A NOTE ON MODIFIED DEGENERATE GAMMA AND LAPLACE TRANSFORMATION

Kim-Kim([9]) 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.

At first, L. Carlitz introduced the degenerate special polynomials (see [6,7]).The recently works which can be cited in this and researchers have studied the degenerate special polynomials and numbers (see [2,[8][9][10][11][12][13][14][15][16][17][18][19]).Recently, the concept of degenerate gamma function and degenerate Laplace transformation was introduced by Kim-Kim [2].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.

Modified Degenerate Gamma Function
In this section, we will define modified degenerate gamma functions which are different to degenerate gamma functions.For each λ ∈ (0, ∞), we define modified degenerate gamma function for the complex variable s with 0 < Re(s) as follows: Let λ ∈ (0, 1).Then, for 0 < Re(s), we have Therefore, by (18), we obtain the following theorem.

Modified Degenerate Laplace Transformation
In this section, we will define modified Laplace transformation which are different to degenerate Laplace transformation.Let λ ∈ (0, ∞) and let f (t) be a function defined for t ≥ 0. Then the integral is said to be the modified degenerate Laplace transformation of f if the integral converges which is also defined by where α and β are constant real numbers.First, we observe that for n ∈ N, (28) Therefore, by (28), we obtain the following theorem.Theorem 4. For k ∈ N and λ ∈ (0, 1), 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 Thus, by ( 32) and (33), we have Therefore, by ( 33) and (34), we obtain the following theorem.
Theorem 5.If f is a function defined t ≥ 0 and L * λ ( f (t)) exists, then we have where is the Heviside function.
Thirdly, we observe the modified degenerate Laplace transformation of f (t − a)U(t − a) as follows: Therefore, by (36), we obtain the following theorem.
Fourthly, we observe the modified degenerate Laplace transformation of the convolution f * g of two function f , g as follows: Therefore, by (38), we obtain the following theorem.
Theorem 8.For λ ∈ (0, 1], we have Fifthly, we observe that the modified degenerate Laplace transformation of derivative of f which is f and By using mathematical induction, we obtain the following theorem.
Theorem 10.For λ ∈ (0, 1] and 0 < Re(s), we have dF 4. Conclusions ) 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.