EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 4, 2023, 2213-2233 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Degenerate Laplace-type Integral Transform Harren J. Campos1, Jezer C. Fernandez1,∗, Jade Bong M. Natuil1 1 Department of Mathematics, Mindanao State University, 9700 Marawi City, Philippines Abstract. This paper is motivated by the work of Taekyun Kim and Dae San Kim on the de- generate Laplace transform and degenerate gamma function, as published in the Russian Journal of Mathematical Physics. We introduce the degenerate Laplace-type integral transform and delve into its properties and interrelations. This paper focuses on the degenerate Laplace-type integral transforms of several fundamental functions, including the degenerate sine, degenerate cosine, de- generate hyperbolic sine, and degenerate hyperbolic cosine functions. Furthermore, we establish crucial connections between the degenerate Laplace-type integral transform and existing degen- erate integral transforms. Specifically, we explore its relationships with the degenerate Laplace transform, the degenerate Elzaki transform, and the degenerate Sumudu transforms. 2020 Mathematics Subject Classifications: 44A99 Key Words and Phrases: Degenerate Sumudu transform, Degenerate Elzaki transform, Degen- erate Laplace transform, Degenerate Laplace-type Integral Transform 1. Introduction Integral transforms have long captivated the mathematical world due to their multi- faceted properties and widespread applications across diverse scientific fields. During the 20th and 21st centuries, the Laplace transform has been extensively studied and employed in various scientific disciplines. Among the noteworthy contributions in this domain is the investigation of work the intrinsic structure and properties of Laplace-typed integral trans- forms by H. Kim [7] and some additional properties of Laplace-type integral transforms by H. Kim et.al[5] . This integral transform is defined as Fα(u) = Gα{f(t)} = uα ∫ ∞ 0 e −t u f(t)dt, where α ∈ Z. In recent years, there has been growing interest in degenerate versions of existing inte- gral transforms. Pioneering work of T. Kim and D. S. Kim [8] introduced the concept of ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i4.4868 Email addresses: harren.campos@msumain.edu.ph (H. Campos), jezercastro.fernandez@msumain.edu.ph (J. Fernandez), natuil.jadebong@gmail.com (J. B. Natuil) https://www.ejpam.com 2213 © 2023 EJPAM All rights reserved. H. J. Campos, J. C. Fernandez, J. B. M. Natuil / Eur. J. Pure Appl. Math, 16 (4) (2023), 2213-2233 2214 degenerate gamma functions and degenerate Laplace transforms, as well as the derivation of fundamental properties. Subsequently, L. M. Upadhyaya [14–16] further delved into properties of the degenerate Laplace transform, while U. Duran [4] investigated the de- generate Sumudu transform and L. M. Upadhyaya et.al [1] defined the degenerate Elzaki transform and its properties. In light of the growing significance of degenerate integral transforms, this article seeks to contribute to this research by introducing the degenerate Laplace-type integral trans- form. The goal is to derive some properties of this transform and explore its relationship with other degenerate integral transforms. 2. Definition and Some Explicit Formulas Taekyun Kim and Dae San Kim [8] introduced the concept of degenerate Laplace transform, where f(t) be a function defined for t ≥ 0 and λ ∈ (0,∞). Then the integral Lλ{f(t)} = ∫ ∞ 0 e−s λ (t)f(t)dt = ∫ ∞ 0 (1 + λt)− s λ f(t)dt, (1) is said to be the degenerate Laplace transform of f if the integral converges. Letting s = 1 u , the degenerate Laplace transform can be rewritten as Lλ{f(t)} = ∫ ∞ 0 e − 1 u λ (t)f(t)dt = ∫ ∞ 0 (1 + λt)− 1 uλ f(t)dt Ugur Duran of Iskenderun Technical University [4] introduced the concept of degener- ate Sumudu transform of f(t) which is defined by the improper integral Sλ{f(t)} = 1 u ∫ ∞ 0 e −1 u λ (t)f(t)dt = 1 u ∫ ∞ 0 (1 + λt)− 1 uλ f(t)dt, where λ ∈ (0,∞), and f(t) be a function defined for t ≥ 0. On the paper of Lalit Mohan Upadhyaya et.al [1], they defined the degenerate of Elzaki transform and its properties. The degenerate Elzaki transform is defined by the integral Eλ{f(t)} = u ∫ ∞ 0 e − 1 u λ (t)f(t)dt = u ∫ ∞ 0 (1 + λt)− 1 uλ f(t)dt where λ ∈ (0,∞), and f(t) be a function defined for t ≥ 0. We now have the following definition: Definition 1. [3, 8–14] For any nonzero real number λ, the degenerate exponential function is defined as follows: exλ(t) = (1 + λt) x λ , eλ(t) = e1λ(t) = (1 + λt) 1 λ (2) H. J. Campos, J. C. Fernandez, J. B. M. Natuil / Eur. J. Pure Appl. Math, 16 (4) (2023), 2213-2233 2215 That is, the degenerate of the exponential function ext is equal to exλ(t) = (1+λt) x λ , where λ ∈ R− {0}. Here, we note that exλ(t) = ∞∑ n=0 (x)n,λ tn n! , where (x)0,λ = 1, (x)n,λ = x(x− λ)(x− 2λ) · · · (x− (n− 1)λ) for n ≥ 1. It is noteworthy to mention that lim λ→0 exλ(t) = lim λ→0 (1 + λt) x λ = ext. Definition 2. [8] A function f(t) is said to be of degenerate exponential order C if there exists C,M > 0 and T > 0 such that |f(t)| ≤ M(1 + λt) C λ = MeCλ (t) for all t > T. Definition 3. [2, 4, 6] The degenerate sine function is defined by the relation sin (x) λ (t) = eixλ (t)− e−ix λ (t) 2i = sin ( x λ log(1 + λt) ) , where i = √ −1. (3) It can be noted that, lim λ→0 sin (x) λ (t) = sinxt. Definition 4. [2, 4, 6] The degenerate cosine function is defined by the relation cos (x) λ (t) = eixλ (t) + e−ix λ (t) 2 = cos ( x λ log(1 + λt) ) , where i = √ −1. (4) It can be noted that, lim λ→0 cos (x) λ (t) = cosxt. Definition 5. [4, 6, 14] The degenerate Euler function is defined by the relation eixλ (t) = cos (x) λ (t) + i sin (x) λ (t) (5) where cos (x) λ (t) = cos ( x λ log(1 + λt) ) and sin (x) λ (t) = sin ( x λ log(1 + λt) ) . It can be noted that, lim λ→0 eixλ (t) = cosxt+ i sinxt. Definition 6. [4, 8, 14] The degenerate hyperbolic sine function is defined by the relation sinh (x) λ (t) = exλ(t)− e−x λ (t) 2 . (6) It can be noted that, lim λ→0 sinh (x) λ (t) = sinhxt. H. J. Campos, J. C. Fernandez, J. B. M. Natuil / Eur. J. Pure Appl. Math, 16 (4) (2023), 2213-2233 2216 Definition 7. [4, 8, 14] The degenerate hyperbolic cosine function is defined by the relation cosh (x) λ (t) = exλ(t) + e−x λ (t) 2 . (7) It can be noted that, lim λ→0 cosh (x) λ (t) = coshxt. 3. Degenerate Laplace-type Integral Transform Definition 8. Let λ ∈ (0,∞), α ∈ Z and let f(t) be a function defined for t ≥ 0. Then the integral Fα,λ(u) = Gα,λ{f(t)} = uα ∫ ∞ 0 e − 1 u λ (t)f(t)dt = uα ∫ ∞ 0 (1 + λt)− 1 uλ f(t)dt, (8) is said to be the degenerate Laplace-type integral transform of f(t). If the improper integral is convergent, then we say that the function f(t) possesses a degenerate Laplace- type integral transform. We note that lim λ→0 Gα,λ{f(t)} = Gα{f(t)}. (9) Theorem 1. Suppose that f(t) is a piecewise-continuous function on the interval [0,∞) and has a degenerate exponential order at infinity with |f(t)| ≤ MeCλ (t) for t > P , where M ≥ 0 and P,C are constants. Then, Gα,λ{f(t)} exists for 1− uC λu > 1. Proof. Suppose that f(t) is a piecewise-continuous function on the interval [0,∞) and has a degenerate exponential order at infinity with |f(t)| ≤ MeCλ (t). Then uα ∫ ∞ 0 e − 1 u λ (t)f(t)dt = uα ∫ P 0 e − 1 u λ (t)f(t)dt+ uα ∫ ∞ P e − 1 u λ (t)f(t)dt. (10) Since the function f(t) is piecewise-continuous in every finite interval 0 ≤ t ≤ P , the first integral on the right-hand side of equation (10) exists. Since∣∣∣∣e− 1 u λ (t)f(t) ∣∣∣∣≤ Me − 1 u λ (t)eCλ (t) for t > P , we have∣∣∣∣uα ∫ ∞ P e − 1 u λ (t)f(t)dt ∣∣∣∣≤uα ∫ ∞ P ∣∣∣∣e− 1 u λ (t)f(t) ∣∣∣∣dt ≤uα ∫ ∞ P e − 1 u λ (t)MeCλ (t)dt =Muα ∫ ∞ P (1 + λt) −1+uC uλ dt H. J. Campos, J. C. Fernandez, J. B. M. Natuil / Eur. J. Pure Appl. Math, 16 (4) (2023), 2213-2233 2217 =Muα lim R→∞ ∫ R P (1 + λt)−( 1−uC uλ )dt = Muα λ lim R→∞ [ (1 + λt)1−( 1−uC uλ ) 1 uλ ( uλ− (1− uC) )]∣∣∣∣R P =Muα+1 lim R→∞ [ (1 + λ(R))1−( 1−uC uλ ) uλ− 1 + uC − (1 + λ(P ))1−( 1−uC uλ ) uλ− 1 + uC ] = Muα+1 1− uλ− uC (1 + Pλ)1−( 1−uC uλ ) < ∞, for 1− uC uλ > 1. Hence, the second integral converges for 1− uC uλ > 1. Since the first integral on the right hand side of equation (10) converges and the second integral on the right hand side of equation (10) also converges for 1− uC uλ > 1. Thus f(t) has a degenerate Laplace-type integral transform, for 1− uC uλ > 1. Theorem 2. Let a, b ∈ R and let f(t) and h(t) be function whose degenerate Laplace-type integral exists. Then Gα,λ{af(t) + bh(t)} = aGα,λ{f(t)}+ bGα,λ{h(t)}. Proof. Let a, b ∈ R and f(t) and h(t) be any function whose degenerate Laplace-type integral exists. Then Gα,λ{af(t) + bh(t)} =uα ∫ ∞ 0 e − 1 u λ (t) [ af(t) + bh(t) ] dt =auα ∫ ∞ 0 e − 1 u λ (t)f(t)dt+ buα ∫ ∞ 0 e − 1 u λ (t)h(t)dt =aGα,λ{f(t)}+ bGα,λ{h(t)}. Thus, linearity property of the degenerate Laplace-type integral transform holds true. 4. Degenerate Laplace-type Integral Transform of Some Elementary Functions In this section the researcher establish the degenerate Laplace-type integral transform of some elementary functions. Theorem 3. The degenerate Laplace-type integral transform of the function f(t) = 1 is given by Gα,λ{1} = uα+1 1− λu , for λu < 1. (11) H. J. Campos, J. C. Fernandez, J. B. M. Natuil / Eur. J. Pure Appl. Math, 16 (4) (2023), 2213-2233 2218 Proof. By Definition 8, for f(t) = 1, we have Gα,λ{1} =uα ∫ ∞ 0 e − 1 u λ (t)dt = uα lim R→∞ ∫ R 0 (1 + λt)− 1 uλdt =uα+1 lim R→∞ [ (1 + λt)1− 1 uλ (λu− 1) ]∣∣∣∣∣ R 0 =uα+1 lim R→∞ [ (1 + λR)1− 1 uλ (λu− 1) − 1 (λu− 1) ] = uα+1 1− λu , for λu < 1. Remark 1. It is clear from Theorem 3 and equation (9) that lim λ→0 Gα,λ{1} = lim λ→0 uα+1 1− λu = uα+1 = Gα{1}. Theorem 4. The degenerate Laplace-type integral transform of the function f(t) = t is given by Gα,λ{t} = uα+2 (1− uλ)(1− 2uλ) , for 2uλ < 1. (12) Proof. By Definition 8, for f(t) = t, we have Gα,λ{t} =uα ∫ ∞ 0 e − 1 u λ (t)tdt = uα λ2 lim R→∞ [ (1 + λt)2− 1 uλ 1 uλ(2uλ− 1) − (1 + λt)1− 1 uλ 1 uλ(uλ− 1) ]∣∣∣∣∣ R 0 = uα+1 λ lim R→∞ [( (1 + λR) 2uλ−1 uλ 2uλ− 1 − (1 + λR) uλ−1 uλ uλ− 1 ) − ( 1 2uλ− 1 − 1 uλ− 1 )] = uα+2 (1− uλ)(1− 2uλ) , for 2uλ < 1. Remark 2. It is clear from Theorem 4 and equation (9) that lim λ→0 Gα,λ{t} = lim λ→0 [ uα+2 (1− uλ)(1− 2uλ) ] = uα+2 = Gα{t}. Theorem 5. The degenerate Laplace-type integral transform of the function f(t) = tn is given by Gα,λ{tn} = n!uα+1+n (1− uλ) · · · (1− (n+ 1)uλ) , for (n− k + 1)uλ < 1. (13) H. J. Campos, J. C. Fernandez, J. B. M. Natuil / Eur. J. Pure Appl. Math, 16 (4) (2023), 2213-2233 2219 Proof. By Definition 8, for f(t) = tn, we obtain Gα,λ{tn} =uα ∫ ∞ 0 tne − 1 u λ (t)dt = uα λn+1 n∑ k=0 ( n k ) (−1)k lim R→∞ [ (1 + λt)n−k− 1 uλ +1 1 uλ(nuλ− kuλ− 1 + uλ) ]∣∣∣∣∣ R 0 = uα+1 λn n∑ k=0 ( n k ) (−1)k [ − 1 nuλ− kuλ− 1 + uλ ] , for (n− k + 1)uλ < 1 = uα+1 λn [ n!unλn (1− (1)uλ)(1− (2)uλ) · · · (1− (n)uλ)(1− (n+ 1)uλ) ] , for (n− k + 1)uλ < 1 = n!uα+1+n (1− uλ)(1− 2uλ) · · · (1− nuλ)(1− (n+ 1)uλ) , for (n− k + 1)uλ < 1. Remark 3. It is clear from Theorem 5 and equation (9) that lim λ→0 Gα,λ{tn} = lim λ→0 [ n!uα+1+n (1− uλ)(1− 2uλ) · · · (1− (n+ 1)uλ) ] = n!uα+1+n = Gα{tn}. Theorem 6. The degenerate Laplace-type integral transform of a function f(t) = eaλ(t) is given by Gα,λ{eaλ(t)} = uα+1 1− u(a+ λ) , for (a+ λ)u < 1. (14) Proof. By Definition 8, for f(t) = eaλ(t), we set Gα,λ{eaλ(t)} =uα ∫ ∞ 0 e − 1 u λ (t) [ eaλ(t) ] dt = uα lim R→∞ ∫ R 0 (1 + λt) ua−1 uλ dt. =uα+1 lim R→∞ [ (1 + λt) ua−1+uλ uλ ua− 1 + uλ ]∣∣∣∣∣ R 0 =uα+1 [ − 1 u(a+ λ)− 1 ] , for (a+ λ)u < 1 = uα+1 1− u(a+ λ) , for (a+ λ)u < 1. Remark 4. It is clear from Theorem 6 and equation (9) that lim λ→0 Gα,λ{eaλ(t)} = lim λ→0 [ uα+1 1− u(a+ λ) ] = uα+1 1− ua = Gα{eat}. H. J. Campos, J. C. Fernandez, J. B. M. Natuil / Eur. J. Pure Appl. Math, 16 (4) (2023), 2213-2233 2220 Theorem 7. The degenerate Laplace-type integral transform of a function f(t) = eiaλ (t) is given by Gα,λ{eiaλ (t)} = uα+1 1− u(ia+ λ) , for uλ < 1, (15) where a is any positive constant. Proof. By Definition 8, for f(t) = eiaλ (t), we set Gα,λ{eiaλ (t)} =uα ∫ ∞ 0 e − 1 u λ (t) [ eiaλ (t) ] dt = uα lim R→∞ ∫ R 0 (1 + λt) uia−1 uλ dt =uα+1 lim R→∞ [ (1 + λt) −1+uλ uλ eiaλ (t) uia− 1 + uλ ]∣∣∣∣∣ R 0 . By the definition of degenerate Euler formula in Definition 5, eiaλ (t) = cos (a) λ (t) + i sin (a) λ (t) = cos ( a λ log ( 1 + λt )) +i sin ( a λ log ( 1 + λt )) . Thus, we have Gα,λ{eiaλ (t)} =uα+1 lim R→∞ [ (1 + λt) −1+uλ uλ eiaλ (t) uia− 1 + uλ ]∣∣∣∣∣ R 0 =uα+1 lim R→∞ [ (1 + λt) −1+uλ uλ [ cos ( a λ log(1 + λt) ) +i sin ( a λ log(1 + λt) )] uia− 1 + uλ ]∣∣∣∣∣ R 0 =uα+1 [ − 1 uia− 1 + uλ ] = uα+1 1− u(ia− λ) , for uλ < 1. Theorem 8. The degenerate Laplace-type integral transform of the degenerate sine func- tion f(t) = sin (a) λ (t) is given by Gα,λ{sin (a) λ (t)} = auα+2 (1− λu)2 + u2a2 . (16) Proof. By the definition of the degenerate sine in Definition 3, we have Gα,λ{sin (a) λ (t)} =Gα,λ { eiaλ (t)− e−ia λ (t) 2i } . Now using Theorem 2 and Theorem 7, we obtain Gα,λ{sin (a) λ (t)} = 1 2i [ Gα,λ{eiaλ (t)} − Gα,λ{e−ia λ (t)} ] H. J. Campos, J. C. Fernandez, J. B. M. Natuil / Eur. J. Pure Appl. Math, 16 (4) (2023), 2213-2233 2221 = 1 2i [ uα+1 1− u(ia+ λ) − uα+1 1− u(−ia+ λ) ] = auα+2 (1− λu)2 + u2a2 . Remark 5. It is clear from Theorem 8 and equation (9) that lim λ→0 Gα,λ{sin (a) λ (t)} = lim λ→0 [ auα+2 (1− λu)2 + u2a2 ] = auα+2 1 + u2a2 = Gα{sin at}. Theorem 9. The degenerate Laplace-type integral transform of the degenerate cosine func- tion f(t) = cos (a) λ (t) is given by Gα,λ{cos (a) λ (t)} = (1− λu)uα+1 (1− λu)2 + u2a2 . (17) Proof. By the definition of the degenerate cosine in Definition 4 and using Theorem 2 and Theorem 7, we obtain Gα,λ{cos (a) λ (t)} = 1 2 [ Gα,λ{eiaλ (t)}+ Gα,λ{e−ia λ (t)} ] = (1− λu)uα+1 (1− λu)2 + u2a2 . Remark 6. It is clear from Theorem 9 and equation (9) that lim λ→0 Gα,λ{cos (a) λ (t)} = lim λ→0 [ (1− λu)uα+1 (1− λu)2 + u2a2 ] = uα+1 1 + u2a2 = Gα{cos at}. Theorem 10. The degenerate Laplace-type integral transform of the degenerate hyperbolic sine function f(t) = sinh (a) λ (t) is given by Gα,λ{sinh (a) λ (t)} = auα+2 (1− λu)2 − u2a2 . (18) Proof. By the definition of the degenerate hyperbolic sine in Definition 6, we have Gα,λ{sinh (a) λ (t)} =Gα,λ { eaλ(t)− e−a λ (t) 2 } . Now, using Theorem 2 and Theorem 6, we obtain Gα,λ{sinh (a) λ (t)} = 1 2 [ Gα,λ{eaλ(t)} − Gα,λ{e−a λ (t)} ] = 1 2 [ uα+1 1− u(a+ λ) − uα+1 1− u(−a+ λ) ] = uα+1 2 [ 2au (1− λu)2 − u2a2 ] = auα+2 (1− λu)2 − u2a2 . H. J. Campos, J. C. Fernandez, J. B. M. Natuil / Eur. J. Pure Appl. Math, 16 (4) (2023), 2213-2233 2222 Remark 7. It is clear from Theorem 10 and equation (9) that lim λ→0 Gα,λ{sinh (a) λ (t)} = lim λ→0 [ auα+2 (1− λu)2 − u2a2 ] = auα+2 1− u2a2 = Gα{sinh at}. Theorem 11. The degenerate Laplace-type integral transform of the degenerate hyperbolic cosine function f(t) = cosh (a) λ (t) is given by Gα,λ{cosh (a) λ (t)} = (1− λu)uα+1 (1− λu)2 − u2a2 . (19) Proof. By the definition of the degenerate hyperbolic cosine in [? ] and using Theorem 2 and Theorem 7, we obtain Gα,λ{cosh (a) λ (t)} = 1 2 [ Gα,λ{eaλ(t)}+ Gα,λ{e−a λ (t)} ] = 1 2 [ uα+1 1− u(a+ λ) + uα+1 1− u(−a+ λ) ] = (1− λu)uα+1 (1− λu)2 − u2a2 . Remark 8. It is clear from Theorem 11 and equation (9) that lim λ→0 Gα,λ{cosh (a) λ (t)} = lim λ→0 [ (1− λu)uα+1 (1− λu)2 − u2a2 ] = uα+1 1− u2a2 = Gα{cosh at}. Theorem 12. The degenerate Laplace-type integral transform of the function f(t) = eaλ(t) sin (b) λ (t) is given by Gα,λ{eaλ(t) sin (b) λ (t)} = buα+2 (1− au− uλ)2 + b2u2 . (20) Proof. By the definition of the degenerate sine in in Definition 3, we have eaλ(t) sin (b) λ (t) =eaλ(t) [ eibλ (t)− e−ib λ (t) 2i ] = ea+ib λ (t)− ea−ib λ (t) 2i . Hence, by Theorem 2 and Theorem 7, we obtain Gα,λ{eaλ(t) sin (b) λ (t)} =Gα,λ { ea+ib λ (t)− ea−ib λ (t) 2i } = 1 2i [ Gα,λ { ea+ib λ (t) } − Gα,λ { ea−ib λ (t) }] = 1 2i [ uα+1 1− u((a+ ib) + λ) − uα+1 1− u((a− ib) + λ) ] = buα+2 (1− au− uλ)2 + b2u2 . H. J. Campos, J. C. Fernandez, J. B. M. Natuil / Eur. J. Pure Appl. Math, 16 (4) (2023), 2213-2233 2223 Remark 9. It is clear from Theorem 12 and equation (9) that lim λ→0 Gα,λ{eaλ(t) sin (b) λ (t)} = lim λ→0 [ buα+2 (1− au− uλ)2 + b2u2 ] = buα+2 (1− au)2 + b2u2 = Gα{eat sin bt}. Theorem 13. The degenerate Laplace-type integral transform of the function f(t) = eaλ(t) cos (b) λ (t) is given by Gα,λ{eaλ(t) cos (b) λ (t)} = (1− au− uλ)uα+1 (1− au− uλ)2 + b2u2 . (21) Proof. By the definition of the degenerate hyperbolic cosine in Definition 7, Theorem 2 and Theorem 7, we obtain Gα,λ{eaλ(t) cos (b) λ (t)} =Gα,λ { ea+ib λ (t) + ea−ib λ (t) 2 } = (1− au− uλ)uα+1 (1− au− uλ)2 + b2u2 . Remark 10. It is clear from Theorem 13 and equation (9) that lim λ→0 Gα,λ{eaλ(t) cos (b) λ (t)} = lim λ→0 [ (1− au− uλ)uα+1 (1− au− uλ)2 + b2u2 ] = (1− au)uα+1 (1− au)2 + b2u2 = Gα{eat cos bt}. 4.1. Degenerate Laplace-type Integral Transform of Derivative Theorem 14. If f(t), f ′(t), ..., f (n−1)(t) are continuous and f (n)(t) is a piecewise-continuous function on [0,∞) and has a degenerate exponential order at infinity with ∣∣f (n)(t) ∣∣ ≤ MeCλ (t) for t ≥ C, where C is a constant, then the following hold: (i.) Gα,λ{f ′(t)} = 1 u Gα,λ { (1 + λt)−1f(t) } − uαf(0) (ii.) Gα,λ{f ′′(t)} = 1 u2 (1 + λu)Gα,λ { (1 + λt)−2f(t) } − uα−1f(0)− uαf ′(0) (iii.) Gα,λ{f (n)(t)} = 1 un Gα,λ { (1 + λt)−nf(t) } n−1∏ l=1 ( 1 + luλ ) − uα+1−n n−1∑ i=0 uif (i)(0) [ n−i−2∏ l=1 ( 1 + luλ )] , where f (n)(t) = ( d dt )n f(t) and n = 1, 2, 3, 4, · · · . H. J. Campos, J. C. Fernandez, J. B. M. Natuil / Eur. J. Pure Appl. Math, 16 (4) (2023), 2213-2233 2224 Proof. First we prove (i.). By Definition 8, we have Gα,λ{f ′(t)} = uα ∫ ∞ 0 (1 + λt)− 1 uλ f ′(t)dt = uα lim R→∞ ∫ R 0 (1 + λt)− 1 uλ f ′(t)dt. Using integration by parts, we get Gα,λ{f ′(t)} =uα lim R→∞ [ (1 + λt)− 1 uλ f(t) ∣∣∣∣∣ R 0 ] + uα u lim R→∞ [∫ R 0 (1 + λt)−1− 1 uλ f(t)dt ] =uα lim R→∞ [ (1 + λR)− 1 uλ f(R)− (1 + λ0)− 1 uλ f(0) ] + 1 u Gα,λ { (1 + λt)−1f(t) } = 1 u Gα,λ { (1 + λt)−1f(t) } − uαf(0). For (ii.), using Definition 8, we have Gα,λ{f ′′(t)} = uα ∫ ∞ 0 (1 + λt)− 1 uλ f ′′(t)dt = uα lim R→∞ ∫ R 0 (1 + λt)− 1 uλ f ′′(t)dt. Using integration by parts, we have Gα,λ{f ′′(t)} =uα lim R→∞ [ (1 + λt)− 1 uλ f ′(t) ∣∣∣∣∣ R 0 − ∫ R 0 ( − 1 u ) (1 + λt)−1− 1 uλ f ′(t)dt ] =− uαf ′(0) + uα u lim R→∞ [∫ R 0 (1 + λt)−1− 1 uλ f ′(t)dt ] =− uαf ′(0) + uα u lim R→∞ [ (1 + λt)−1− 1 uλ f(t) ∣∣∣∣∣ R 0 − ∫ R 0 ( − λ− 1 u ) (1 + λt)−2− 1 uλ f(t)dt ] =− uαf ′(0)− uα−1f(0) + 1 u2 (1 + λu)uα lim R→∞ [∫ R 0 (1 + λt)−2− 1 uλ f(t)dt ] = 1 u2 (1 + λu)Gα,λ { (1 + λt)−2f(t) } − uα−1f(0)− uαf ′(0). For (iii.), we prove Gα,λ{f (n)(t)} = 1 un Gα,λ { (1 + λt)−nf(t) } n−1∏ l=1 ( 1 + luλ ) − uα+1−n n−1∑ i=0 uif (i)(0) [ n−i−2∏ l=1 ( 1 + luλ )] (22) H. J. Campos, J. C. Fernandez, J. B. M. Natuil / Eur. J. Pure Appl. Math, 16 (4) (2023), 2213-2233 2225 by induction. From results (i.) and (ii.), equation (22) holds for n = 1 and n = 2. Assume that equation (22) is true for n = k. Let g(t) = f (k)(t), then by the result of (i.), Gα,λ{f (k+1)(t)} =Gα,λ{g′(t)} = 1 u Gα,λ { (1 + λt)−1g(t) } − uαg(0) = 1 u Gα,λ { (1 + λt)−1f (k)(t) } − uαf (k)(0). (23) Now, Gα,λ { (1 + λt)−1f (k)(t) } =uα ∫ ∞ 0 e − 1 u λ (t)(1 + λt)−1f (k)(t)dt =uα ∫ ∞ 0 (1 + λt)−( 1+uλ uλ )f (k)(t)dt =(1 + uλ)α ( u 1 + uλ )α∫ ∞ 0 e − 1 u 1+uλ λ (t)f (k)(t)dt. (24) By inductive hypothesis, we have( u 1 + uλ )α∫ ∞ 0 e − 1 u 1+uλ λ (t)f (k)(t)dt = 1 ( u 1+uλ ) k Gα,λ { (1 + λt)−kf(t) } k−1∏ l=1 ( 1 + luλ 1 + uλ ) − ( u 1 + uλ )α+1−kk−1∑ i=0 ( u 1 + uλ )i f (i)(0)[ k−i−2∏ l=1 ( 1 + luλ 1 + uλ )] . Observe that Gα,λ { (1 + λt)−kf(t) } = ( u 1 + uλ )α∫ ∞ 0 e − 1 u 1+uλ λ (t)(1 + λt)−kf(t)dt = 1 (1 + uλ)α [ Gα,λ { (1 + λt)−(k+1)f(t) }] . Thus, the RHS of equation (23) becomes RHS = (1 + uλ)k uk [ 1 (1 + uλ)α Gα,λ { (1 + λt)−(k+1)f(t) }]k−1∏ l=1 (1 + (l + 1)uλ) 1 + uλ − uα+1−k (1 + uλ)α+1−k k−1∑ i=0 ui (1 + uλ)i f (i)(0) k−i−2∏ l=1 (1 + (l + 1)uλ) 1 + uλ = 1 uk(1 + uλ)α [ Gα,λ { (1 + λt)−(k+1)f(t) }][ (1 + uλ) k−1∏ l=1 ( 1 + (l + 1)uλ )] − ( uα−k (1 + uλ)α−k ) k−1∑ i=0 ( u 1 + uλ )i+1 f (i)(0) [ 1 (1 + uλ)k−i−2 k−i−2∏ l=1 ( 1 + (l + 1)uλ )] H. J. Campos, J. C. Fernandez, J. B. M. Natuil / Eur. J. Pure Appl. Math, 16 (4) (2023), 2213-2233 2226 = 1 uk(1 + uλ)α [ Gα,λ { (1 + λt)−(k+1)f(t) }][ k∏ l=1 ( 1 + luλ )] − ( uα−k (1 + uλ)α ) k−1∑ i=0 ui+1f (i)(0) [ k−i−1∏ l=1 ( 1 + luλ )] . Hence, the RHS of equation (22) becomes RHS =(1 + uλ)α [ 1 uk(1 + uλ)α Gα,λ { (1 + λt)−(k+1)f(t) } k∏ l=1 ( 1 + luλ ) − uα−k (1 + uλ)α k−1∑ i=0 ui+1f (i)(0) k−i−1∏ l=1 ( 1 + luλ )] = 1 uk Gα,λ { (1 + λt)−(k+1)f(t) } k∏ l=1 ( 1 + luλ ) − uα−k k−1∑ i=0 ui+1f (i)(0) k−i−1∏ l=1 ( 1 + luλ ) . So, the RHS of equation (24) becomes RHS = 1 u [ 1 uk Gα,λ { (1 + λt)−(k+1)f(t) } k∏ l=1 ( 1 + luλ ) − uα−k k−1∑ i=0 ui+1f (i)(0) k−i−1∏ l=1 (1 + luλ) ] −uαfk(0) = 1 uk+1 Gα,λ { (1 + λt)−(k+1)f(t) } k∏ l=1 (1 + luλ) − uα−k k∑ i=0 uif (i)(0) k−i−1∏ l=1 ( 1 + luλ ) . Hence, Gα,λ{f (k+1)(t)} holds. Therefore, by induction equation (22) holds for all n ≥ 1. 4.2. Degenerate Laplace-type Integral Transform of an Integral Theorem 15. Let Gα,λ{f(t)} = Fαλ(u). Then Gα,λ{f(t)} = 1 u Gα,λ { (1 + λt)−1 ∫ t 0 f(s)ds } . H. J. Campos, J. C. Fernandez, J. B. M. Natuil / Eur. J. Pure Appl. Math, 16 (4) (2023), 2213-2233 2227 Proof. Let g(t) = ∫ t 0 f(s)ds. Then g′(t) = d dt {∫ t 0 f(s)ds } = f(t) and g(0) = 0. Note that by Theorem 14 (i.) we have Gα,λ{g′(t)} = 1 u Gα,λ { (1 + λt)−1g(t) } − uαg(0) = 1 u Gα,λ { (1 + λt)−1 ∫ t 0 f(s)ds } . 4.3. The First Translation Theorem for the Degenerate Laplace-type In- tegral Transform Theorem 16. If Gα,λ{f(t)} = Fαλ(u) then Gα,λ{eaλ(t)f(t)} = (1− au)αFα,λ ( u 1− ua ) , for a ̸= 1 u . Proof. From Definition 8, it follows that Gα,λ{eaλ(t)f(t)} =uα ∫ ∞ 0 eaλ(t)e − 1 u λ (t)f(t)dt = (1− au)α (1− au)α uα ∫ ∞ 0 (1 + λt) − 1 ( u 1−au )λ f(t)dt =(1− au)α ( u 1− au )α∫ ∞ 0 (1 + λt) − 1 ( u 1−au )λ f(t)dt =(1− au)αFα,λ ( u 1− ua ) , for a ̸= 1 u . 4.4. The Change of Scale Property for the Degenerate Laplace-type In- tegral Transform Theorem 17. If Gα,λ{f(t)} = Fα,λ(u) then Gα,λ{f(at)} = 1 aα+1 Fα,λ a (au), for a > 0. Proof. From Definition 8, it follows that Gα,λ{f(at)} =uα ∫ ∞ 0 e − 1 u λ (t)f(at)dt = uα ∫ ∞ 0 (1 + λt)− 1 uλ f(at)dt. H. J. Campos, J. C. Fernandez, J. B. M. Natuil / Eur. J. Pure Appl. Math, 16 (4) (2023), 2213-2233 2228 Let w = at, then dw = adt and t = w a . Hence Gα,λ{f(at)} =uα ∫ ∞ 0 (1 + λt)− 1 uλ f(at)dt = 1 aα+1 [ (ua)α ∫ ∞ 0 ( 1 + (λ a ) w )− 1 (λa )au f(w)dw ] = 1 aα+1 Fα,λ a (au), for a > 0. 5. Generalization of the Degenerate Laplace, Degenerate Sumudu and Degenerate Elzaki Transform Definition 9. The degenerate Laplace-type integral transform Gα,λ{f(t)} = Fα,λ(u) = uα ∫ ∞ 0 e − 1 u λ (t)f(t)dt = uα ∫ ∞ 0 (1 + λt)− 1 uλ f(t)dt, is the generalization of the degenerate Laplace, degenerate Sumudu and degenerate Elzaki transform, where α = 0, α = −1 and α = 1, respectively. That is, when α = 0, we can have the degenerate Laplace transform, that is G0,λ{f(t)} = F0,λ(u) =u0 ∫ ∞ 0 e − 1 u λ (t)f(t)dt = ∫ ∞ 0 (1 + λt)− 1 uλ f(t)dt = ∫ ∞ 0 e − 1 u λ (t)f(t)dt = Lλ{f(t)}. When α = −1, we can have the degenerate Sumudu transform, that is G−1,λ{f(t)} = F−1,λ(u) =u−1 ∫ ∞ 0 e − 1 u λ (t)f(t)dt = 1 u ∫ ∞ 0 (1 + λt)− 1 uλ f(t)dt = 1 u ∫ ∞ 0 e − 1 u λ (t)f(t)dt = Sλ{f(t)}. Lastly, when α = 1, we can have the degenerate Elzaki transform, given by G1,λ{f(t)} = F1,λ(u) =u1 ∫ ∞ 0 e − 1 u λ (t)f(t)dt = u ∫ ∞ 0 (1 + λt)− 1 uλ f(t)dt =u ∫ ∞ 0 e − 1 u λ (t)f(t)dt = Eλ{f(t)}. . H. J. Campos, J. C. Fernandez, J. B. M. Natuil / Eur. J. Pure Appl. Math, 16 (4) (2023), 2213-2233 2229 Table 1 gives the summary of some elementary functions of the degenerate Laplace- type and degenerate Laplace [8, 14–16]. f(t) Gα,λ{f(t)} G0,λ{f(t)} = Lλ{f(t)}, α = 0 1 uα+1 1− λu u 1− λu t uα+2 (1− uλ)(1− 2uλ) u2 (1− uλ)(1− 2uλ) tn(n = 0, 1, 2, · · · ) n!uα+1+n (1− uλ) · · · (1− (n+ 1)uλ) n!u1+n (1− uλ) · · · (1− (n+ 1)uλ) eaλ(t) uα+1 1− u(a+ λ) u 1− u(a+ λ) sin (a) λ (t) auα+2 (1− λu)2 + u2a2 au2 (1− λu)2 + u2a2 cos (a) λ (t) (1− λu)uα+1 (1− λu)2 + u2a2 (1− λu)u (1− λu)2 + u2a2 sinh (a) λ (t) auα+2 (1− λu)2 − u2a2 au2 (1− λu)2 − u2a2 cosh (a) λ (t) (1− λu)uα+1 (1− λu)2 − u2a2 (1− λu)u (1− λu)2 − u2a2 eaλ(t) sin (b) λ (t) buα+2 (1− au− uλ)2 + b2u2 bu2 (1− au− uλ)2 + b2u2 eaλ(t) cos (b) λ (t) (1− au− uλ)uα+1 (1− au− uλ)2 + b2u2 (1− au− uλ)u (1− au− uλ)2 + b2u2 Table 1: The degenerate Laplace-type and degenerate Laplace transform. H. J. Campos, J. C. Fernandez, J. B. M. Natuil / Eur. J. Pure Appl. Math, 16 (4) (2023), 2213-2233 2230 Table 2 gives the summary of some elementary functions degenerate Laplace-type and degenerate Sumudu transforms[4]. f(t) Gα,λ{f(t)} G−1,λ{f(t)} = Sλ{f(t)}, α = −1 1 uα+1 1− λu 1 1− λu t uα+2 (1− uλ)(1− 2uλ) u (1− uλ)(1− 2uλ) tn(n = 0, 1, 2, · · · ) n!uα+1+n (1− uλ) · · · (1− (n+ 1)uλ) n!un (1− uλ) · · · (1− (n+ 1)uλ) eaλ(t) uα+1 1− u(a+ λ) 1 1− u(a+ λ) sin (a) λ (t) auα+2 (1− λu)2 + u2a2 au (1− λu)2 + u2a2 cos (a) λ (t) (1− λu)uα+1 (1− λu)2 + u2a2 1− λu (1− λu)2 + u2a2 sinh (a) λ (t) auα+2 (1− λu)2 − u2a2 au (1− λu)2 − u2a2 cosh (a) λ (t) (1− λu)uα+1 (1− λu)2 − u2a2 1− λu (1− λu)2 − u2a2 eaλ(t) sin (b) λ (t) buα+2 (1− au− uλ)2 + b2u2 bu (1− au− uλ)2 + b2u2 eaλ(t) cos (b) λ (t) (1− au− uλ)uα+1 (1− au− uλ)2 + b2u2 1− au− uλ (1− au− uλ)2 + b2u2 Table 2: The degenerate Laplace-type and degenerate Sumudu transform. H. J. Campos, J. C. Fernandez, J. B. M. Natuil / Eur. J. Pure Appl. Math, 16 (4) (2023), 2213-2233 2231 Table 3 gives the summary of some elementary functions degenerate Laplace-type and degenerate Elzaki transforms [1]. f(t) Gα,λ{f(t)} G1,λ{f(t)} = Eλ{f(t)}, α = 1 1 uα+1 1− λu u2 1− λu t uα+2 (1− uλ)(1− 2uλ) u3 (1− uλ)(1− 2uλ) tn(n = 0, 1, 2, · · · ) n!uα+1+n (1− uλ) · · · (1− (n+ 1)uλ) n!u2+n (1− uλ) · · · (1− (n+ 1)uλ) eaλ(t) uα+1 1− u(a+ λ) u2 1− u(a+ λ) sin (a) λ (t) auα+2 (1− λu)2 + u2a2 au3 (1− λu)2 + u2a2 cos (a) λ (t) (1− λu)uα+1 (1− λu)2 + u2a2 (1− λu)u2 (1− λu)2 + u2a2 sinh (a) λ (t) auα+2 (1− λu)2 − u2a2 au3 (1− λu)2 − u2a2 cosh (a) λ (t) (1− λu)uα+1 (1− λu)2 − u2a2 (1− λu)u2 (1− λu)2 − u2a2 eaλ(t) sin (b) λ (t) buα+2 (1− au− uλ)2 + b2u2 bu3 (1− au− uλ)2 + b2u2 eaλ(t) cos (b) λ (t) (1− au− uλ)uα+1 (1− au− uλ)2 + b2u2 (1− au− uλ)u2 (1− au− uλ)2 + b2u2 Table 3: The degenerate Laplace-type and degenerate Elzaki transform. 6. Conclusion and Recommendations The concept of degenerate Laplace-type Integral Transform is introduced in this work, and it includes three essential degenerate integral transforms: the degenerate Laplace Inte- gral transform, the degenerate Sumudu Integral transform, and the degenerate Elzaki In- tegral transform. These transformations offer potentially powerful mathematical tools for addressing a wide range of problems in engineering, physics, and other scientific fields. The degenerate Laplace-type Integral Transform is a unifying framework from which the degen- erates of several current Integral Transforms may be derived. This degenerate Laplace-type Integral Transform has a lot of promise and is still being researched and developed. As a result, more study may uncover new applications, features, and generalizations of this groundbreaking concept. 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