Open AccessArticle

A Note on Modified Degenerate Gamma and Laplace Transformation

YunJae Kim

¹,

Byung Moon Kim

²,

Lee-Chae Jang

³ and

Jongkyum Kwon

^4,*

Department of Mathematics, Dong-A University, Busan 49315, Korea

Department of Mechanical System Engineering, Dongguk University, Gyungju-si, Gyeongsangbukdo 38066, Korea

Graduate School of Education, Konkuk University, Seoul 139-701, Korea

⁴

Department of Mathematics Education and ERI, Gyeongsang National University, Jinju, Gyeongsangnamdo 52828, Korea

Author to whom correspondence should be addressed.

Symmetry 2018, 10(10), 471; https://doi.org/10.3390/sym10100471

Submission received: 8 September 2018 / Revised: 1 October 2018 / Accepted: 3 October 2018 / Published: 10 October 2018

(This article belongs to the Special Issue Current Trends in Symmetric Polynomials with their Applications)

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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.

Keywords:

the degenerate gamma function; the modified degenerate gamma function; the degenerate Laplace transform; the modified degenerate Laplace transform

1. Introduction

It is well known that gamma function is defied by

Γ (s) = \int_{0}^{\infty} e^{- t} t^{s - 1} d t, where s \in C with R e (s) > 0,

(1)

(see [1,2]). From (1), we note that

Γ (s + 1) = s Γ (s), and Γ (n + 1) = n!, where n \in N .

(2)

Let

f (t)

be a function defined for

t \geq 0

. Then, the integral

L (f (t)) = \int_{0}^{\infty} e^{- s t} f (t) d t,

(3)

(see [1,2,3,4]), is said to be the Laplace transform of f, provided that the integral converges. For

λ \in (0, \infty)

. Kim-Kim [2] introduced the degenerate gamma function for the complex variable s with

0 < R e (s) < \frac{1}{λ}

as follows:

Γ_{λ} (s) = \int_{0}^{\infty} {(1 + λ t)}^{- \frac{1}{λ}} t^{s - 1} d t,

(4)

(see [2]) and degenerate Laplace transformation which was defined by

L_{λ} (f (t)) = \int_{0}^{\infty} {(1 + λ t)}^{- \frac{s}{λ}} f (t) d t,

(5)

(see [2,5]), if the integral converges. The authors obtained some properties and interesting formulas related to the degenerate gamma function. For examples, For

λ \in (0, 1)

and

0 < R e (s) < \frac{1 - λ}{λ}

Γ_{λ} (s + 1) = \frac{s}{{(1 - λ)}^{s - 1}} Γ_{\frac{λ}{1 - λ}} (s),

(6)

and

λ \in (0, \frac{1}{k + s})

with

k \in N

and

0 < R e (s) < \frac{1 - λ}{λ}

Γ_{λ} (s + 1) = \frac{s (s - 1) \dots (s - (k + 1) + 1)}{(1 - λ) (1 - 2 λ) \dots (1 - k λ) (1 - (k + 1) λ)} Γ_{\frac{λ}{1 - (k + 1) λ}} (s - k),

(7)

and for

k \in N

and

λ \in (0, \frac{1}{k})

Γ_{λ} (k) = \frac{(k - 1)!}{(1 - λ) (1 - 2 λ) \dots (1 - k λ)} .

(8)

The authors obtained some formulas related to the degenerate Laplace transformation. For examples,

L_{λ} (1) = \frac{1}{s - λ}, if s > λ,

(9)

and

L_{λ} ({(1 + λ t)}^{- \frac{a}{λ}}) = \frac{1}{s + a - λ}, if s > - a + λ,

(10)

and

L_{λ} ({cos}_{λ} (a t)) = \frac{s - λ}{{(s - λ)}^{2} + a^{2}},

(11)

and

L_{λ} ({sin}_{λ} (a t)) = \frac{a}{{(s - λ)}^{2} + a^{2}},

(12)

where

{cos}_{λ} (t) = \frac{1}{2} ({(1 + λ t)}^{\frac{i t}{λ}} + {(1 + λ t)}^{- \frac{i t}{λ}})

and

{sin}_{λ} (t) = \frac{1}{2 i} ({(1 + λ t)}^{\frac{i t}{λ}} - {(1 + λ t)}^{- \frac{i t}{λ}})

Furthermore, the authors obtained that

L_{λ} (t^{n}) = \frac{n!}{(s - λ) (s - 2 λ) \dots (s - n λ) (s - (n + 1) λ)},

(13)

for

n \in N

and

s > (n + 1) λ

, and

\begin{matrix} L_{λ} (f^{(n)} (t)) & = s (s + λ) (s + 2 λ) \dots (s + (n - 1) λ) L_{λ} ({(1 + λ t)}^{- n} f (t)) \\ - \sum_{i = 0}^{n - 1} f^{(i)} (0) (\prod_{t = 1}^{n - i - 1} s + (l - 1) λ) . \end{matrix}

(14)

where

f, f^{(1)}, \dots, f^{(n - 1)}

are continuous on

(0, \infty)

and are of degenerate exponential order and

f^{(n)} (t)

is piecewise continuous on

(0, \infty)

, and

L_{λ} ({(log (1 + λ t))}^{n} f (t)) = {(- 1)}^{n} λ^{n} {(\frac{d}{d s})}^{n} L_{λ} (s),

(15)

for

n \in N

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.

L_{λ} (f * g) = L_{λ} (f) L_{λ} (g)

(16)

2. Modified Degenerate Gamma Function

In this section, we will define modified degenerate gamma functions which are different to degenerate gamma functions. For each

λ \in (0, \infty)

, we define modified degenerate gamma function for the complex variable s with

0 < R e (s)

as follows:

Γ_{λ}^{*} (s) = \int_{0}^{\infty} {(1 + λ)}^{- \frac{t}{λ}} t^{s - 1} d t .

(17)

Let

λ \in (0, 1)

. Then, for

0 < R e (s)

, we have

\begin{matrix} Γ_{λ}^{*} (s + 1) & = \int_{0}^{\infty} {(1 + λ)}^{- \frac{t}{λ}} t^{s} d t \\ = \frac{1}{(log {(1 + λ)}^{- \frac{1}{λ}}} {(1 + λ)}^{- \frac{t}{λ}} t^{s} ∣_{0}^{\infty} + \frac{λ}{log (1 + λ)} \int_{0}^{\infty} s {(1 + λ)}^{- \frac{t}{λ}} t^{s - 1} d t \\ = \frac{λ}{log (1 + λ)} s Γ_{λ}^{*} (s) . \end{matrix}

(18)

Therefore, by (18), we obtain the following theorem.

Theorem 1.

Let

λ \in (0, 1)

. Then, for

0 < R e (s)

, we have

Γ_{λ}^{*} (s + 1) = \frac{λ s}{log (1 + λ)} Γ_{λ}^{*} (s) .

(19)

Then, for

0 < R e (s)

and

λ \in (0, 1)

, repeatly we calculate

Γ_{λ}^{*} (s + 1) = \frac{λ s}{log (1 + λ)} Γ_{λ}^{*} (s) = \frac{λ^{2} (s - 1)}{{(log (1 + λ))}^{2}} Γ_{λ}^{*} (s - 1) .

(20)

Thus, continuing this process, for

0 < R e (s)

and

λ \in (0, 1)

, we have

Γ_{λ}^{*} (s + 1) = \frac{λ^{k} (s - 1) \dots (s - k + 1)}{{(log (1 + λ))}^{k}} Γ_{λ}^{*} (s - k) .

(21)

Therefore, by (21), we obtain the following theorem.

Theorem 2.

Let

λ \in (0, 1)

. Then, for

0 < R e (s)

, we have

Γ_{λ}^{*} (s + 1) = \frac{λ^{k} (s - 1) \dots (s - k + 1)}{{(log (1 + λ))}^{k}} Γ_{λ}^{*} (s - k) .

(22)

Let us take

s = k + 1

. Then, by Theorem 2, we get

\begin{matrix} Γ_{λ}^{*} (k + 2) & = \frac{λ^{k + 1} k \dots 2}{{(log (1 + λ))}^{k + 1}} Γ_{λ}^{*} (1) \\ = \frac{λ^{k + 1} k!}{{(log (1 + λ))}^{k + 1}} Γ_{λ}^{*} (1) \end{matrix}

(23)

and

\begin{matrix} Γ_{λ}^{*} (1) & = \int_{0}^{\infty} {(1 + λ)}^{- \frac{t}{λ}} d t \\ = - \frac{λ}{(log (1 + λ)} {(1 + λ)}^{- \frac{t}{λ}} ∣_{0}^{\infty} \\ = \frac{λ}{(log (1 + λ)} . \end{matrix}

(24)

Therefore, by (23) and (24), we obtain the following theorem.

Theorem 3.

For

k \in N

and

λ \in (0, 1)

, we have

Γ_{λ}^{*} (k + 1) = \frac{λ^{k + 1} k!}{{(log (1 + λ))}^{k + 1}} .

(25)

3. Modified Degenerate Laplace Transformation

In this section, we will define modified Laplace transformation which are different to degenerate Laplace transformation. Let

λ \in (0, \infty)

and let

f (t)

be a function defined for

t \geq 0

. Then the integral

L_{λ}^{*} (f (t)) = \int_{0}^{\infty} {(1 + λ s)}^{- \frac{t}{λ}} f (t) d t .

(26)

is said to be the modified degenerate Laplace transformation of f if the integral converges which is also defined by

L_{λ}^{*} (f (t)) = F_{λ} (s)

From (26), we get

L_{λ}^{*} (α f (t) + β g (t)) = α L_{λ}^{*} (f (t)) + β L_{λ}^{*} (g (t)),

(27)

where α and β are constant real numbers.

First, we observe that for

n \in N

\begin{matrix} L_{λ}^{*} (t^{n}) & = \int_{0}^{\infty} {(1 + λ s)}^{- \frac{t}{λ}} t^{n} d t \\ = - \frac{λ}{log (1 + λ s)} {(1 + λ s)}^{- \frac{t}{λ}} t^{n} ∣_{0}^{\infty} + \frac{λ n}{log (1 + λ s)} \int_{0}^{\infty} {(1 + λ s)}^{- \frac{t}{λ}} t^{n - 1} d t \\ = \frac{λ n}{log (1 + λ s)} L_{λ}^{*} (t^{n - 1}) \\ = \frac{λ n}{log (1 + λ s)} (- \frac{λ}{log (1 + λ s)} {(1 + λ s)}^{- \frac{t}{λ}} t^{n - 1} ∣_{0}^{\infty} + \frac{λ (n - 1)}{log (1 + λ s)} \int_{0}^{\infty} {(1 + λ s)}^{- \frac{t}{λ}} t^{n - 2} d t) \\ = {(\frac{λ}{log (1 + λ s)})}^{2} n (n - 1) L_{λ}^{*} (t^{n - 2}) \\ = \dots \\ = {(\frac{λ}{log (1 + λ s)})}^{n} n! L_{λ}^{*} (1) \\ = {(\frac{λ}{log (1 + λ s)})}^{n + 1} n! . \end{matrix}

(28)

Therefore, by (28), we obtain the following theorem.

Theorem 4.

For

k \in N

and

λ \in (0, 1)

, we have

L_{λ}^{*} (t^{n}) = {(\frac{λ}{log (1 + λ s)})}^{n + 1} n! .

(29)

Secondly, we note that if f is a periodic function with a period T.

\begin{matrix} L_{λ}^{*} (f (t)) & = \int_{0}^{\infty} {(1 + λ s)}^{- \frac{t}{λ}} f (t) d t \\ = \int_{0}^{T} {(1 + λ s)}^{- \frac{t}{λ}} f (t) d t + \int_{T}^{\infty} {(1 + λ s)}^{- \frac{t}{λ}} f (t) d t \\ = \int_{0}^{T} {(1 + λ s)}^{- \frac{t}{λ}} f (t) d t + \int_{0}^{\infty} {(1 + λ s)}^{- \frac{t + T}{λ}} f (t + T) d t \\ = \int_{0}^{T} {(1 + λ s)}^{- \frac{t}{λ}} f (t) d t + {(1 + λ s)}^{- \frac{T}{λ}} \int_{0}^{\infty} {(1 + λ s)}^{- \frac{t}{λ}} f (t) d t \end{matrix}

(30)

By (30), we get

(1 - {(1 + λ s)}^{- \frac{T}{λ}}) L_{λ}^{*} (f (t)) = \int_{0}^{T} {(1 + λ s)}^{- \frac{t}{λ}} f (t) d t .

(31)

Thus, by (31), we get

L_{λ}^{*} (f (t)) = \frac{1}{(1 - {(1 + λ s)}^{- \frac{T}{λ}})} \int_{0}^{T} {(1 + λ s)}^{- \frac{t}{λ}} f (t) d t .

(32)

We recall that the degenerate Bernoulli numbers are introduced as

\frac{t}{{(1 + λ)}^{- \frac{t}{λ}}} = \sum_{n = 0}^{\infty} B_{n, λ} \frac{t^{n}}{n!},

(33)

Thus, by (32) and (33), we have

\begin{matrix} \frac{1}{1 - {(1 + λ S)}^{- \frac{T}{λ}}} & = - \frac{1}{T S} \frac{S T}{{(1 + λ s)}^{- \frac{T S}{λ S}} - 1} \\ = - \frac{1}{T S} \sum_{n = 0}^{\infty} B_{n, λ S} {(- 1)}^{n} S^{n} \frac{T^{n}}{n!} . \end{matrix}

(34)

Therefore, by (33) and (34), we obtain the following theorem.

Theorem 5.

If f is a function defined

t \geq 0

and

L_{λ}^{*} (f (t))

exists, then we have

\begin{matrix} L_{λ}^{*} (f (t)) & = - \frac{1}{T S} \sum_{n = 0}^{\infty} B_{n, λ S} {(- 1)}^{n} S^{n} \int_{0}^{T} {(1 + λ s)}^{- \frac{t}{λ}} f (t) d t \frac{T^{n}}{n!} \\ = - \frac{1}{T S} \sum_{n = 0}^{\infty} B_{n, λ S} {(- 1)}^{n} S^{n} L_{λ}^{*} (U (t - T) f (t)), \end{matrix}

(35)

where

U (t - a) = \{\begin{matrix} 0, for 0 \geq t \geq a, \\ 1, for t \leq a . \end{matrix}

is the Heviside function.

Thirdly, we observe the modified degenerate Laplace transformation of

f (t - a) U (t - a)

as follows:

\begin{matrix} L_{λ}^{*} (f (t - a) U (t - a)) & = \int_{0}^{\infty} {(1 + λ s)}^{- \frac{t}{λ}} f (t - a) U (t - a) d t \\ = \int_{a}^{\infty} {(1 + λ s)}^{- \frac{t}{λ}} f (t - a) d t \\ = \int_{0}^{\infty} {(1 + λ s)}^{- \frac{t + a}{λ}} f (t) d t \\ = {(1 + λ s)}^{- \frac{a}{λ}} \int_{0}^{\infty} {(1 + λ s)}^{- \frac{t}{λ}} f (t) d t \\ = {(1 + λ s)}^{- \frac{a}{λ}} L_{λ}^{*} (f (t)) . \end{matrix}

(36)

Therefore, by (36), we obtain the following theorem.

Theorem 6.

For

λ \in (0, 1)

and

a \in (0, \infty)

we have

L_{λ}^{*} (f (t - a) U (t - a)) = {(1 + λ s)}^{- \frac{a}{λ}} L_{λ}^{*} (f (t)),

(37)

where

U (t - 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:

\begin{matrix} L_{λ}^{*} (f) L_{λ}^{*} (g) & = (\int_{0}^{\infty} {(1 + λ s)}^{- \frac{t}{λ}} f (t) d t) (\int_{0}^{\infty} {(1 + λ s)}^{- \frac{τ}{λ}} g (τ) d τ) \\ = \int_{0}^{\infty} \int_{0}^{\infty} {(1 + λ s)}^{- \frac{t + τ}{λ}} f (t) g (τ) d t d τ \\ = \int_{0}^{\infty} f (t) \int_{τ}^{\infty} {(1 + λ s)}^{- \frac{μ}{λ}} g (μ - τ) d μ d τ \\ = \int_{0}^{\infty} \int_{τ}^{\infty} f (t) {(1 + λ s)}^{- \frac{μ}{λ}} g (μ - τ) d μ d τ \\ = \int_{0}^{\infty} (f * g) {(1 + λ s)}^{- \frac{μ}{λ}} d μ \\ = L_{λ} (f * g) . \end{matrix}

(38)

Therefore, by (38), we obtain the following theorem.

Theorem 7.

For

λ \in (0, 1]

, we have

L_{λ}^{*} (f * g) = L_{λ}^{*} (f) L_{λ}^{*} (g) .

(39)

We note that

\begin{matrix} L_{λ}^{*} (1) & = \int_{0}^{\infty} {(1 + λ s)}^{- \frac{t}{λ}} 1 d t \\ = - \frac{λ}{log (1 + λ s)} {(1 + λ s)}^{- \frac{t}{λ}} ∣_{0}^{\infty} \\ = \frac{λ}{log (1 + λ s)} . \end{matrix}

(40)

By (40), we have

\begin{matrix} L_{λ}^{*} (f * 1) = L_{λ}^{*} (f) L_{λ}^{*} (1) = L_{λ}^{*} (f) \frac{λ}{log (1 + λ s)} . \end{matrix}

(41)

Therefore, by (41), we obtain the following theorem.

Theorem 8.

For

λ \in (0, 1]

, we have

L_{λ}^{* - 1} (L_{λ}^{*} (f) \frac{λ}{log (1 + λ s)}) = f * 1 (t) = \int_{0}^{t} f (t) d t .

(42)

Fifthly, we observe that the modified degenerate Laplace transformation of derivative of f which is

f (t) = 0 ({(1 + λ s)}^{- \frac{t}{λ}})

, where

f (t) = 0 (u (t))

means

\begin{matrix} L_{λ}^{*} (f^{'}) & = \int_{0}^{\infty} {(1 + λ s)}^{- \frac{t}{λ}} f^{'} d t \\ = {(1 + λ s)}^{- \frac{t}{λ}} f (t) ∣_{0}^{\infty} + \int_{0}^{\infty} \frac{log (1 + λ s)}{λ} {(1 + λ s)}^{- \frac{t}{λ}} f (t) d t \\ = - f (0) + \frac{log (1 + λ s)}{λ} L_{λ}^{*} (f) . \end{matrix}

(43)

and

\begin{matrix} L_{λ}^{*} (f^{(2)}) & = \int_{0}^{\infty} {(1 + λ s)}^{- \frac{t}{λ}} f^{(2)} d t \\ = {(1 + λ s)}^{- \frac{t}{λ}} f^{'} (t) ∣_{0}^{\infty} + \frac{log (1 + λ s)}{λ} \int_{0}^{\infty} {(1 + λ s)}^{- \frac{t}{λ}} f^{'} (t) d t \\ = - f (0) + \frac{log (1 + λ s)}{λ} (- f (0) + \frac{log (1 + λ s)}{λ} L_{λ}^{*} (f)) \\ = {(\frac{log (1 + λ s)}{λ})}^{2} L_{λ}^{*} (f) - \frac{log (1 + λ s)}{λ} f (0) - f^{'} (0) . \end{matrix}

(44)

By using mathematical induction, we obtain the following theorem.

Theorem 9.

For

λ \in (0, 1]

, we have

L_{λ}^{*} (f^{(n)}) = {(\frac{log (1 + λ s)}{λ})}^{n} L_{λ}^{*} (f) - \sum_{i = 0}^{n - 1} {(\frac{log (1 + λ s)}{λ})}^{n - 1 - i} f^{(i)} (0) .

(45)

Finally, we observe

\begin{matrix} \frac{d F_{λ}^{*}}{d s} & = \int_{0}^{\infty} \frac{λ}{1 + λ s} (- \frac{t}{λ}) {(1 + λ s)}^{- \frac{t}{λ}} f (t) d t \\ = - \frac{1}{1 + λ s} \int_{0}^{\infty} {(1 + λ s)}^{- \frac{t}{λ}} t f (t) d t \\ = - \frac{1}{1 + λ s} L_{λ}^{*} (t f (t)) . \end{matrix}

(46)

By (46), we obtain the following theorem.

Theorem 10.

For

λ \in (0, 1]

and

0 < R e (s)

, we have

\frac{d F_{λ}^{*}}{d s} = - \frac{1}{1 + λ s} L_{λ}^{*} (t f (t)) .

(47)

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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Kim, Y.; Kim, B.M.; Jang, L.-C.; Kwon, J. A Note on Modified Degenerate Gamma and Laplace Transformation. Symmetry 2018, 10, 471. https://doi.org/10.3390/sym10100471

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Kim Y, Kim BM, Jang L-C, Kwon J. A Note on Modified Degenerate Gamma and Laplace Transformation. Symmetry. 2018; 10(10):471. https://doi.org/10.3390/sym10100471

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Kim, YunJae, Byung Moon Kim, Lee-Chae Jang, and Jongkyum Kwon. 2018. "A Note on Modified Degenerate Gamma and Laplace Transformation" Symmetry 10, no. 10: 471. https://doi.org/10.3390/sym10100471

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Kim, Y., Kim, B. M., Jang, L. -C., & Kwon, J. (2018). A Note on Modified Degenerate Gamma and Laplace Transformation. Symmetry, 10(10), 471. https://doi.org/10.3390/sym10100471

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A Note on Modified Degenerate Gamma and Laplace Transformation

Abstract

1. Introduction

2. Modified Degenerate Gamma Function

3. Modified Degenerate Laplace Transformation

4. Conclusions

Author Contributions

Funding

Conflicts of Interest

References

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