Hacettepe Journal of Hacet. J. Math. Stat. Volume 49 (1) (2020), 8 – 18 & Statistics DOI : 10.15672/HJMS.2018.652 Research Article

Some identities on fractional and transforms

A. Neşe Dernek∗, Gülçin Bozkurt

Department of Mathematics, Marmara University, 34722 Kadıköy, İstanbul, Turkey

Abstract In this paper, we introduce various theorems that associate the generalized Riemann- Liouville fractional integral and the generalized Weyl fractional integral oper- ator with some well-known integral transforms including generalized , Widder potential transform, generalized Widder transform, and Bessel transform. We evaluate certain integrals of some elementary functions and some special functions as applications of these theorems and their results.

Mathematics Subject Classification (2010). 26A33, 44A10, 44A15, 33C10

Keywords. Riemann-Liouville fractional integral, Weyl fractional integral, fractional , Laplace transform, Widder potential transform, Hankel transform, Bessel transform, error , exponential integral function

1. Introduction The name fractional is used for integrals and derivatives of arbitrary order. The idea of the first arises in the 17th century and rapidly improves from that day on. In the last two decades, in addition to theoretical development, fractional integrals and derivatives are widely used in applied science and engineering studies such as viscoelastic systems, , control processing, fractional stochastic systems, and ecology [10–12]. Unlike the classical calculus, there are various definitions of fractional order integrals and fractional order derivatives. In this paper, we give place to two of the main definitions of fractional integrals which are Riemann-Liouville fractional integral and Weyl fractional integral. Also, we define two new fractional integrals motivated from these definitions and we present some new identities that include fractional integrals, fractional derivatives and some integral transforms such as Laplace transform, L2 transform, Widder potential transform and Bessel transform.

∗ Corresponding Author. Email addresses: [email protected] (A.N. Dernek), [email protected] (G. Bozkurt) Received: 11.06.2018; Accepted: 28.08.2018 Some identities on fractional integrals and integral transforms 9

2. Preliminaries Before giving the main results, we introduce some definitions about fractional integrals, fractional derivatives and integral transforms. Definition 2.1. Weyl fractional integral operator of order µ is defined in the form ∫ 1 ∞ W −µf(x) = (x − y)µ−1f(x)dx, (2.1) Γ(µ) y where y ≥ 0, µ ∈ C and Re(µ) > 0 [11]. Definition 2.2. Riemann-Liouville fractional integral operator of order µ is defined in the form ∫ 1 y D−µf(x) = (y − x)µ−1f(x)dx, (2.2) Γ(µ) 0 where y ≥ 0 and, µ ∈ C and Re(µ) > 0 [11,14]. In the following, we introduce two new fractional integrals namely the generalized Weyl fractional integral and the generalized Riemann-Liouville fractional integral. Definition 2.3. The generalized Weyl fractional integral can be defined as follows: ∫ ∞ 1 2 2 µ−1 Wµ,2{f(x); y} = x(x − y ) f(x)dx, (2.3) Γ(µ) y where y ≥ 0 and, µ ∈ C and Re(µ) > 0. Definition 2.4. The generalized Riemann-Liouville fractional integral is defined as fol- lows: ∫ y 1 2 2 µ−1 Rµ,2{f(x); y} = x(y − x ) f(x)dx, (2.4) Γ(µ) 0 where y ≥ 0, µ ∈ C and Re(µ) > 0. In the definitions (2.1)−(2.4), Γ(z) is the Gamma Euler function given by the following integral, ∫ ∞ Γ(z) = e−ttz−1dt, (2.5) 0 where z ∈ C and Re(z) > 0. Definition 2.5. Weyl fractional of order α is defined by the formula dn W αf(x) = W −(n−α)f(x), (2.6) dxn where n ∈ N, α ∈ C, Re(α) > 0 and n − 1 < Re(α) ≤ n [11]. Definition 2.6. Riemann-Liouville fractional derivative of order α is defined by the for- mula dn Dαf(x) = D−(n−α)f(x), (2.7) 0 y dxn where n ∈ N, α ∈ C, Re(α) > 0 and n − 1 < Re(α) ≤ n [11]. In the formulas (2.6) and (2.7), fractional derivatives are defined by means of fractional integral operators. Definition 2.7. The Laplace transform of f(x) is defined by the following formula [1], ∫ ∞ L{f(x); y} = e−xyf(x)dx. (2.8) 0

Definition 2.8. The L2 transform of f(x) is defined by the following formula [14, 15], ∫ ∞ −x2y2 L2{f(x); y} = xe f(x)dx. (2.9) 0 10 A.N. Dernek and G. Bozkurt

The L2 transform and the Laplace transform are related by the equation [3], 1 L {f(x); y} = L{f(x1/2); y2}. (2.10) 2 2 Definition 2.9. The Widder Potential transform of f(x) is defined by [13], ∫ ∞ P{ } xf(x) f(x); y = 2 2 dx. (2.11) 0 x + y Definition 2.10. The Stieltjes integral transform is defined by [1,8], ∫ ∞ f(x) S{f(x); y} = dx. (2.12) 0 x + y The Widder Potential transform and the Stieltjes transform are related by the following relation [7], 1 P{f(x); y} = S{f(x1/2); y2}. (2.13) 2 Definition 2.11. The 2n-Generalized Widder potential transform of f(x) is defined by [2, 3], ∫ ∞ P { } 2n−1 f(x) ∈ N 2n f(x); y = x 2n 2n dx, where n . (2.14) 0 x + y

The P4 transform, which is the special case of 2n-Generalized Widder potential trans- form that appears with the choice of n = 2, and the Steiltjes transform are related by the following relation [6], { } 1 P {f(x); y} = S f(x1/4); y4 . (2.15) 4 4 Definition 2.12. The generalized Widder potential transform of f(x) is defined by [5], ∫ ∞ P { } xf(x) ∈ C ν,2 f(x); y = 2 2 ν dx, where ν . (2.16) 0 (x + y ) The Widder potential transform is the special case of generalized Widder potential transform that appears with the choice of ν = 1. Definition 2.13. The Hankel transform of f(x) is defined by [4], ∫ ∞ 1/2 Hν{f(x); y} = (xy) Jv(xy)f(x)dx, (2.17) 0 where Jv(x) is the Bessel function of the first kind that has the following represen- tation, ∞ ( ) ∑ x ν+2n 1 J (x) = (−1)n , where Re(ν) > −(n + 1). (2.18) ν 2 n!Γ(ν + n + 1) n=0 Definition 2.14. The Bessel transform of f(x) is defined by the identity [4], ∫ ∞ 1/2 Kν{f(x); y} = (xy) Kv(xy)f(x)dx (2.19) 0 where Kv(x) is the modified Bessel function of the second kind and defined as π I− (x) − I (x) K (x) = ν ν (2.20) ν 2 sin(πν) in which Iν(x) is the modified Bessel function of the first kind and defined by the series representation ∞ ( ) ∑ x ν+2n 1 I (x) = , where Re(ν) > −(n + 1). (2.21) ν 2 n!Γ(ν + n + 1) n=0 Some identities on fractional integrals and integral transforms 11

Definition 2.15. The error function Erf(x) is defined by the equation [9], ∫ x 2 2 Erf(x) = √ e−t dt. (2.22) π 0 Definition 2.16. The complementary error function Erfc(x) is defined by the equation [9], ∫ ∞ 2 2 Erfc(x) = √ e−t dt. (2.23) π x

3. Identities on generalized fractional integrals and the L2 transform By the following lemma, we give a relation between the generalized Weyl fractional integral and the generalized Laplace transform. Lemma 3.1. If Re(µ) > 0 for µ ∈ C, the following identity holds true, provided that the integrals involved converge absolutely, { } 1 f(x) W {L {f(x); u}; y} = L ; y . (3.1) µ,2 2 2 2 x2µ Proof. By definitions (2.3) and (2.9), we have ∫ ∞ [∫ ∞ ] 1 2 2 µ−1 −u2x2 Wµ,2{L2{f(x); u}; y} = u(u − y ) xe f(x)dx du. (3.2) Γ(µ) y 0 By making the change of u2 − y2 = t2 in (3.2), we get that ∫ ∞ [∫ ∞ ] 1 2 µ−1 −(t2+y2)x2 Wµ,2{L2{f(x); u}; y} = t(t ) xe f(x)dx dt. Γ(µ) 0 0 Then, by changing the order of integration that is permissible by absolute convergence of the integrals involved, we get ∫ ∞ [∫ ∞ ] 1 −x2y2 2(µ−1) −t2x2 Wµ,2{L2{f(x); u}; y} = xe f(x) tt e dt dx Γ(µ) 0 0 ∫ ∞ [ ] 1 −x2y2 2(µ−1) = xe f(x) L2{t ; x} dx. Γ(µ) 0 Applying the relation (2.10) and using the equation (1) of [7, p. 137], we obtain

∫ ∞ [ ] 1 −x2y2 1 µ−1 2 Wµ,2{L2{f(x); u}; y} = xe f(x) L{t ; x } dx Γ(µ) 0 2 ∫ ∞ [ ] 1 2 2 1 = xe−x y f(x) Γ(µ)x−2µ dx Γ(µ) 0 2 ∫ ∞ { } 1 −x2y2 f(x) 1L f(x) = xe 2µ dx = 2 2µ ; y 2 0 x 2 x which proves the statement. 

Next theorem gives a Parseval-Goldstein type relation between the L2 transform and the generalized Riemann-Liouville fractional integral defined by (2.4). Theorem 3.2. If Re(µ) > 0 for µ ∈ C, the following Parseval-Goldstein type relation holds true, provided that the integrals involved converge absolutely, ∫ ∫ { } ∞ − µ−1 ∞ R { } L { } ( 1) L g(u) t µ,2 f(x); t 2 g(u); t dt = xf(x) 2 2µ ; x dx. (3.3) 0 2 0 u 12 A.N. Dernek and G. Bozkurt

Proof. By the definition (2.4), we have ∫ ∞ I = tRµ,2 {f(x); t} L2 {g(u); t} dt 0 ∫ [ ∫ ] ∞ t 1 2 2 µ−1 = t x(t − x ) f(x)dx L2 {g(u); t} dt. (3.4) 0 Γ(µ) 0 By changing the order of integration that is permissible under the absolute convergence assumption, we obtain ∫ ∞ [∫ ∞ ] 1 2 2 µ−1 I = xf(x) t(t − x ) L2 {g(u); t} dt dx Γ(µ) 0 x ∫ [∫ ] − µ−1 ∞ ∞ ( 1) 2 2 µ−1 = xf(x) t(x − t ) L2 {g(u); t} dt dx. Γ(µ) 0 x

Then, by using the definition (2.3) and Lemma (3.1), we get

∫ ∞ µ−1 I = (−1) xf(x)Wµ,2 {L2 {g(u); t} ; x} dx ∫0 { } − µ−1 ∞ ( 1) L g(u) = xf(x) 2 2µ ; x dx 2 0 u which proves the statement. 

Theorem 3.3. The following Parseval-Goldstein type relation holds true, provided that each of the integrals involved converge absolutely, { { } } √ ∫ ( ) 1 1 π ∞ y L2 R1,2 f(x); ; y = xf(x) Erf dx. (3.5) u u 2y 0 x Proof. By setting µ = 1 and g(u) = sin(2uy) in (3.3), we obtain

∫ ∞ ∫ ∞ { } R { } L { } 1 L sin(2uy) t 1,2 f(x); t 2 sin(2uy); t dt = xf(x) 2 2 ; x dx. (3.6) 0 2 0 u Now, by the relation (2.10) and the relation (32) of [7, p.153], we have { } 1 √ L {sin(2uy); t} = L sin(2 uy); t2 2 2 1√ 2 2 = πyt−3e−y /t , Re(t2) > 0. (3.7) 2 Likewise, by the relation (2.10) and the relation (34) of [7, p.154], we have { } { √ } sin(2uy) 1 sin(2 uy) L ; t = L ; t2 2 u2 2 u ( ) 1 y = π Erf , Re(t2) > 0. (3.8) 2 x Then, by substituting (3.7) and (3.8) in (3.6), we get ∫ ( ) ∫ ( ( )) ∞ √ ∞ 1 −3 −y2/t2 1 1 y tR1,2 {f(x); t} πyt e dt = xf(x) π Erf dx 0 2 2√ 0 2 x ∫ ∞ ∫ ∞ ( ) −3 −y2/t2 π y tR1,2 {f(x); t} t e dt = xf(x) Erf dx. 0 2y 0 x Some identities on fractional integrals and integral transforms 13

Finally, by making change of variable t = 1/u on the left hand side of equation above and using the definition (2.9), we obtain √ ∫ ∞ { } ∫ ∞ ( ) 1 1 2 2 π y R f(x); ue−y u du = xf(x) Erf dx u 1,2 u 2y x 0 { { } } √ ∫0 ( ) 1 1 π ∞ y L2 R1,2 f(x); ; y = xf(x) Erf dx, u u 2y 0 x as desired. 

Example 3.4. Following equation holds true under the conditions of Theorem (3.3), ∫ ( ) ∞ 1 ν tν−2 Erf(t)dt = √ Γ , where 0 < Re(ν) < 1. (3.9) 0 π(1 − ν) 2

Proof. By choosing f(x) = x−ν−1 in Theorem (3.3), we get √ { { } } ∫ ∞ ( ) 1 −ν−1 1 π −ν y L2 R1,2 x ; ; y = x Erf dx. (3.10) u u 2y 0 x Now, by using the definitions (2.4) and (2.9) on the left hand side of the equation (3.10), we get { { } } { (∫ ) } 1/u 1 −ν−1 1 1 −ν L2 R1,2 x ; ; y = L2 x dx ; y u u u 0 { ( ) } 1 uν−1 = L ; y 2 u 1 − ν { } 1 = L uν−2; y . (1 − ν) 2

Then, by using the relation (2.10) and the equation (1) of [7, p.137], we have { { } } { } 1 1 1 L R x−ν−1; ; y = L uν/2−1; y2 (3.11) 2 u 1,2 u 2(1 − ν) ( ) 1 ν = Γ y−ν 2(1 − ν) 2 and by making change of variable x = yt−1 on the right hand side of the equation (3.10), we get √ ∫ ( ) √ ∫ π ∞ y π ∞ x−ν Erf dx = (yt−1)−ν Erf(t)(yt−2) dt 2y x 2y 0 √ 0 ∫ πy−ν ∞ = tν−2 Erf(t) dt. (3.12) 2 0 From the equations (3.11) and (3.12) we have, ( ) √ ∫ 1 ν πy−ν ∞ Γ y−ν = tν−2 Erf(t) dt 2(1 − ν) 2 2 0 which proves the statement.  14 A.N. Dernek and G. Bozkurt

4. An identity on fractional integrals and the 2n-generalized widder po- tential transform

Theorem 4.1. If Re(t2) > |Im(y2)|, then the identity { } ∫ ( ) ∞ 2 P 1 R { } 1 −1 y 4 2 1,2 f(x); t ; y = 2 xf(x) tan 2 dx (4.1) t 2y 0 x holds true, provided that each of the integrals involved converges absolutely. Proof. By choosing µ = 1 and g(u) = sin(u2y2) in the equation (3.3), we obtain ∫ ∞ { } 2 2 tR1,2 {f(x); t}L2 sin(u y ); t dt (4.2) 0 ∫ { } ∞ 2 2 1 L sin(u y ) = xf(x) 2 2 ; x dx. 2 0 u Now, by the relation (2.10) and the formula (1) of [7, p.150], we have { } { } 1 L sin(u2y2); t = L sin(uy2); t2 (4.3) 2 2 1 y2 = , Re(t2) > |Im(y2)|. 2 y4 + t4 Likewise, by using the relation (2.10) and the equation (16) of [7, p.152], we have { } { } sin(u2y2) 1 sin(uy2) L ; x = L ; x2 (4.4) 2 u2 2 u ( ) 1 y2 = tan−1 . 2 x2 By substituting (4.3) and (4.4) in (4.2) and using definition (2.14), we get ∫ ∫ ( ) ∞ ∞ 2 t R { } 1 −1 y 4 4 1,2 f(x); t dt = 2 xf(x) tan 2 dx 0 y + t 2y 0 x { } ∫ ( ) ∞ 2 P 1 R { } 1 −1 y 4 2 1,2 f(x); t ; y = 2 xf(x) tan 2 dx t 2y 0 x



Example 4.2. If −3 < Re(ν) < 1, then the following equation holds true under the conditions of Theorem (4.1), ∫ ( ) ( ) ∞ 2 − ν+1 − ν −1 y πy π(ν 1) x tan 2 dx = csc . (4.5) 0 x 2(ν + 1) 4 Proof. If we choose f(x) = xν−1 on the left hand side of equation (4.1), we get { } { (∫ ) } { } t P 1 R ν−1 P 1 ν 4 2 1,2 x ; t ; y = 4 2 x dx ; y . t t 0 Then, by using the definitions (2.4) and (2.14) respectively, we obtain { ( ) } { { } } 1 1 tν+1 P R xν−1; t ; y = P ; y (4.6) 4 t2 1,2 4 t2 ν + 1 { } 1 = P tν−1; y , ν + 1 4 Some identities on fractional integrals and integral transforms 15 and by using the identity (2.15) and the equation (5) of [8, p.216], we get { { } } { } 1 1 P R xν−1; t ; y = S t(ν−1)/4; y4 4 t2 1,2 4(ν + 1) ( ) −πyν−1 π(ν − 1) = csc . (4.7) 4(ν + 1) 4 Finally, by substituting (4.7) in equation (4.1), we obtain ( ) ∫ ( ) − ν−1 − ∞ 2 πy π(ν 1) 1 ν −1 y csc = 2 x tan 2 dx, 4(ν + 1) 4 2y 0 x which proves the statement.  5. Identities for various integral transforms and fractional integrals

Lemma 5.1. If Re(µ) < 1, Re(t2) > −Re(s2) and Re(x2) > −Re(s2), then the identity (−1)µ−1 P {R {f(x); t} ; s} = Γ(1 − µ)P − {f(x); s} , (5.1) µ,2 2 1 µ,2 holds true, provided that each of the integrals involved converges absolutely. Proof. By choosing g(u) = e−s2u2 in the equation (3.3), we get ∫ ∞ { } −s2u2 tRµ,2 {f(x); t}L2 e ; t dt (5.2) 0 ∫ { } − µ−1 ∞ −s2u2 ( 1) L e = xf(x) 2 2µ ; x dx. 2 0 u Now by (2.10) and the relation (1) of [7, p.143], we have { } { } 2 2 1 2 L e−s u ; t = L e−s u; t2 (5.3) 2 2 1 = (t2 + s2)−1, Re(t2) > −Re(s2). 2 Likewise, by (2.10) and the relation (3) of [7, p.144], we have { } { } e−s2u2 1 e−s2u L ; x = L ; x2 (5.4) 2 u2µ 2 uµ 1 = Γ(1 − µ)(x2 + s2)µ−1, Re(x2) > −Re(s2). 2 By substituting results (5.3) and (5.4) into the equation (5.2), we get ∫ ∞ 2 2 −1 tRµ,2 {f(x); t}(t + s ) dt (5.5) 0 ∫ (−1)µ−1 ∞ = Γ(1 − µ) xf(x)(x2 + s2)µ−1dx. 2 0 Then, the equation (5.1) can be shown directly by using the definitions (2.11), (2.16) and the relation (5.5). 

Theorem 5.2. If Re(µ) < 1, Re(t2) > −Re(s2) and Re(x2) > −Re(s2) for µ, s, t ∈ C, then the Parseval-Goldstein type identity ∫ ∞ tP {f(x); t} Rµ,2 {g(u); t} dt (5.6) 0 ∫ (−1)µ−1 ∞ = Γ(1 − µ) xf(x)P1−µ,2 {g(u); x} dx 2 0 16 A.N. Dernek and G. Bozkurt holds true, provided that each of the integrals involved converges absolutely. Proof. Firstly, if we apply the definition (2.11) on the left-hand side of equation (5.6) and then change the order of integration under the absolute convergence condition, we obtain ∫ ∞ tP {f(x); t}Rµ,2 {g(u); t} dt 0 ∫ ∞ ∫ ∞ xf(x) R { } = t 2 2 µ,2 g(u); t dx dt 0 0 x + t ∫ ∞ ∫ ∞ t R { } = xf(x) 2 2 µ,2 g(u); t dt dx 0 0 x + t ∫ ∞ = xf(x)P {Rµ,2 {g(u); t} ; x} dx. (5.7) 0 On the other hand, we have the following relation by Lemma (5.1), ∫ ∞ tP {f(x); t}Rµ,2 {g(u); t} dt 0 ∫ ( ) ∞ (−1)µ−1 = xf(x) Γ(1 − µ)P1−µ,2 {g(u); x} dx 0 2 ∫ (−1)µ−1 ∞ = Γ(1 − µ) xf(x)P1−µ,2 {g(u); x} dx. (5.8) 2 0 The relation (5.6) is a direct consequence of equations (5.7) and (5.8). 

Corollary 5.3. The following equation holds true under the conditions of Theorem (5.2), ∫ − 1−µ ∞ ( 1) y2t2 2 2 L2 {P1−µ,2 {g(u); x} ; y} = te E1(y t )Rµ,2 {g(u); t} dt, (5.9) Γ(1 − µ) 0 where Re(µ) < 1 and E1(x) is the exponential integral function which is defined by the following identity [15], ∫ ∞ e−u E1(x) = du. (5.10) x u Proof. If we choose f(x) = e−x2y2 in the equation (5.6), we get ∫ ∞ { } −x2y2 tP e ; t Rµ,2 {g(u); t} dt (5.11) 0 ∫ − µ−1 ∞ ( 1) −x2y2 = Γ(1 − µ) xe P1−µ,2 {g(u); x} dx. 2 0 By the equation (2.13) and the relation (11) of [8, p.217], we have { } { } 2 2 1 2 P e−x y ; t = S e−xy ; t2 2 1 2 2 = ey t E (y2t2). (5.12) 2 1 Then, the equation (5.11) can be written as, ∫ ∞ ( ) y2t2 2 2 t e E1(y t ) Rµ,2 {g(u); t} dt 0 µ−1 = (−1) Γ(1 − µ)L2 {P1−µ,2 {g(u); x} ; y} , by using the definition (2.9) and the relation (5.12).  Some identities on fractional integrals and integral transforms 17

Corollary 5.4. The following equation holds true under the conditions of Theorem (5.2), { } −1 L2 x P1−µ,2 {g(u); x} ; y dx (5.13) ∫ − µ−1 ∞ π( 1) y2t2 = e Erfc(yt)Rµ,2 {g(u); t} dt, Re(µ) < 1. Γ(1 − µ) 0 Proof. If we choose f(x) = x−1e−x2y2 in the equation (5.6) and use the formula (15) of [8, p.217], we get { } { } 2 2 1 2 P x−1e−x y ; t = S x−1/2e−xy ; t2 2 1 2 2 = πt−1ey t Erfc(yt). (5.14) 2 Now, by substituting the equation (5.14) in the equation (5.6) and using the relation (2.9), we obtain ∫ ∞ π 2 2 ey t Erfc(yt)R {g(u); t} dt 2 µ,2 0 ∫ − µ−1 ∞ ( 1) −x2y2 = Γ(1 − µ) e P1−µ,2 {g(u); x} dx 2 0 − µ−1 { } ( 1) −1 = Γ(1 − µ)L x P − {g(u); x} ; y dx. 2 2 1 µ,2 

Corollary 5.5. If y > 0, −k − 1 < Re(ν) < −2k + 3/2 for k = 0, 1, 2, ... and Re(µ) < 1 then the identity − { } 2(−1)k µ+1 K tν+2k+1/2R {g(u); t} ; y (5.15) Γ(1 − µ) ν µ,2 { } ν+2k+1/2 = Hν x P1−µ,2 {g(u); x} ; y , holds true, provided that each of the integrals involved converges absolutely.

ν+2k Proof. By taking f(x) = x Jν(xy) in Theorem (5.2), we obtain ∫ ∞ { } ν+2k t P x Jν(xy); t Rµ,2 {g(u); t} dt (5.16) 0 ∫ − µ−1 ∞ ( 1) ν+2k+1 = Γ(1 − µ) x Jν(xy)P1−µ,2 {g(u); x} dx. 2 0 Now, by the relation (2.13) and the identity (10) of [8, p.225], we have { } { } 1 P xν+2kJ (xy); t = S xν/2+kJ (x1/2y); t2 ν 2 ν k ν+2k = (−1) t Kν(yt), (5.17) where y > 0 and −k − 1 < Re(ν) < −2k + 3/2 for k ∈ N. Now, by using the equation (5.17) and the definition (2.19), we obtain ∫ ∞ { } ν+2k t P x Jν(xy); t Rµ,2 {g(u); t} dt 0 ∫ ∞ k ν+2k+1 = (−1) t Kν(yt)Rµ,2 {g(u); t} dt 0 { } (−1)k = K tν+2k+1/2R {g(u); t} ; y . (5.18) y1/2 ν µ,2 18 A.N. Dernek and G. Bozkurt

On the other hand, by using the definition (2.17), we get ∫ ∞ ν+2k+1 x Jν(xy)P1−µ,2 {g(u); x} dx (5.19) 0 { } 1 ν+2k+1/2 = H x P − {g(u); x} ; y . y1/2 ν 1 µ,2 So, the equation (5.15) follows directly from the equations (5.18) and (5.19). 

References [1] L. Debnath and D. Bhatta, Integral Transforms and Their Applications 2nd Edition, Chapman and Hall CRC, Boca Raton, London and New York, 2007. [2] A.N. Dernek and F. Aylıkçı, Identities for the Ln-Transform, the L2n-Transform, and the P2n-Transform and Their Applications, Journal of Inequaities and Special Functions, 5 (4), 1–16, 2014. [3] A.N. Dernek and F. Aylıkçı, Some Results on the Pν,2n, Kν,n, Hν,n Integral trans- forms, Turkish J. Math. 41, 337–349, 2017. [4] A. Dernek, N. Dernek and O. Yürekli, Identities for the Henkel Transform and Their Applications, J. Math. Anal. Appl. 354, 165–176, 2009. [5] A.N. Dernek, V. Kurt, Y. Şimşek and O. Yürekli, A generalization of Widder Potential Transform and Applications, Integral Transforms Spec. Funct. 22, 391–401, 2011. [6] A.N. Dernek, H.M. Srivasta and O. Yürekli, Parseval-Goldstein Type Identities In- volving The L4 Transform and P4 Transform and Their Applications, Integral Trans- forms Spec. Funct. 18, (6), 397–408, 2007. [7] A. Erdelyi, W. Magnus, F. Oberhettinger and F. Tricomi, Tables of Integral Trans- forms Vol: 1, Mc-Graw-Hill Book Company, New York, Toronto and London, 1954. [8] A. Erdelyi, W. Magnus, F. Oberhettinger and F. Tricomi, Tables of Integral Trans- forms Vol: 2, Mc-Graw-Hill Book Company, New York, Toronto and London, 1954. [9] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products, Academic Press, New York, 1996. [10] C. Li, D. Qian and Y. Chen, On Riemann Liouville and Caputo Derivatives, Discrete Dyn. Nat. Soc. 2011, 2011. [11] K.S. Miller and B. Ross, An Introduction to Fractional Calculus and Fractional Dif- ferential Equations, John Wiley-Interscience Publication, New York, Toronto, Singa- pore, 1993. [12] I. Podlubny, An Introduction to Fractional Derivatives, Fractional Differential Equa- tions, to Methods of their Solution and Some of their Applications, Academic Press, San Diego and London, 1990. [13] D.V. Widder, An Introduction to Transform Theory, Academic Press, New York, 1971. [14] O. Yürekli and I. Sadek, A Parseval-Goldstein Type Theorem on the Widder Potential Transform and Its Applications, International J. Math. Math. Sci. 14 (3), 517–524, 1991. [15] O. Yürekli, Identities on Fractional Integrals and Various Integral Transforms, Appl. Math. Comput. 187, 559–566, 2007.