𝑥 − 𝑐)𝑛 

     = (𝑥 − 𝑐)𝑂𝑝𝑠𝑐{𝑎𝑛} 
 
 
 
Hence by induction we can conclude as corollary below 
 
Corollary 1.  For each {𝑎𝑛}  sequence, we have 

𝑂𝑝𝑠𝑐 {0,0,0, . . ,0⏟      
𝑘−𝑒𝑛𝑡𝑟𝑖𝑒𝑠

𝑎0,𝑎1,…} = (𝑥 − 𝑐)
𝑘𝑂𝑝𝑠𝑐{𝑎𝑛}. 

 
Proof: For n = 1 the statement is true due to theorem 7. Lets assume for n = k, the 
statement is also true, i.e (17) holds. For n = k+1, we have 
 

𝑂𝑝𝑠𝑐 { 0,0,0, . . ,0⏟      
𝑘+1−𝑒𝑛𝑡𝑟𝑖𝑒𝑠

𝑎0,𝑎1,…} = (𝑥 − 𝑐) 𝑂𝑝𝑠
𝑐 {0,0,0, . . ,0⏟      

𝑘−𝑒𝑛𝑡𝑟𝑖𝑒𝑠

𝑎0,𝑎1,…} 

  = (𝑥 − 𝑐)((𝑥 − 𝑐)𝑘𝑂𝑝𝑠𝑐{𝑎𝑛}) 
  = (𝑥 − 𝑐)𝑘+1𝑂𝑝𝑠𝑐{𝑎𝑛} 
 
Theorem 8. For each {𝑎𝑛}  sequence, we have  

(16) 

(17) 

(18) 



C-Type Ops Transformation 

Ahmad Lazwardi 406 

𝑂𝑝𝑠𝑐({𝑎𝑛})− 𝑎0 = (𝑥 − 𝑐)𝑂𝑝𝑠
𝑐{𝑎𝑛+1}. 

  
Proof: Observe that 
 

𝑂𝑝𝑠𝑐({𝑎𝑛})− 𝑎0 = ∑𝑎𝑛(𝑥 − 𝑐)
𝑛

∞

𝑛=1

 

= ∑𝑎𝑛+1

∞

𝑛=0

(𝑥 − 𝑐)𝑛+1 

= (𝑥 − 𝑐)∑𝑎𝑛+1

∞

𝑛=0

(𝑥 − 𝑐)𝑛 

 
= (𝑥 − 𝑐)𝑂𝑝𝑠𝑐{𝑎𝑛+1} 

 
 
Fortunately we also sucess to keep linearity properties of ops transformation[16]. 
  
Theorem 9. For each {𝑎𝑛},{𝑏𝑛} sequences and any real numbers  𝛼, 𝛽, we have 

𝑂𝑝𝑠𝑐(𝛼{𝑎𝑛} + 𝛽{𝑏𝑛}) = 𝛼𝑂𝑝𝑠
𝑐{𝑎𝑛}+ 𝛽𝑂𝑝𝑠

𝑐{𝑏𝑛}. 
 
 
Proof: Lets observe 
 

𝑂𝑝𝑠𝑐(𝛼{𝑎𝑛}+ 𝛽{𝑏𝑛}) = ∑(𝛼𝑎𝑛 + 𝛽𝑏𝑛)(𝑥 − 𝑐)
𝑛

∞

𝑛=0

 

 

= 𝛼 ∑𝑎𝑛

∞

𝑛=0

(𝑥 −𝑐)𝑛 + 𝛽 ∑𝑎𝑛

∞

𝑛=0

(𝑥 − 𝑐)𝑛 

 
= 𝛼𝑂𝑝𝑠𝑐{𝑎𝑛}+ 𝛽𝑂𝑝𝑠

𝑐{𝑏𝑛} 
Although we success to prove linearity of ops transformation, but unfortunately that 
linearity of upper index, i.e 
 

𝑂𝑝𝑠𝛼𝑐+𝛽𝑑{𝑎𝑛} ≠ 𝑂𝑝𝑠
𝛼𝑐{𝑎𝑛} +𝑂𝑝𝑠

𝛽𝑑{𝑏𝑛}. 
 
 
Next we shall observe properties of c-type ops transformation for product of two power 
series. Its still works similarily as previous result. 
 
 
Theorem 10. For each {𝑎𝑛},{𝑏𝑛} sequences, we have 

𝑂𝑝𝑠𝑐{𝑎𝑛}𝑂𝑝𝑠
𝑐{𝑏𝑛} = 𝑂𝑝𝑠

𝑐 {∑𝑎𝑘𝑏𝑛−𝑘

𝑛

𝑘=0

}. 

 
Proof: Let  {𝑎𝑛},{𝑏𝑛}  any two sequences, we have 

(19) 

(20) 



C-Type Ops Transformation 

Ahmad Lazwardi 407 

}{}{
n

c

n

c
bOpsaOps  = ...))()(...)()()((

2

210

2

210
 cxbcxbbcxacxaa  

 ......))()((...))((( 101100  cxbbcxacxbba  

 ...)(.)()()(
2

11

2

20011000
cxbacxbacxbacxbaba   

 ...))(())((
2

021120011000
 cxbababacxbababa  

 
n

n

n

k

knk
cxba )(

0 0









  



 

  

 








 




n

k

knk

c
baOps

0

 

 
From above theorem, we can conclude easily the following fact  
 
Example 3. (𝑂𝑝𝑠𝑐{1})2 = 𝑂𝑝𝑠𝑐{𝑛 + 1}. 
 
 
Translating the notation into sigma notation we get 
 

∑(𝑛 +1)(𝑥 − 𝑐)𝑛
∞

𝑛=0

= (
1

1 −(𝑥 − 𝑐)
)
2

 

 
 
From example 1 we can observe that how c-type ops transformations helps us to 
calculate, or manipulate some power series. 
Here is another example 
 

Example 4. (𝑂𝑝𝑠𝑐 {
𝟏

𝒏!
})
𝟐

= 𝑂𝑝𝑠𝒄 {∑
𝟏

𝒌!(𝒏−𝒌)!

𝒏
𝒌=𝟎 }. 

Therefore we concluce that 
 

∑ ∑(
1

𝑘!(𝑛 − 𝑘)!
)(𝑥 − 𝑐)𝑛

𝑛

𝑘=0

∞

𝑛=0

= 𝑒2𝑥−2𝑐 

 
Hence we can take another form of 𝑒𝑥 as following equation 
 

𝑒𝑥 = (∑ ∑(
𝑒𝑐

𝑘!(𝑛 − 𝑘)!
)(𝑥 − 𝑐)𝑛

𝑛

𝑘=0

∞

𝑛=0

)

1

2

 

 
As for last discussion we shall observe how c-type ops transformation properties when 
we take its derivatives. We still use 𝐷𝑥𝑂𝑝𝑠

𝑐{𝑎𝑛} to notate the derivative of ops 
transformation on its radius convergence.  
 
Consider the fact that the form 𝑥 − 𝑐 has the same derivative with x itself [17]. Therefore 
we still able to adapt the formula for derivative of previous ops transformation as below 
 
 



C-Type Ops Transformation 

Ahmad Lazwardi 408 

Theorem 11. For each {𝑎𝑛} sequence, we have  
𝐷𝑥𝑂𝑝𝑠

𝑐({𝑎𝑛}) = 𝑂𝑝𝑠
𝑐({(𝑛 + 1)𝑎𝑛+1}). 

 
Proof: Observe that 
 

𝐷𝑥𝑂𝑝𝑠
𝑐({𝑎𝑛}) =

𝑑

𝑑𝑥
∑𝑎𝑛(𝑥 − 𝑐)

𝑛

∞

𝑛=0

 

= ∑𝑛𝑎𝑛(𝑥 −𝑐)
𝑛−1

∞

𝑛=1

 

= ∑(𝑛 + 1)𝑎𝑛+1(𝑥 − 𝑐)
𝑛

∞

𝑛=0

 

= 𝑂𝑝𝑠𝑐{(𝑛 + 1)𝑎𝑛+1} 
 
 
Trivialy we also can conclude the next modification theorem 
 
Theorem 12. For each {𝑎𝑛} sequence, we have 

(𝑥 − 𝑐)𝐷𝑥𝑂𝑝𝑠
𝑐({𝑎𝑛}) = 𝑂𝑝𝑠

𝑐{𝑛𝑎𝑛}. 
 
Proof: Observe that 

(𝑥 − 𝑐)𝐷𝑥𝑂𝑝𝑠
𝑐({𝑎𝑛}) })1{()( 1 n

c
anOpscx   

  







0

1
)()1()(

n

n

n
cxancx  

  









0

1

1
)()1(

n

n

n
cxan  

  





0

)(
n

n

n
cxna  

  }{
n

c
naOps  

From the last two theorems, we also can inductively conclude the following two 
corollaries. 
 
Corollary 2. For each {𝑎𝑛} sequence, we have 
 

𝐷𝑥
𝑘𝑂𝑝𝑠𝑐{𝑎𝑛} = 𝑂𝑝𝑠

𝑐 {
(𝑛 + 𝑘)!

𝑛!
𝑎𝑛+𝑘}. 

where 𝐷𝑥
𝑘 is kth-derivative of ops transformation [15]. 

 
Corollary 3. For each {𝑎𝑛} sequence, we have 
 

𝑂𝑝𝑠𝑐{𝑛𝑘𝑎𝑛} = (𝑥 − 𝑐)𝐷𝑥((𝑥 − 𝑐)𝐷𝑥(…))(𝑥 −𝑐)𝐷𝑥𝑂𝑝𝑠
𝑐{𝑎𝑛}.⏟                              

𝑘−𝑡𝑖𝑚𝑒𝑠

 

 
At the end of this research, we will try our transformation to solve some ordinary 
differential equation with variable coefficient. This solution must be exist due to [18] For 
example the equation 

(21) 

(22) 

(23) 

(24) 



C-Type Ops Transformation 

Ahmad Lazwardi 409 

 02')1(2''  yyxy                                               (25) 

We will find the solution of (25) near c = 1 as following: 

Step 1: Assume the solution has the form 





0

)1(
n

n

n
xCy . 

Step 2: Transform (25) to ops transformation equation. 

}0{}{2}{)1(2}{
11112

OpsCOpsCOpsDxCOpsD
nnxnx
  

Step 3: Solve the equation 

}{2}{)1(2}{
2

nnxn

c

x
COpsCOpsDxCOpsD  }2{})1)(2{(

1

2

1

nn
nCOpsCnnOps 


 

 }2{
1

n
COps  

 }22)1)(2{(
2

1

nnn
CnCCnnOps 


 

 }0{
1

Ops  

Step 4: Remove ops transformation from the equation, we have  
(𝑛 + 2)(𝑛 + 1)𝐶𝑛+2 = (2𝑛 − 2)𝐶𝑛   for    n = 0,1,2, ... 

 
Step 5: By evaluating n one by one, we have the solution 

𝑦 = 𝐶0 (1 − (𝑥 −1)
2 −

1

6
(𝑥 −1)4 + ⋯) +𝐶1(𝑥 −1) 

 

CONCLUSIONS 

Based on the discussion above, it can be concluded that c-type ops transformation is a 
generalization of the ordinary ops transformation with additional c as the upper index. 
All properties of ordinary ops transformations can still apply in c-type ops 
transformations. The c-type ops transformation can also be applied to solve ordinary 
differential equations for variable coefficients.  

 

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