A NOTE ON MODIFIED DEGENERATE GAMMA AND LAPLACE TRANSFORMATION. 1. Introduction. It is well known that gamma function is defied by

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1 Preprints NOT PEER-REVIEWED Posted: 1 September 218 doi:1.2944/preprints v1 Peer-reviewed version available at Symmetry 218, 1, 471; doi:1.339/sym11471 A NOTE ON MODIFIED DEGENERATE GAMMA AND LAPLACE TRANSFORMATION YUNJAE KIM 1, BYUNG MOON KIM 2, LEE-CHAE JANG 3, AND JONGKYUM KWON 4 Abstract. 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. 1. Introduction It is well known that gamma function is defied by Γs e t t s 1 dt, where s C with Res >, see [5, 9]. 1.1 From 1.1, we note that Γs + 1 sγs, andγn + 1 n!, where n N. 1.2 Let ft be a function defined for t. Then, the integral Lft e st ftdt, see [3, 5, 9, 14] 1.3 is said to be the Laplace transform of f, provided that the integral converges. For,. Kim-Kim[9] introduced the degenerate gamma function for the complex variable s with < Res < 1 as follows: Γ s 1 + t 1 t s 1 dt, see [9] Mathematics Subject Classification. 11B68, 11S4, 11S8. Key words and phrases. The degenerate gamma function, The modified degenerate gamma function, The degenerate Laplace Transformation, The modified degenerate Laplace Transformation. 4 corresponding author by the authors. Distributed under a Creative Commons CC BY license.

2 Preprints NOT PEER-REVIEWED Posted: 1 September 218 doi:1.2944/preprints v1 Peer-reviewed version available at Symmetry 218, 1, 471; doi:1.339/sym YunJae Kim, Byung Moon Kim, Lee-Chae Jang, Jongkyum Kwon and degenerate Laplace transformation which was defined by L ft 1 + t s ftdt, see [9, 15] 1.5 if the integral converges. The authors obtained some properties and interesting formulas related to the degenerate gamma function. For examples, For, 1 and < Res < 1, s Γ s s 1 Γ s, and, k+s with k N and < Res <, Γ s + 1 ss 1 s k k1 k + 1 Γ s k, k+1 and for k N and, 1 k, Γ k k 1! k. 1.8 The authors obtained some formulas related to the degenerate Laplace transformation. For examples, and and and L 1 1, if s >, 1.9 s L 1 + t a 1, if s > a +, 1.1 s + a L cos at where cos t t it t it Furthermore, the authors obtained that L t n for n N and s > n + 1, and s s 2 + a 2, 1.11 a L sin at s 2 + a 2, 1.12 and sin t 1 2i 1 + t it 1 + t it n! s s 2 s ns n + 1, 1.13 L f n t ss + s + 2 s + n 1L 1 + t n ft n 1 n i 1 f i s + l 1. i t

3 Preprints NOT PEER-REVIEWED Posted: 1 September 218 doi:1.2944/preprints v1 Peer-reviewed version available at Symmetry 218, 1, 471; doi:1.339/sym11471 A note on modified degenerate gamma and Laplace Transformation 3 where f, f 1,, f n 1 are continuous on, and are of degenerate exponential order and f n t is piecewise continuous on,, and for n N. L log1 + t n ft 1 n n d ds n L s, 1.15 At first, L. Carlitz introduced the degenerate special polynomials see [1,2]. The recently works which can be cited in this and researchers have studied the degenerate special polynomials and numbers see [4,6-13,16,17]. Recently, the concept of degenerate gamma function and degenerate Laplace transform was introduced by Kim-Kim217. They studied some properties of the degenerate gamma and degenerate Laplace transformation and obtained their properties. We observe that does L f g L fl g 1.16 holds? Thus, we consider the modified degenerate Laplace transform which are satisfied 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 < Res as follows: Γ s Let, 1. Then, for < Res, we have Γ s t t s dt t log1 + 1 t s + log1 + log1 + sγ s. Therefore, by 2.2, we obtain the following theorem. 1 + t t s 1 dt. 2.1 s1 + t t s 1 dt 2.2

4 Preprints NOT PEER-REVIEWED Posted: 1 September 218 doi:1.2944/preprints v1 Peer-reviewed version available at Symmetry 218, 1, 471; doi:1.339/sym YunJae Kim, Byung Moon Kim, Lee-Chae Jang, Jongkyum Kwon Theorem 2.1. Let, 1. Then, for < Res, we have Γ s + 1 s log1 + Γ s. 2.3 Then, for < Res and, 1, repeatly we calculate Γ s + 1 s log1 + Γ s 2 s 1 log1 + 2 Γ s Thus, continuing this process, for < Res and, 1, we have Γ s + 1 k s 1 s k + 1 log1 + k Γ s k. 2.5 Therefore, by 2.5, we obtain the following theorem. Theorem 2.2. Let, 1. Then, for < Res, we have Γ s + 1 k s 1 s k + 1 log1 + k Γ s k. 2.6 Let us take s k + 1. Then, by Theorem 2.2, we get and Γ k + 2 k+1 k 2 log1 + k+1 Γ 1 Γ 1 k+1 k! log1 + k+1 Γ t dt log t log Therefore, by 2.7 and 2.8, we obtain the following theorem. Theorem 2.3. For k N and, 1, we have Γ k + 1 k+1 k!. 2.9 log1 + k+1

5 Preprints NOT PEER-REVIEWED Posted: 1 September 218 doi:1.2944/preprints v1 Peer-reviewed version available at Symmetry 218, 1, 471; doi:1.339/sym11471 A note on modified degenerate gamma and Laplace Transformation 5 3. Modified degenerate Laplace transformation In this section, we will define modified Laplace transformation which are different to degenerate Laplace transformation. Let, and let ft be a function defined for t. Then the integral L ft 1 + s t ftdt. 3.1 is said to be the modified degenerate Laplace transformation of f if the integral converges which is also defined by L ft F s. From 3.1, we get L αft + βgt αl ft + βl gt, 3.2 where α and β are constant real numbers. First, we observe that for n N, L t n 1 + s t t n dt log1 + s 1 + s t t n n s t t n 1 dt log1 + s n log1 + s L t n 1 n log1 + s log1 + s 1 + s t t n 1 n s t t n 2 dt log1 + s 2 nn 1L log1 + s t n 2 n n!l log1 + s 1 n+1 n!. log1 + s 3.3 Therefore, by 3.3, we obtain the following theorem. Theorem 3.1. For k N and, 1, we have n+1 L t n n!. 3.4 log1 + s

6 Preprints NOT PEER-REVIEWED Posted: 1 September 218 doi:1.2944/preprints v1 Peer-reviewed version available at Symmetry 218, 1, 471; doi:1.339/sym YunJae Kim, Byung Moon Kim, Lee-Chae Jang, Jongkyum Kwon Secondly, we note that if f is a periodic function with a period T. L ft 1 + s t ftdt 1 + s t ftdt s t ftdt T 1 + s t ftdt s t+t ft + T dt 1 + s t ftdt s T By 3.5, we get s T L ft Thus, by 3.6, we get L 1 ft s T 1 + s t ftdt s t ftdt s t ftdt. 3.7 We recall that the degenerate Bernoulli numbers are introduced as t t n B 1 + t n, n!, 3.8 Thus, by 3.7 and 3.8, we have S T 1 T S 1 T S n ST 1 + s T S S 1 n B n,s 1 n S n T n n!. Therefore, by 3.8 and 3.9, we obtain the following theorem. 3.9 Theorem 3.2. If f is a function defined t and L ft exists, then we have L ft 1 T S B n,s 1 n S n 1 + s t T n ftdt n 1 B n,s 1 n S n L T S Ut T ft, n {, for t a, where Ut a is the Heviside function. 1, for t a. n! 3.1

7 Preprints NOT PEER-REVIEWED Posted: 1 September 218 doi:1.2944/preprints v1 Peer-reviewed version available at Symmetry 218, 1, 471; doi:1.339/sym11471 A note on modified degenerate gamma and Laplace Transformation 7 Thirdly, we observe the modified degenerate Laplace transformation of ft aut a as follows: L ft aut a a 1 + s t ft aut adt 1 + s t ft adt 1 + s t+a 1 + s a ftdt 1 + s a L ft. 1 + s t ftdt 3.11 Therefore, by 3.11, we obtain the following theorem. Theorem 3.3. For, 1 and a, we have L ft aut a 1 + s a L ft, 3.12 where Ut a is the Heviside function. Fourthly, we observe the modified degenerate Laplace transformation of the convolution f g of two function f, g as follows: L fl g 1 + s t ftdt 1 + s τ gτdτ ft τ L f g. 1 + s t+τ τ ftgτdtdτ 1 + s µ gµ τdµdτ ft1 + s µ gµ τdµdτ f g 1 + s µ dµ Therefore, by 3.13, we obtain the following theorem. Theorem 3.4. For, 1], we have 3.13 L f g L fl g. 3.14

8 Preprints NOT PEER-REVIEWED Posted: 1 September 218 doi:1.2944/preprints v1 Peer-reviewed version available at Symmetry 218, 1, 471; doi:1.339/sym YunJae Kim, Byung Moon Kim, Lee-Chae Jang, Jongkyum Kwon We note that By 3.15, we have L s t 1dt log1 + s 1 + s t log1 + s L f 1 L fl 1 L f log1 + s Therefore, by 3.16, we obtain the following theorem. Theorem 3.5. For, 1], we have L 1 L f log1 + s f 1t t ftdt Fifthly, we observe that the modified degenerate Laplace transformation of derivative of f which is ft 1 + s t, where ft ut means and L f 1 + s t f dt 1 + s t ft log1 + s s t ftdt log1 + s f + L f L f s t f 2 dt 1 + s t f t log1 + s + log1 + s f + f + 2 log1 + s L f 1 + s t f tdt log1 + s L f log1 + s f f. By using mathematical induction, we obtain the following theorem. 3.19

9 Preprints NOT PEER-REVIEWED Posted: 1 September 218 doi:1.2944/preprints v1 Peer-reviewed version available at Symmetry 218, 1, 471; doi:1.339/sym11471 A note on modified degenerate gamma and Laplace Transformation 9 Theorem 3.6. For, 1], we have n n 1 log1 + s n 1 i log1 + s L f n L f f i. 3.2 Finally, we observe df ds s i 1 + s t 1 + s t ftdt s L tft. By 3.21, we obtain the following theorem. 1 + s t tftdt Theorem 3.7. For, 1] and < Res, we have 3.21 df ds s L tft Conclusion 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 Theorem 3.4 and Theorem 3.5 for the modified degenerate Laplace transformation. We investigated some results which are Theorem 2.1 and Theorem 2.3 for modified degenerate gamma functions. Furthermore, Theorem 3.3 and Theorem 3.6 are some interesting properties which are applied to differential equations in engineering mathematics. Competing interests The authors declare that they have no competing interests. Authors 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. 217R1E1A1A37882.

10 Preprints NOT PEER-REVIEWED Posted: 1 September 218 doi:1.2944/preprints v1 Peer-reviewed version available at Symmetry 218, 1, 471; doi:1.339/sym YunJae Kim, Byung Moon Kim, Lee-Chae Jang, Jongkyum Kwon References 1. L. Carlitz, A degenerate Staudt-Clausen Theorem, Arch. Math.Basel , L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Util. Math., , W. S. Chung, T. Kim, H. I. Kwon, On the q-analog of the Laplace transform, Russ. J. Math. Phys., , no. 2, D. V. Dolgy, T. Kim, J. -J. Seo, On the symmetric identities of modified degenerate Bernoulli polynomials, Proc. Jangjeon Math. Soc., , no. 2, E. Kreyszig, H. Kreyszig, E.J. Norminton, Advanced Engineering Mathematics, John Wiley & Sons Inc., New Jersey, D. S. Kim, T. Kim, Some identities of degenerate Euler polynomials arising from p-adic fermionic integral on Z p, Integral Transforms Spec. Funct., , T. Kim, Degenerate Euler Zeta function, Russ. J. Math. Phys., , no. 4, T. Kim, On the degenerate q-bernoulli polynomials, Bull. Korean Math. Soc., , no.4, T. Kim, D. S. Kim, Degenerate Laplace transform and degenerate Gamma function, Russ. J. Math. Phys., , no. 2, T. Kim, D. S. Kim, D. V. Dolgy, Degenerate q-euler polynomials, Adv. Difference Equ., , T. Kim, D. V. Dolgy, D. S. Kim, Symmetric identities for degenerate generalized Bernoulli polynomials, J. Nonlinear Sci. and Appl., 9 216, T. Kim, D. S. Kim, J. J. Seo, Differential equations associated with degenerate Bell polynomials, Inter. J. Pure and Appl. Math., , no. 3, H. I. Kwon, T. Kim, J. -J. Seo, A note on degenerate Changhee numbers and polynomials, Proc. Jangjeon Math. Soc., , no. 3, M.R. Spiegel, Laplace TransformsSchaum s Outlines, McGraw Hill, New York, L. M. Upadhyaya, On the degenerate Laplace transform, Int. J. of Eng. & Sci. Research, 6 218, no. 2, D. S. Kim, T. Kim, Some p-adic integrals on Z p associated with trigonometric functions, Russ. J. Math.Phys , no.3, T. Kim, D. S. Kim Identities for degenerate Bernoulli polynomials and Korobov polynomials, Science China Mathematics 1.17/s see: 1 Department of Mathematics, Dong-A University, Busan, 49315, Republic of Korea address: kimholzi@gmail.com 2 Department of Mechanical System Engineering, Dongguk University, Gyungjusi,Gyeongsangbukdo,3866, Republic of Korea address: kbm713@dongguk.ac.kr 3 Graduate School of Education, Konkuk University, Seoul, , Republic of Korea. address: lcjang@konkuk.ac.kr 4 Department of Mathematics Education and ERI, Gyeongsang National University, Jinju, Gyeongsangnamdo, 52828, Republic of Korea address: mathkjk26@gnu.ac.kr