Abstract
Kim-Kim studied some properties of the degenerate gamma and degenerate Laplace transformation and obtained their properties. In this paper, we define modified degenerate gamma and modified degenerate Laplace transformation and investigate some properties and formulas related to them.
1. Introduction
Let be a function defined for . Then, the integral
(see [,,,]), is said to be the Laplace transform of f, provided that the integral converges. For . Kim-Kim [] introduced the degenerate gamma function for the complex variable s with as follows:
(see []) and degenerate Laplace transformation which was defined by
(see [,]), if the integral converges. The authors obtained some properties and interesting formulas related to the degenerate gamma function. For examples, For and ,
and with and ,
and for and ,
The authors obtained some formulas related to the degenerate Laplace transformation. For examples,
and
and
and
where and .
Furthermore, the authors obtained that
for and , and
where are continuous on and are of degenerate exponential order and is piecewise continuous on , and
for .
At first, L. Carlitz introduced the degenerate special polynomials (see [,]). The recently works which can be cited in this and researchers have studied the degenerate special polynomials and numbers (see [,,,,,,,,,,,,]). Recently, the concept of degenerate gamma function and degenerate Laplace transformation was introduced by Kim-Kim []. They studied some properties of the degenerate gamma and degenerate Laplace transformation and obtained their properties. We observe whether or not that holds. Thus, we consider the modified degenerate Laplace transform which are satisfied (16). The degenerate gamma and degenerate Laplace transformation applied to engineer’s mathematical toolbox as they make solving linear ODEs and related initial value problems. This paper consists of two sections. The first section contains the modified degenerate gamma function and investigate the properties of the modified gamma function. The second part of the paper provide the modified degenerate Laplace transformation and investigate interesting results of the modified degenerate Laplace transformation.
2. Modified Degenerate Gamma Function
In this section, we will define modified degenerate gamma functions which are different to degenerate gamma functions. For each , we define modified degenerate gamma function for the complex variable s with as follows:
Let . Then, for , we have
Therefore, by (18), we obtain the following theorem.
Theorem 1.
Let . Then, for , we have
Then, for and , repeatly we calculate
Thus, continuing this process, for and , we have
Therefore, by (21), we obtain the following theorem.
Theorem 2.
Let . Then, for , we have
Let us take . Then, by Theorem 2, we get
and
Theorem 3.
For and , we have
3. Modified Degenerate Laplace Transformation
In this section, we will define modified Laplace transformation which are different to degenerate Laplace transformation. Let and let be a function defined for . Then the integral
is said to be the modified degenerate Laplace transformation of f if the integral converges which is also defined by .
From (26), we get
where α and β are constant real numbers.
First, we observe that for ,
Therefore, by (28), we obtain the following theorem.
Theorem 4.
For and , we have
Secondly, we note that if f is a periodic function with a period T.
By (30), we get
Thus, by (31), we get
We recall that the degenerate Bernoulli numbers are introduced as
Theorem 5.
If f is a function defined and exists, then we have
where is the Heviside function.
Thirdly, we observe the modified degenerate Laplace transformation of as follows:
Therefore, by (36), we obtain the following theorem.
Theorem 6.
For and we have
where is the Heviside function.
Fourthly, we observe the modified degenerate Laplace transformation of the convolution of two function f, g as follows:
Therefore, by (38), we obtain the following theorem.
Theorem 7.
For , we have
We note that
By (40), we have
Therefore, by (41), we obtain the following theorem.
Theorem 8.
For , we have
Fifthly, we observe that the modified degenerate Laplace transformation of derivative of f which is , where means
and
By using mathematical induction, we obtain the following theorem.
Theorem 9.
For , we have
Finally, we observe
By (46), we obtain the following theorem.
Theorem 10.
For and , we have
4. Conclusions
Kim-Kim ([9]) defined a degenerate gamma function and a degenerate Laplace transformation. The motivation of this paper is to define modified degenerate gamma functions and modified degenerate Laplace transformations which are different to degenerate gamma function and degenerate Laplace transformation and to obtain more useful results which are Theorems 7 and 8 for the modified degenerate Laplace transformation. We do not obtain these result from the degenerate Laplace transformation. Also, we investigated some results which are Theorems 1 and 3 for modified degenerate gamma functions. Furthermore, Theorems 6 and 9 are some interesting properties which are applied to differential equations in engineering mathematics.
Author Contributions
All authors contributed equally to this work. All authors read and approved the final manuscript.
Funding
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2017R1E1A1A03070882).
Conflicts of Interest
The authors declare that they have no competing interests.
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