EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6136 ISSN 1307-5543 – ejpam.com Published by New York Business Global On the Degenerate Sadik Transform Jamilon B. Mohamadali1,∗, Normalah S. Abdulcarim1 1 Department of Mathematics, College of Natural Sciences and Mathematics, Mindanao State University Main Campus, 9700 Marawi City, Philippines Abstract. In this paper, the authors introduce the degenerate Sadik transform and investigates the transform of some elementary functions. Also, sufficient condition for the existence of the said transform is also presented. Furthermore, this paper concludes that degenerate sadik transform is a unification of some other degenerate transforms such as degenerate Laplace transform, degenerate Sumudu transform, degenerate Elzaki integral transform and Laplace-type integral transform. 2020 Mathematics Subject Classifications: 44A99 Key Words and Phrases: Degenerate Laplace Transform, Degenerate Elzaki Transform, Degen- erate Tarig Integral Transform, Degenerate Sumudu Transform, Degenerate Laplace-type Integral Transform 1. Introduction The integral transformation method is widely utilized in solving various types of differ- ential equations due to its ability to simplify complex problems. By converting differential equations into algebraic equations, integral transforms streamline the problem-solving pro- cess, making it significantly easier and more efficient [1]. Over time, numerous integral transforms have been developed, including the Sumudu transform [2], Tarig transform [3], Elzaki transform [4], Aboodh transform [5], Kamal transform [6], and Laplace-Carson transform [7]. Among these, the Laplace transform, introduced by Pierre-Simon Laplace, remains one of the most prominent and widely applied tools in mathematics, physics, and engineering [8]. The Laplace transform of a function f(t), denoted by L{f(t)} is defined as F (s) = L { f(t) } = ∫ ∞ 0 e−stf(t)dt, for all real numbers t ≥ 0 where s ∈ C. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6136 Email addresses: mohamadali.jb57@s.msumain.edu.ph (J. Mohamadali), normalah.abdulcarim@msumain.edu.ph (N. Abdulcarim) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) J. Mohamadali, N. Abdulcarim / Eur. J. Pure Appl. Math, 18 (3) (2025), 6136 2 of 19 Throughout the centuries, integral transforms have played a vital role in the develop- ment of new mathematical tools across various scientific disciplines. In 2018, mathemati- cian Sadikali Latif Shaikh introduced a new integral transform, named the Sadik transform [9], which generalizes and unifies several existing integral transforms. This transform is denoted by S{f(t)} and defined by F{(uα, β)} = S{f(t)} = 1 uβ ∫ ∞ 0 e−uα (t)f(t)dt. (1) where α, β ∈ R and u is a parameter. The Sadik transform provides a flexible framework that encompasses a wide range of existing transforms by assigning specific values to the parameters α and β. That is, when α = 0 and β = 1, we get S{f(t)} = ∫ ∞ 0 f(ut)e−tdt, u ∈ (−τ1, τ2), popularly known as the Sumudu transform [2], when α = −2 and β = 1, we acquire T {f(t)} = 1 u ∫ ∞ 0 e− t u2 f(t)dt, commonly called as the Tarig integral transform [3], when α = −1 and β = −1, we obtain Ef{(t)} = u ∫ ∞ 0 e− t u f(t)dt, t ≤ 0, k1 ≤ u ≤ k2, well-known as the Elzaki integral transform [10], when α = 1 and β = 1, we get A{f(t)} = K(u) = 1 u ∫ ∞ 0 f(t)e−utdt, t ≥ 0, k1 ≤ u,≤ k2, familiarly named as the Aboodh transform [5], when α = −1 and β = 0, we derive K{F (t)} = ∫ ∞ 0 f(t)e −t u tdt = G(u), t ≥ 0, k1 ≤ u ≤ k2, recognized as the Kamal transform [6]; and when α = 1 and β = −1, we get LC{f(t)} = G{p} = p ∫ ∞ 0 e−ptg(t)dt = pL{g(t)}, identified as as the Laplace-Carson transform [7]. These instances illustrate the remarkable generality and versatility of the Sadik transform as a unifying structure for various well- known transforms. In mathematics the degenerate refers to the simplification of solutions, particularly in limiting scenarios where specific parameters, denoted as λ, approach critical values like zero. Recently, researchers have shown growing interest in exploring degenerate versions of classical integral transforms. Some of these degenerate transforms are the degenerate J. Mohamadali, N. Abdulcarim / Eur. J. Pure Appl. Math, 18 (3) (2025), 6136 3 of 19 Laplace integral transform [11], degenerate Laplace-type integral transform [12], degener- ate Elzaki integral transform [13] and degenerate Sumudu Transform [14] defined by Lλ{f(t)} = ∫ ∞ 0 e−s λ (t)f(t)dt = ∫ ∞ 0 (1 + λt)− s λ f(t)dt, Fαλ(u) = Gαλ{f(t)} = uα ∫ ∞ 0 e − 1 u λ (t) f(t)dt = uα ∫ ∞ 0 (1 + λt)− 1 uλ f(t)dt, Eλ{f(t)} = u ∫ ∞ 0 e − 1 u λ (t)f(t)dt = u ∫ ∞ 0 (1 + λt) 1 uλ f(t)dt, Gλ{(u)} = Sλ{f(t)} = 1 u ∫ ∞ 0 e −1 u λ (t)dt, u ∈ (−τ1, τ2), respectively. In light of these developments, the authors are motivated to introduce and analyze a degenerate version of the Sadik transform and investigates the transform of some elemen- tary functions. Also, sufficient condition for the existence of the said transform is also presented. 2. Preliminaries Taekyon Kim and Dae S. Kim [11] defined the degenerate Laplace transform by the integral Lλf(t) = ∫ ∞ 0 e−s λ (t)f(t)dt = ∫ ∞ 0 (1 + λt)− s λ f(t)dt if the integral converges. The following are some results of the degenerate Laplace integral transform of some elementary functions f(t): f(t) Lλ{f(t)} 1 1 s−λ cos (a) λ (t) s−λ (s−λ)2+a2 t 1 s2−3sλ+2λ2 sinh (a) λ (t) a (s−λ)2−a2 f(t) Lλ{f(t)} tn(n = 0, 1, ...) n! sn+1 1( 1−λ s ) ... ( 1− (n+1)λ s ) cosh (a) λ (t) s−λ (s−λ)2−a2 eaλ(t) 1 s−λ−a sin (a) λ (t) a (s−λ)2+a2 Now, 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. Consequently, the degenerate Laplace integral transform [14] of some elementary functions f(t) are as follows: J. Mohamadali, N. Abdulcarim / Eur. J. Pure Appl. Math, 18 (3) (2025), 6136 4 of 19 f(t) Lλ{f(t)} 1 u 1−λu t u2 (1−λ)(1−2λu) sinh (a) λ (t) au2 (1−λu)2−u2a2 cosh (a) λ (t) (1−uλ)u (1−λu)2−u2a2 f(t) Lλ{f(t)} tn(n = 0, 1, ...) n!un+1 (1−λu)...(1(n+1)λu) eaλ(t) u 1−u(a+λ) cos (a) λ (t) (1−uλ)u (1−uλ)2+u2a2 sin (a) λ (t) au2 (1−uλ)2+u2a2 Note that lim λ→0 Lλ{f(t)} = L{f(t)}. In 2023, Jade Bong M. Natuil, Harren J. Campos and Jezer C. Fernandez [12] defined the degenerate of Laplace-type integral transform by the integral Fαλ(u) = Gαλ{f(t)} = uα ∫ ∞ 0 e − 1 u λ (t) f(t)dt = uα ∫ ∞ 0 (1 + λt)− 1 uλ f(t)dt. Presented below are the Laplace-type integral transform of some elementary functions that are well-discussed in: f(t) Gαλ{f(t)} 1 uα+1 1−uλ tn(0, 1, 2, ...) n!uα+1+n (1−uλ)...(1−(n+1)uλ) t uα+2 (1−uλ)(1−2uλ) eaλt uα+1 1 −u(a+λ) f(t) Gαλ{f(t)} sin (a) λ t auα+2 (1−λu)2+u2a2 cos (a) λ t (1−λu)uα+1 (1−λu)2+u2a2 sinh (a) λ t auα+2 (1−λu)2−u2a2 cosh (a) λ t (1−λu)uα+1 (1−λu)2−u2a2 In 2021, A. Kalavati, T. Kohali, and L.M. Upadhyaya [13] defined the degenerate of Elzaki transform 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. Below are the degenerate Elzaki integral transform of some elementary functions f(t): f(t) Eλ{f(t)} 1 u2 1−λu cos (a) λ (t) (1−uλ)u2 (1−uλ)2+u2a2 t u3 (1−λ)(1−2λu) tn(n = 0, 1, ...) n!un+2 (1−λu)...(1−(n+1)λu) f(t) Eλ{f(t)} cosh (a) λ (t) (1−uλ)u2 (1−λu)2−u2a2 eaλ(t) u2 1−u(a+λ) sin (a) λ (t) au3 (1−uλ)2+u2a2 sinh (a) λ (t) au3 (1−λu)2−u2a2 J. Mohamadali, N. Abdulcarim / Eur. J. Pure Appl. Math, 18 (3) (2025), 6136 5 of 19 Note that limλ→0 Eλ{f(t)} = E{f(t)}. In 2020, Duran of Iskenderun Technical University [14] defined the degenerate of Sumudu Transform by the integral Gλ{(u)} = Sλ{f(t)} = 1 u ∫ ∞ 0 e −1 u λ (t)dt, u ∈ (−τ1, τ2). The degenerate Sumudu transform satisfies the following operational properties and the transform of some elementary function f(t): f(t) Sλ{f(t)} 1 1 1−λu tn n!un (1−λu)(1−2λu)...(1−(n+1)λu) t u (1−λu)(1−2λu) eaλt 1 1 −u(a+λ) f(t) Sλ{f(t)} sin (a) λ t au (1−λu)2+u2a2 cos (a) λ t 1−uλ (1−λu)2+u2a2 sinh (a) λ t au (1−λu)2−u2a2 cosh (a) λ t 1−uλ (1−λu)2−u2a2 Furthermore, in 2017, Kim et al [15] defined the degenerate exponential function as exλ(t) = (1 + λt) x λ , eλ(t) = e1λ(t) = (1 + λt) 1 λ (2) for λ ∈ R, . 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 and that limλ→0 e x λ(t) = limλ→0(1+λt) x λ = ext. In [16], the degenerate sine and degenerate cosine functions are defined by the relations sin (x) λ (t) = eixλ (t)− e−ix λ (t) 2i = sin (x λ log(1 + λt) ) , (3) cos (x) λ (t) = eixλ (t) + e−ix λ (t) 2 = cos (x λ log(1 + λt) ) , (4) respectively, where i = √ −1. In [11], 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) ) . J. Mohamadali, N. Abdulcarim / Eur. J. Pure Appl. Math, 18 (3) (2025), 6136 6 of 19 In [14], the degenerate hyperbolic sine and degenerate hyperbolic cosine functions are defined by the relations sinh (x) λ (t) = exλ(t)− e−x λ (t) 2 (6) cosh (x) λ (t) = exλ(t) + e−x λ (t) 2 , (7) respectively. 3. Main Results This section presents definition of the degenerate Sadik transform. Moreover, discus- sions on some elementary functions of the degenerate Sadik transform are also provided. Definition 1. Let λ ∈ (0,∞) and let f(t) be a function defined for t ≥ 0. Then the integral Fλ(u α, β) = Sλ{f(t)} = 1 uβ ∫ ∞ 0 e−uα λ (t)f(t)dt = 1 uβ ∫ ∞ 0 (1 + λt)− uα λ f(t)dt (8) is said to be the degenerate Sadik transform of f(t). If the improper integral is con- vergent, then we say that the function f(t) possesses as a degenerate Sadik transform. Now, observe that, if the degenerate Sadik transform of f(t) exists, then from the above definition, lim λ→0 Sλ{f(t)} = lim λ→0 [ 1 uβ ∫ ∞ 0 e−uα λ (t)f(t)dt ] = 1 uβ ∫ ∞ 0 e−uα (t)f(t)dt = S{f(t)}. Remark 1. m 1. When β = 0 and α = 1 in equation (8), Sλ{f(t)} = 1 u0 ∫ ∞ 0 e−u λ (t)f(t) dt = Lλ{f(t)}. 2. When β = −1 and α = −1 in equation (8), Sλ{f(t)} = 1 u−1 ∫ ∞ 0 e−u−1 λ (t)f(t) dt = Eλ{f(t)}. 3. When β = 1 and α = −1 in equation (8), Sλ{f(t)} = 1 u1 ∫ ∞ 0 e−u−1 λ (t)f(t) dt = Sλ{f(t)}. J. Mohamadali, N. Abdulcarim / Eur. J. Pure Appl. Math, 18 (3) (2025), 6136 7 of 19 4. When β = −α and α = −1 in equation (8), Gαλ{f(t)} = 1 u−α ∫ ∞ 0 e−u−1 λ (t)f(t) dt = Gαλ{f(t)}. The following theorem is a sufficient condition for the existence of Degenerate Sadik Transform Theorem 1. Suppose that f(t) is a piecewise-continuous function on the interval [0,∞) and of degenerate exponential order at infinity with |f(t)| ≤ Mecλ(t) for t > P, where M ≥ 0 and P,C are constants. Then Sλ{f(t)} exists for −uα + c λ + 1 < 0. Proof. Suppose that f(t) is a piecewise-continuous function on the interval [0,∞) and has degenerate exponential order at infinity with |f(t)| ≤ Mecλ(t). Then, 1 uβ ∫ ∞ 0 e−uα λ (t)f(t) dt = 1 uβ ∫ P 0 e−uα λ (t)f(t) dt+ 1 uβ ∫ ∞ P e−uα λ (t)f(t) dt. (9) 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 (9) exists. Now, we will show that the second integral on the right hand side also exists. Note that for t > P,∣∣e−uα λ (t)f(t) ∣∣ ≤ e−uα λ (t) Mecλ(t). Thus, 1 uβ ∫ ∞ P ∣∣e−uα λ (t)f(t) ∣∣ dt ≤ 1 uβ ∫ ∞ P e−uα λ (t)Mecλ(t) dt = M uβ lim R→∞ ∫ R P (1 + λt) −uα+c λ dt. Evaluation the right-hand side of the above equation, we can see that the integral converges for −uα + c λ + 1 < 0. Since both the integrals on the right hand equation (9) converges for −uα + c λ + 1 < 0, f(t) has a degenerate Sadik transform for −uα + c λ + 1 < 0. The following theorem is the linearity property of the Degenerate Sadik Transform Theorem 2. Let a, b ∈ R and let f(t) and g(t) be functions whose degenerate Sadik transform exist, then Sλ[af(t) + bg(t)] = aSλ[f(t)] + bSλ[g(t)] Proof. Let a, b ∈ R and let f(t) and g(t) be any function whose degenerate Sadik transform exist, then Sλ{af(t) + bg(t)} = 1 uβ ∫ ∞ 0 e−uα λ (t) {af(t) + bg(t)} dt J. Mohamadali, N. Abdulcarim / Eur. J. Pure Appl. Math, 18 (3) (2025), 6136 8 of 19 =a 1 uβ ∫ ∞ 0 e−uα λ (t) f(t) dt+ b 1 uβ ∫ ∞ 0 e−uα λ (t) g(t) dt =a Sλ{f(t)}+ b Sλ{g(t)}. The following results are some of the elementary functions of the degenerate Sadik Transform. Theorem 3. The degenerate Sadik transform of the function f(t) = 1 is given by Sλ{1} = 1 uα+β − uβλ , for −uα λ + 1 < 0. (10) Proof. From equation (8), when f(t) = 1, we have Sλ{1} = 1 uβ ∫ ∞ 0 e−uα λ (t){1} dt = 1 uβ ∫ ∞ 0 (1 + λt) −uα λ dt = 1 uβ lim R→∞ ∫ R 0 (1 + λt) −uα λ dt = 1 uβ lim R→∞ [( (1 + λR) −uα λ +1 −uα + λ ) − ( (1) −uα λ +1 −uα + λ )] = 1 uβ [ − ( 1 −uα + λ )] , for −uα λ + 1 < 0 = 1 uβ ( 1 uα − λ ) = 1 uαuβ − uβλ . Hence, the degenerate Sadik transform of the function f(t) = 1 is given by Sλ{1} = 1 uα+β − uβλ , for −uα λ + 1 < 0. Remark 2. Observe that as λ → 0, Sλ{1} tends to S{1}. That is, lim λ→0 Sλ{1} = lim λ→0 [ 1 uα+β − uβλ ] = 1 uα+β = S{1}. Remark 3. m 1. When β = 0 and α = 1 in equation (10), Sλ{1} = 1 u1+0 − u0λ = 1 u− λ = Lλ{1}. J. Mohamadali, N. Abdulcarim / Eur. J. Pure Appl. Math, 18 (3) (2025), 6136 9 of 19 2. When β = −1 and α = −1 in equation (10), Sλ{1} = 1 u−1+−1 − u−1λ = u2 1− uλ = Eλ{1}. 3. When β = 1 and α = −1 in equation (10), Sλ{1} = 1 u−1+1 − u1λ = 1 1− uλ = Sλ{1}. 4. When β = −α and α = −1 in equation (10), Sλ{1} = 1 u−1+(−α) − u−αλ = u(α+1) 1− uλ = Gαλ{1}. Theorem 4. The degenerate Sadik transform of the function f(t) = t is given by Sλ{t} = u−β (uα − 2λ)(uα − λ) , for −uα λ + 2 < 0. (11) Proof. By Definition for f(t) = t, we have Sλ{t} = 1 uβ ∫ ∞ 0 e−uα λ (t) {t} dt = 1 uβ lim R→∞ ∫ R 0 (1 + λt) −uα λ t dt. = 1 uβλ lim R→∞ [( (1 + λ(R)) −uα λ +2 −uα + 2λ − (1 + λ(R)) −uα λ +1 −uα + λ ) − ( (1) −uα λ +2 −uα + 2λ − (1) −uα λ +1 −uα + λ )] = 1 uβλ − ( 1 −uα + 2λ − 1 −uα + λ ) for −uα λ + 2 < 0 = 1 uβλ ( 1 uα − 2λ − 1 uα − λ ) = u−β (uα − 2λ)(uα − λ) . Remark 4. Observe that as λ → 0, Sλ{t} tends to S{t}. That is, lim λ→0 Sλ{t} = u−β (uα)(uα) = u−β u2α = S{t}. Remark 5. m 1. When β = 0 and α = 1 in equation (11), Sλ{t} = u−0 (u1 − 2λ)(u1 − λ) = 1 u2 − 3uλ+ 2λ2 = Lλ{t}. J. Mohamadali, N. Abdulcarim / Eur. J. Pure Appl. Math, 18 (3) (2025), 6136 10 of 19 2. When β = −1 and α = −1 in equation (11), Sλ{t} = u−(−1) (u−1 − 2λ)(u−1 − λ) = u3 (1− uλ)(1− 2uλ) = Eλ{t}. 3. When β = 1 and α = −1 in equation (11), Sλ{t} = u−1 (u−1 − 2λ)(u−1 − λ) = u (1− uλ)(1− 2uλ) = Sλ{t}. 4. When β = −α and α = −1 in equation (11), Sλ{t} = u−(−α) (u−1 − 2λ)(u−1 − λ) = uα+2 (1− uλ)(1− 2uλ) = Gαλ{t}. Theorem 5. The degenerate Sadik transform of the function f(t) = eaλ(t) is given by Sλ{eaλ(t)} = u−β uα − a− λ for −uα + a λ + 1 < 0. (12) Proof. From equation (8), when f(t) = eaλ(t), we have Sλ{eaλ(t)} = 1 uβ ∫ ∞ 0 e−uα λ (t) eaλ(t) dt = 1 uβ lim R→∞ ∫ R 0 (1 + λt) −uα+a λ dt. = 1 uβ lim R→∞ [( (1 + λR) −uα+a λ +1 −uα + a+ λ ) − ( (1) −uα+a λ +1 −uα + a+ λ )] = 1 uβ [ − ( 1 −uα + a+ λ )] , for −uα + a λ + 1 < 0 = 1 uβ ( 1 uα − a− λ ) = u−β uα − a− λ . Remark 6. Observe that as λ → 0, Sλ{eaλ(t)} tends to S{ea(t)}. That is, lim λ→0 Sλ{eaλ(t)} = u−β uα − a− 0 = S{ea(t)}. Remark 7. m 1. When β = 0 and α = 1 in equation (12), Sλ{eaλ(t)} = u−0 u1 − a− λ = 1 u− a− λ = Lλ{eaλ(t)}. J. Mohamadali, N. Abdulcarim / Eur. J. Pure Appl. Math, 18 (3) (2025), 6136 11 of 19 2. When β = −1 and α = −1 in equation (12), Sλ{eaλ(t)} = u−(−1) u−1 − a− λ = u2 1− u(a+ λ) = Eλ{eaλ(t)}. 3. When β = 1 and α = −1 in equation (12), Sλ{eaλ(t)} = u−1 u−1 − a− λ = 1 1− u(a+ λ) = Sλ{eaλ(t)}. 4. When β = −α and α = −1 in equation (12), Sλ{eaλ(t)} = u−(−α) u−1 − a− λ = uα+1 1− u(a+ λ) = Gαλ{eaλ(t)}. Corollary 1. The degenerate Sadik transform of the function is f(t) = e−a λ (t) is given by Sλ{e−a λ (t)} = u−β uα + a− λ , for −uα − a λ + 1 < 0. (13) Theorem 6. The degenerate Sadik transform of the function f(t) = eiaλ (t) is given by Sλ{eiaλ (t)} = u−β uα − ia− λ , for 0 < u−αλ < 1. (14) Proof. When f(t) = eiaλ (t) in equation (8), we have Sλ{eiaλ (t)} = 1 uβ ∫ ∞ 0 e−uα λ (t) {eiaλ (t)} dt = 1 uβ lim R→∞ ∫ R 0 (1 + λt) −uα+ia λ dt. = 1 uβ lim R→∞ [ (1 + λR) −uα+λ λ (1 + λR) ia λ −uα + ia+ λ − (1) −uα+λ λ (1) ia λ −uα + ia+ λ ] = 1 uβ lim R→∞ [ (1 + λR) −uα+λ λ eiaλ R −uα + ia+ λ − 1 −uα + ia+ λ ] = 1 uβ lim R→∞ [(1 + λR) −uα+λ λ ( cos ( a λ log(1 + λR) ) + i sin ( a λ log(1 + λR) )) −uα + ia+ λ ] − [ 1 −uα + ia+ λ ] If u−αλ = 1, then the first limit on the right-hand side of the above equation does not exist since the value of the function J. Mohamadali, N. Abdulcarim / Eur. J. Pure Appl. Math, 18 (3) (2025), 6136 12 of 19 cos (a λ log(1 + λR) ) and i sin (a λ log(1 + λR) ) is oscillate between 1 and −1. Thus, Sλ{eiaλ } is not defined. Similarly, Sλ{eiaλ } is not de- fined for u−αλ > 1. However, observe that for 0 < u−αλ < 1, (1+λ(R)) −uα+λ λ approaches to 0 as R approaches to ∞. Hence, Sλ{eiaλ } = 1 uβ [ − ( 1 −uα + ia+ λ )] = u−β uα − ia− λ , for u−αλ < 1. Corollary 2. The degenerate Sadik transform of the function f(t) = e−ia λ (t) is given by Sλ{e−ia λ (t)} = u−β uα + ia− λ , for 0 < u−αλ < 1. (15) Theorem 7. The degenerate Sadik transform of the function f(t) = sinaλ(t) is given by Sλ{sinaλ(t)} = au−β u2α − 2uαλ+ a2 + λ2 . (16) Proof. Note that from Definition 3 for f(t) = sin (a) λ (t), we have Sλ{sinaλ(t)} = Sλ { eiaλ (t)− e−ia λ (t) 2i } . (17) Now, applying Theorems 2, 6 and Corollary 2 equation (17) deduces Sλ{sinaλ(t)} = ( 1 2i )[ Sλ { eiaλ (t) } − Sλ { e−ia λ (t) }] = ( 1 2i )[ u−β uα − ia− λ − u−β uα + ia− λ ] = ( u−β 2i )[ uα + ia− λ− uα + ia+ λ (uα)2 + uαia− uαλ− uαia− (ia)2 + iaλ− uαλ− iaλ+ λ2 ] = ( u−β 2i )[ 2ia u2α − 2uαλ+ a2 + λ2 ] = [ au−β u2α − 2uαλ+ a2 + λ2 ] . Therefore, the degenerate Sadik transform of the function f(t) = sin (a) λ (t) is given by Sλ{sinaλ(t)} = au−β u2α − 2uαλ+ a2 + λ2 . J. Mohamadali, N. Abdulcarim / Eur. J. Pure Appl. Math, 18 (3) (2025), 6136 13 of 19 Remark 8. Observe that as λ → 0, Sλ{sin (a) λ (t)} tends to S{sin at}. That is, lim λ→0 Sλ{sinaλ(t)} = lim λ→0 [ au−β u2α − 2uαλ+ a2 + λ2 ] = au−β u2α + a2 = S{sin(at)}. Remark 9. m 1. When β = 0 and α = 1 in equation (16), Sλ{sinaλ(t)} = au−0 u2(1) − 2u1λ+ a2 + λ2 = a (u− λ)2 + a2 = Lλ{sinaλ(t)}. 2. When β = −1 and α = −1 in equation (16), Sλ{sinaλ(t)} = au−(−1) u2(−1) − 2u−1λ+ a2 + λ2 = au3 (1− λu)2 + a2u2 = Eλ{sinaλ(t)}. 3. When β = 1 and α = −1 in equation (16), Sλ{sinaλ(t)} = au−1 u2(−1) − 2u−1λ+ a2 + λ2 = au (1− λu)2 + a2u2 = Sλ{sinaλ(t)}. 4. When β = −α and α = −1 in equation (16), Sλ{sinaλ(t)} = au−(−α) u2(−1) − 2u−1λ+ a2 + λ2 = auα+2 (1− λu)2 + a2u2 = Gαλ{sinaλ(t)}. Theorem 8. The degenerate Sadik transform of the function f(t) = cosaλ(t) is given by Sλ{cosaλ(t)} = auα−β − u−βλ u2α − 2uαλ+ a2 + λ2 (18) Remark 10. Observe that as λ → 0, Sλ{cos (a) λ (t)} tends to S{cos at}. That is, lim λ→0 Sλ{cosaλ(t)} = uα−β − u−β(0) u2α − 2uα(0) + a2 + (0)2 = S{cos(at)}. Remark 11. m 1. When β = 0 and α = 1 in equation (18), Sλ{cosaλ(t)} = u1−0 − u−(1)λ u2(1) − 2u1λ+ a2 + λ2 = u− λ u2 − 2uλ+ λ2 + a2 = Lλ{cosaλ(t)}. 2. When β = −1 and α = −1 in equation (18), Sλ{cosaλ(t)} = u1−(−1) − u−(−1)λ u2(−1) − 2u−1λ+ a2 + λ2 = (1− uλ)u2 (1− λ)2 + a2u2 = Eλ{cosaλ(t)}. J. Mohamadali, N. Abdulcarim / Eur. J. Pure Appl. Math, 18 (3) (2025), 6136 14 of 19 3. When β = 1 and α = −1 in equation (18), Sλ{cosaλ(t)} = u−1−(1) − u−(1)λ u2(−1) − 2u−1λ+ a2 + λ2 = (u−2 − u−1λ)(u2) 1− 2uλ+ λ2u2 + a2u2 = 1− uλ (1− λu)2 + a2u2 = Sλ{cosaλ(t)}. 4. When β = −α and α = −1 in equation (18), Sλ{cosaλ(t)} = u−1−(−α) − u−(−α)λ u2(−1) − 2u−1λ+ a2 + λ2 = (1− uλ)u1+α (1− λu)2 + a2u2 = Gαλ{cosaλ(t)}. Theorem 9. The degenerate Sadik transform of the function f(t) defined by f(t) = sinh (a) λ (t) is given by Sλ{sinhaλ(t)} = au−β u2α − 2uαλ− a2 + λ2 . (19) Remark 12. Observe that as λ → 0, Sλ{sinh (a) λ (t)} tends to S{sinh at}. That is, lim λ→0 Sλ{sinhaλ(t)} = lim λ→0 [ au−β u2α − 2uαλ− a2 + λ2 ] = S{sinh(at)}. Remark 13. m 1. When β = 0 and α = 1 in equation (19), Sλ{sinhaλ(t)} = au−0 u2(1) − 2u1λ− a2 + λ2 = a (u− λ)2 − a2 = Lλ{sinhaλ(t)}. 2. When β = −1 and α = −1 in equation (19), Sλ{sinhaλ(t)} = au−(−1) u2(−1) − 2u−1λ− a2 + λ2 = au3 (1− λu)2 − a2u2 = Eλ{sinhaλ(t)}. 3. When β = 1 and α = −1 in equation (19), = au−1 u2(−1) − 2u−1λ− a2 + λ2 = au (1− λu)2 − a2u2 = Sλ{sinhaλ(t)}. 4. When β = −α and α = −1 in equation (19), = au−(−α) u2(−1) − 2u−1λ− a2 + λ2 = auα+2 (1− λu)2 − a2u2 = Gαλ{sinhaλ(t)}. J. Mohamadali, N. Abdulcarim / Eur. J. Pure Appl. Math, 18 (3) (2025), 6136 15 of 19 Theorem 10. The degenerate Sadik transform of the function f(t) defined by f(t) = cosh (a) λ (t) is given by Sλ{coshaλ(t)} = uα−β − u−βλ u2α − 2uαλ− a2 + λ2 . (20) Proof. Observe that by Definition 7, for f(t) = cosh (a) λ (t) , we have Sλ{cosh (a) λ (t)} = Sλ { eaλ(t) + e−a λ (t) 2 } . Next, applying Theorems 2, 5 and Corollary 1, simplifying the resulting equation resulted to Sλ{coshaλ(t)} =Sλ { eaλ(t) + e−a λ (t) 2 } = ( 1 2 )[ Sλ { eaλ(t) } + Sλ { e−a λ (t) }] = ( 1 2 )[ u−β uα − a− λ + u−β uα + a− λ ] = ( u−β 2 )[ 2uα − 2λ u2α − 2uαλ− a2 + λ2 ] = u−βuα − u−βλ u2α − 2uαλ− a2 + λ2 = uα−β − u−βλ u2α − 2uαλ− a2 + λ2 . Remark 14. Observe that as λ → 0, Sλ{cosh(a) λ(t)} tends to S{cosh at}. That is, lim λ→0 Sλ{coshaλ(t)} = lim λ→0 [ uα−β − u−βλ u2α − 2uαλ− a2 + λ2 ] = [ uα−β − u−β(0) u2α − 2uα(0)− a2 + (0)2 ] = S{cosh(at)}. Remark 15. m 1. When β = 0 and α = 1 in equation (20), Sλ{coshaλ(t)} = u1−0 − u−0λ u2(1) − 2u1λ− a2 + λ2 = u− λ u2 − 2uλ+ λ2 − a2 = Lλ{coshaλ(t)}. 2. When β = −1 and α = −1 in equation (20), Sλ{coshaλ(t)} = u−1−(−1) − u−(−1)λ u2(−1) − 2u−1λ− a2 + λ2 = (1− uλ)u2 1− 2uλ+ λ2u2 − a2u2 = Eλ{coshaλ(t)}. J. Mohamadali, N. Abdulcarim / Eur. J. Pure Appl. Math, 18 (3) (2025), 6136 16 of 19 3. When β = 1 and α = −1 in equation (20), Sλ{coshaλ(t)} = u−1−1 − u−1λ u2(−1) − 2u−1λ− a2 + λ2 = 1− uλ (1− λu)2 − a2u2 = Sλ{coshaλ(t)}. 4. When β = −α and α = −1 in equation (20), Sλ{coshaλ(t)} = u−1−(−α) − u−(−α)λ u2(−1) − 2u−1λ− a2 + λ2 = (1− uλ)uα+1 (1− λu)2 − a2u2 = Gαλ{coshaλ(t)}. Theorem 11. The degenerate Sadik transform of the function f(t) = tn is given by, Sλ{tn} = n! uβ(uα − λ)(uα − 2λ)....(uα − nλ)(uα − (n+ 1)λ) (21) for (n− k + 1)λ− uα λ < 0. Proof. Directly from the Definition 1, for f(t) = tn, we have Sλ{tn} = 1 uβ ∫ ∞ 0 e−uα λ (t) tn dt = 1 uβλn+1 n∑ k=0 ( n k ) (−1)k lim R→∞ [ λ v −uα+λn−λk+λ λ −uα + λn− λk + λ ] ∣∣∣∣1+λR 1 = 1 uβλn n∑ k=0 ( n k ) (−1)k [ − 1 −uα + λ(n− k + 1) ] , for (n− k + 1)λ− uα λ < 0 = 1 uβλn [ 1 uα − λ(n+ 1) − n uα − nλ + ...+ n(−1)n−1 ( 1 uα − 2λ ) + (−1)n ( 1 uα − λ )] = n!λn uβλn(uα − λ)(uα − 2λ)....(uα − nλ)(uα − (n+ 1)λ) = n! uβ(uα − λ)(uα − 2λ)....(uα − nλ)(uα − (n+ 1)λ) for (n− k + 1)λ− uα λ < 0. Remark 16. Observe that as λ → 0, Sλ{tn} tends to S{tn}. That is, lim λ→0 Sλ{tn} = lim λ→0 [ n! uβ(uα − λ)(uα − 2λ)....(uα − nλ)(uα − (n+ 1)λ) ] = S{tn}. Remark 17. m J. Mohamadali, N. Abdulcarim / Eur. J. Pure Appl. Math, 18 (3) (2025), 6136 17 of 19 1. When β = 0 and α = 1 in equation (21), Sλ{tn} = n! u0(u1 − λ)(u1 − 2λ)....(u1 − nλ)(u1 − (n+ 1)λ) = n! un+1(1− λ u)(1− 2λ u )....(1− nλ u )(1− (n+1)λ u ) =Lλ{tn}. 2. When β = −1 and α = −1 in equation (21), Sλ{tn} = n! u−1(u−1 − λ)(u−1 − 2λ)....(u−1 − nλ)(u−1 − (n+ 1)λ) = n!un+2 (1− uλ)(1− 2uλ)....(1− nuλ)(1− (n+ 1)uλ) =Eλ{tn}. 3. When β = 1 and α = −1 in equation (21), Sλ{tn} = n! u1(u−1 − λ)(u−1 − 2λ)....(u−1 − nλ)(u−1 − (n+ 1)λ) = n!un (1− uλ)(1− 2uλ)....(1− nuλ)(1− (n+ 1)uλ) =Sλ{tn}. 4. When β = −α and α = −1 in equation (21), Sλ{tn} = n! u−α(u−1 − λ)(u−1 − 2λ)....(u−1 − nλ)(u−1 − (n+ 1)λ) = n!u−α+(n+1) (1− uλ)(1− 2uλ)....(1− nuλ)(1− (n+ 1)uλ) =Gαλ{tn}. Conclusion This study introduced the Degenerate version of Sadik Transform, a generalization of the Sadik Transform that sum up some known degenerate transforms, including the degen- erate Laplace, Sumudu, Tarig, Elzaki, and Laplace-type integral transforms. By defining the degenerate Sadik transform and establishing its existence under specific conditions, the paper demonstrated its capacity to serve as a unifying framework for multiple de- generate transforms. The transform of several elementary functions was derived, and the results validated that degenerate Sadik transform certainly converges to the natural Sadik transform as the degeneracy parameter approaches zero. This highlights the degenerate Sadik transform’s flexibility, depth, and potential for broader application across various mathematical and applied areas. J. Mohamadali, N. Abdulcarim / Eur. J. Pure Appl. Math, 18 (3) (2025), 6136 18 of 19 Recommendation The following are recommended for further investigations: 1. 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