Generalized fractional integration of k\-Bessel function
aa r X i v : . [ m a t h . C A ] D ec GENERALIZED FRACTIONAL INTEGRATION OF k -BESSEL FUNCTION G. RAHMAN, K.S. NISAR ∗ , S. MUBEEN, AND M. ARSHAD Abstract.
In this present paper our aim is to deal with two integraltransforms which involving the Gauss hypergeometric function as itskernels. We prove some compositions formulas for such a generalizedfractional integrals with k -Bessel function. The results are establishedin terms of generalized Wright type hypergeometric function and gen-eralized hypergeometric series. Also, the authors presented some corre-sponding assertions for RiemannLiouville and Erd´elyiKober fractionalintegral transforms. Keywords and phrases. fractional integral operator, Bessel function,generalized Wright function, generalized hypergeometric function. Introduction and Preliminaries
The Gauss hypergeometric function is defined as:(1) F ( a, b ; c ; z ) = ∞ X n =0 ( a ) n ( b ) n ( c ) n z n n ! , where a, b, c ∈ C , c = 0 , − , − , · · · and ( λ ) n is the Pochhammer symboldefined for λ ∈ C and n ∈ N as:(2) ( λ ) = 1 , ( λ ) n = λ ( λ + 1)( λ + 2) · · · ( λ + n − n ∈ N . The series defined in (1) is absolutely convergent for | z | < | z | = 1(see [2]). Saigo [9] introduced the following left and right sided generalizedintegral transforms defined for x > (cid:16) I α,β,η f (cid:17) ( x ) = x − α − β Γ( α ) × x Z ( x − t ) α − F (cid:0) α + β, − η ; α ; 1 − tx (cid:1) f ( t ) dx, (3) *corresponding author. ∗ , S. MUBEEN, AND M. ARSHAD and (cid:16) I α,β,η − f (cid:17) ( x ) = 1Γ( α ) × ∞ Z x ( x − t ) α − t − α − β F (cid:0) α + β, − η ; α ; 1 − xt (cid:1) f ( t ) dx, (4)where α, β, η ∈ C and R e ( α ) > F ( a, b ; c ; z ) is Gauss hypergeometricfunction defined in (1). When β = − α , then (2) and (4) will lead to theclassical Riemann-Liouville left and right-sided fractional integrals of order α ∈ C , R ( α ) >
0, (see [11]): (cid:16) I α,β,η f (cid:17) ( x ) = x − α − β Γ( α ) x Z ( x − t ) α − f ( t ) dx ( x > , (5)and (cid:16) I α,β,η f (cid:17) ( x ) = 1Γ( α ) ∞ Z x ( x − t ) α − t − α − β f ( t ) dx ( x > . (6)If β = 0, then equations (3) and (4) will reduce to the well known Erd´elyi-Kober fractional defined as: (cid:0) I α, ,η f (cid:1) ( x ) = (cid:0) K + η,α f (cid:1) ( x ) = x − α − β Γ( α ) x Z ( x − t ) α − t η f ( t ) dx (7)and (cid:0) I α, ,η f (cid:1) ( x ) = (cid:0) K − η,α f (cid:1) ( x ) = x η Γ( α ) ∞ Z x ( x − t ) α − t − α − η f ( t ) dx, (8)where α, η ∈ C , R ( α ) > k -Bessel function defined in [10] as: W kv,c ( z ) = ∞ X n =0 ( − c ) n Γ k ( nk + v + k ) n ! ( z n + vk , (9)where k > v > −
1, and c ∈ R and Γ k ( z ) is the k -gamma function definedin [1] as: Γ k ( z ) = ∞ Z t z − e − tkk dt, z ∈ C . (10)By inspection the following relation holds:Γ k ( z + k ) = z Γ k ( z )(11)and Γ k ( z ) = k zk − Γ( zk ) . (12)If k → c = 1, then the generalized k -Bessel function defined in (9)reduces to the well known classical Bessel function J v defined in [3]. For ENERALIZED FRACTIONAL INTEGRATION OF k -BESSEL FUNCTION 3 further detail about k -Bessel function and its properties (see [4]-[6]).The generalized hypergeometric function p F q ( z ) is defined in [2] as: p F q ( z ) = p F q, ( α ) , ( α ) , · · · ( α p ) ; z ( β ) , ( β ) , · · · ( β q ) = ∞ X n =0 ( α ) n ( α ) n · · · ( α p ) n ( β ) n ( β ) n · · · ( β q ) n z n n ! , (13)where α i , β j ∈ C ; i = 1 , , · · · , p , j = 1 , , · · · , q and b j = 0 , − , − , · · · and( z ) n is the Pochhammer symbols. The gamma function is defined as:Γ( µ ) = ∞ Z t µ − e − t dt, µ ∈ C , (14) Γ( z + n ) = ( z ) n Γ( z ) , z ∈ C , (15)and beta function is defined as: B ( x, y ) = Z t x − (1 − t ) y − dt. (16)Also, the following identity of Gauss hypergeometric function holds: F ( a, b ; c ; 1) = Γ( c )Γ( c − a − b )Γ( c − a )Γ( c − b ) ; R ( c − a − b ) > , (17)(see [2], [11]).The Wright type hypergeometric function is defined (see [12]-[14]) by thefollowing series as: p Ψ q ( z ) = p Ψ q ( α i , A i ) ,p ; z ( β j , B j ) ,q = ∞ X n =0 Γ( α + A n ) · · · Γ( α p + A p n )Γ( β + B n ) · · · Γ( β q + B q n ) z n n !(18)where β r and µ s are real positive numbers such that1 + q X s =1 β s − p X r =1 α r > . Equation (18) differs from the generalized hypergeometric function p F q ( z )defined (13) only by a constant multiplier. The generalized hypergeometricfunction p F q ( z ) is a special case of p Ψ q ( z ) for A i = B j = 1, where i =1 , , · · · , p and j = 1 , , · · · , q :1 q Q j =1 Γ( β j ) p F q ( α ) , · · · ( α p ) ; z ( β ) , · · · ( β q ) = 1 p Q i =1 Γ( α i ) p Ψ q ( α i , ,p ; z ( β j , ,q . (19) G. RAHMAN, K.S. NISAR ∗ , S. MUBEEN, AND M. ARSHAD For various properties of this function see [7].
Lemma 1.1 (A. A. Kilbas and N. Sebastian [8]) Let α, β, η ∈ C , R ( α ) > λ > max[0 , β − n ], then the following relation holds: (cid:16) I α,β,η t λ − (cid:17) ( x ) = Γ( λ )Γ( λ + η − β )Γ( λ − β )Γ( λ + α + η ) x λ − β − (20) Lemma 1.2 (A. A. Kilbas and N. Sebastian [8]) Let α, β, η ∈ C , R ( α ) > λ > max[0 , β − n ], then the following relation holds: (cid:16) I α,β,η − t λ − (cid:17) ( x ) = Γ( η − λ + 1)Γ( β − λ + 1)Γ(1 − λ )Γ( α + β + η − λ + 1) x λ − β − (21)In the same paper, they define the following left and right sided Erd´elyi-Kober fractional integral as: (cid:0) K + η,α t λ − (cid:1) ( x ) = Γ( λ + n )Γ( λ + α + η ) x λ − , (22)where R ( α ) > R ( λ ) > − R ( η ), and (cid:0) K − η,α t λ − (cid:1) ( x ) = Γ( η − λ + 1)Γ( α + η − λ + 1) x λ − , (23)where R ( λ ) < R ( η ).2. Representation of Generalized fractional integrals in termof Wright functions
In this section, we introduce the generalized left-sided fractional integra-tion (3) of the k -Bessel functions (9). It is given by the following result. Theorem 2.1.
Assume that α , β , η , λ , v ∈ C be such that R ( v ) > − , R ( α ) > , R ( λ + v ) > max[0 , R ( β − η )] , (24) then the following result holds: (cid:16) I α,β,η t λk − W kv,c ( t ) (cid:17) ( x ) = x λk + vk − β − (2 k ) vk × Ψ ( λk + vk , , ( λk + vk + η − β, k ) | − cx k ( λk + vk − β, , ( λk + vk + α + η, , ( vk + 1 , . (25) Proof.
Note that the condition1 + q X j =1 β j − p X i =1 α i > Ψ ( z ) is defined. Now, from (3) and (9), we have (cid:16) I α,β,η t λk − W kv,c ( t ) (cid:17) ( x ) = ∞ X n =0 ( − c ) n ( ) vk +2 n Γ k ( v + k + nk ) n ! (cid:16) I α,β,η ,k t λ + vk +2 n − (cid:17) ( x ) ENERALIZED FRACTIONAL INTEGRATION OF k -BESSEL FUNCTION 5 By equation (24) and for any n = 0 , , , · · · , R ( λ + v + 2 nk ) ≥ R ( λ + v ) > max[0 , R ( β − η )]. Applying equation (21), we obtain (cid:16) I α,β,η ,k t λk − W kv,c ( t ) (cid:17) ( x ) = x λ + vk − β − vk × ∞ X n =0 Γ( vk + λk + 2 n )Γ( vk + λk + η − β + 2 n )Γ( vk + λk − β + 2 n )Γ( vk + λk + α + η + 2 n )Γ k ( vk + 1 + n ) k vk × ( − cx ) n (4 k ) n n !(26)By equation (18), we obtain (cid:16) I α,β,η ,k t λk − W kv,c ( t ) (cid:17) ( x )= x vk + λk − β − (2 k ) vk Ψ ( vk + λk , , ( vk + λk + η − β, | − cx k ( vk + λk − β, , ( vk + λk + α + η, , ( vk + 1 , . This is the required proof of (25). (cid:3)
Corollary 2.2.
Assume that α , λ , v ∈ C be such that R ( v ) > − , R ( α ) > , R ( λ + v ) > , then the following result holds: (cid:16) I α t λk − W kv,c ( t ) (cid:17) ( x ) = x vk + λk + α − (2 k ) vk × Ψ ( v + λ, k ) | − cx k ( vk + λk + α, , ( vk + 1 , k ) . (27) Proof.
By substituting β = − α in (25), we obtain the required result. (cid:3) Corollary 2.3.
Assume that α , η , λ , v ∈ C be such that R ( v ) > − , R ( α ) > , R ( λ + v ) > , then the following formula holds: (cid:16) K + α,η t λk − W kv,c ( t ) (cid:17) ( x ) = x vk + λk − (2 k ) vk × Ψ ( vk + λk + η, | − cx k ( vk + λk + α + η, , ( vk + 1 , . (28) Proof.
By setting β = 0 in (25), we get the desired result. (cid:3) Theorem 2.4.
Assume that α , β , η , λ , v ∈ C and k > be such that R ( v ) > − , R ( α ) > , R ( λ − v ) < R ( β ) , R ( η )] , (29) then the following result holds: (cid:16) I α,β,η − t λk − W kv,c ( t ) (cid:17) ( x ) = x λ − vk − β − (2 k ) vk G. RAHMAN, K.S. NISAR ∗ , S. MUBEEN, AND M. ARSHAD (30) × Ψ (1 + β − λk + vk , , (1 − λk + vk + η, | − c x (1 − λk + vk , , (1 + β + α + η − λk + vk , , ( vk + 1 , k ) . Proof.
Note that the condition1 + q X j =1 β j − p X i =1 α i > i.e., > −
1) is satisfied so therefore Ψ ( z ) is defined. Now, from (4) and(9), we have (cid:16) I α,β,η − t λk − W kv,c ( t ) (cid:17) ( x ) = ∞ X n =0 ( − c ) n ( ) vk +2 n Γ k ( v + k + nk ) n ! (cid:16) I α,β,η − t λk + vk − n − (cid:17) ( x )By equation (29) and for any k > n = 0 , , , · · · , R ( λ − v − n − ≤ R ( λ − v − < β, R ( η )]. Applying equation (21), we obtain (cid:18) I α,β,η − t λk − W kv,c ( 1 t ) (cid:19) ( x ) = x λk + vk − β − (2 k ) vk × ∞ X n =0 Γ( β − λk + vk + 1 + 2 n )Γ( η − λk + vk + 1 + 2 n )Γ(1 − λk + vk + 2 n )Γ( α + β + η − λk + vk + 1 + 2 n )Γ( vk + 1 + n ) × ( − c ) n (4 kx ) n n !(31)By equation (18), we obtain (cid:18) I α,β,η − t λk − W kv,c ( 1 t ) (cid:19) ( x ) = x λk − vk − βk − (2 k ) vk × Ψ ( β − λk + vk + 1 , , ( η − λk + vk + 1 , | − c kx (1 − λk + vk , , ( α + β + η − λk + vk + 1 , , ( vk + 1 , . This is the required proof of (30). (cid:3)
Corollary 2.5.
Assume that α , η λ , v ∈ C and k > be such that R ( v ) > − , < R ( α ) < − R ( λ − v ) , then the following result holds: (cid:16) I α t λk − W kv,c ( t ) (cid:17) ( x ) = x λk − vk + α − (2 k ) vk × Ψ (1 − α − λk + vk , | − c kx (1 − λk + vk , , ( vk + 1 , . (32) Corollary 2.6.
Assume that α , η , λ , v ∈ C and k > be such that R ( v ) > − , R ( α ) > , R ( λ + v ) < , R ( η )] , then the following formulaholds: (cid:16) K − α,η t λk − W kv,c ( t ) (cid:17) ( x ) = x λk − vk − (2 k ) vk ENERALIZED FRACTIONAL INTEGRATION OF k -BESSEL FUNCTION 7 × Ψ (1 + − λk + vk + η, | − c kx (1 − λk + vk + α + η, , ( vk + 1 , . (33)3. Representation in terms of generalized hypergeometricfunction
In this section, we introduce the generalized fractional integrals of k -Besselfunction in term of generalized hypergeometric function. First we considerthe following well known results:Γ(2 z ) = 2 z − √ π Γ( z )Γ( z + 12 ); z ∈ C (34)and ( z ) n = 2 n ( z n ( z + 12 ) n , z ∈ C , n ∈ N . (35)We represent the following theorems containing the generalized hypergeo-metric function. Theorem 3.1.
Assume that α , β , η , λ , v ∈ C be such that R ( v ) > − , R ( α ) > , R ( λ + v ) > max[0 , R ( β − η )] , (36) and let λk + vk , λk + vk + η − β = 0 , − , · · · , then the following result holds: (cid:16) I α,β,η t λk − W kv,c ( t ) (cid:17) ( x ) = x λk + vk − β − (2 k ) vk Γ( λk + vk )Γ( λk + vk + η − β )Γ( λk + vk − β )Γ( λk + vk + α + η )Γ( vk + 1) × F λ k + v k , λ k + v k + , λ k + v k + η − β , λ k + v k + η − β +12 | − cx kvk + 1 , λ k + v k − β , λ k + v k − β +12 , λk + vk + α + η , λ k + v k + α + η +12 . (37) Proof.
Note that F defined in (37) exit as the series is absolutely conver-gent. Now, using (15) with z = vk + 1 and (26) and applying (35) with z being replaced by λk + vk , λk + vk + η − β and λk + vk + α + η , we have (cid:16) I α,β,η t λk − W kv,c ( t ) (cid:17) ( x ) = x λ + vk − β − (2 k ) vk × ∞ X n =0 Γ( vk + λk )Γ( vk + λk + η − β )Γ( vk + λk − β )Γ( vk + λk + α + η )Γ( vk + 1) × ( vk + λk ) n ( vk + λk + η − β ) n ( vk + λk − β ) n ( vk + λk + α + β ) n ( − cx ) n (4 k ) n n != x λ + vk − β − (2 k ) vk Γ( vk + λk )Γ( vk + λk + η − β )Γ( vk + λk − β )Γ( vk + λk + α + η )Γ( vk + 1) × ∞ X n =0 ( v k + λ k ) n ( v k + λ k + ) n ( v k + λ k + η − β ) n ( v k + λ k + η − β ) n ( vk + 1)( v k + λ k − β ) n ( v k + λ k − β ) n ( v k + λ k + α + η ) n ( v k + λ k + α + η +12 ) n × ( − cx ) n (4 k ) n n ! . G. RAHMAN, K.S. NISAR ∗ , S. MUBEEN, AND M. ARSHAD Thus, in accordance with equation (13), we get the required result (37). (cid:3)
Corollary 3.2.
Assume that α , λ , v ∈ C be such that R ( v ) > − , R ( α ) > , R ( λ + v ) > and λk + vk = 0 , − , · · · , then the following result holds: (cid:16) I α,β,η t λk − W kv,c ( t ) (cid:17) ( x ) = x λk + vk + α − (2 k ) vk Γ( λk + vk )Γ( λk + vk − β )Γ( vk + 1) × F λ k + v k , λ k + v k + , | − cx kvk + 1 , λ k + v k − β , λ k + v k − β +12 . (38) Proof.
By substituting β = − α in (37), we obtain the required result. (cid:3) Corollary 3.3.
Assume that α , η , λ , v ∈ C be such that R ( v ) > − , R ( α ) > , R ( λ + v ) > and let λk + vk + η − β = 0 , − , · · · , then thefollowing result holds: (cid:16) K + α,η t λk − W kv,c ( t ) (cid:17) ( x ) = x λk + vk − (2 k ) vk Γ( λk + vk + η )Γ( λk + vk + α + η )Γ( vk + 1) × F λ k + v k + η , λ k + v k + η +12 | − cx kvk + 1 , λk + vk + α + η , λ k + v k + α + η +12 . (39) Proof.
By setting β = 0 in (37), we get the desired result. (cid:3) Theorem 3.4.
Assume that α , β , η , λ , v ∈ C and k > be such that R ( v ) > − , R ( α ) > , R ( λ − v ) < R ( β ) , R ( η )] , (40) and let β − λk + vk + 1 , η − λk + vk + 1 = 0 , − , · · · , then the following resultholds: (cid:18) I α,β,η − t λk − W kv,c ( 1 t ) (cid:19) ( x ) = x λk − vk − β − (2 k ) vk × Γ( β − λk + vk + 1)Γ( η − λk + vk + 1)Γ(1 − λk + vk )Γ( α + β + η − λk + vk + 1)Γ( vk + 1) × F β +12 − λ k + v k , β +22 − λ k + v k , η +12 − λ k + v k , η +22 − λ k + v k | − c kx vk + 1 , − λ k + v k , − λ k + v k , α + β + η +12 − λk + vk , α + η +22 − λ k + v k . (41) ENERALIZED FRACTIONAL INTEGRATION OF k -BESSEL FUNCTION 9 Proof.
Using (15) with z = vk + 1 and (31) and applying (35) with z beingreplaced by β − λk + vk + 1, 1 − λk + vk and β − λk + vk + α + η + 1, we have (cid:18) I α,β,η − t λk − W kv,c ( 1 t ) (cid:19) ( x ) = x λk − vk − β − (2 k ) vk Γ( β − λk + vk + 1)Γ( η − λk + vk + 1)Γ(1 − λk + vk )Γ( α + β + η − λk + vk + 1)Γ( vk + 1) × ∞ X n =0 ( β +12 − λ k + v k ) n ( β − λ k + v k + 1) n ( η +12 − λ k + v k ) n ( η − λ k + v k + 1) n ( vk + 1) n (( − λ k + v k ) n )(1 − λ k + v k ) n ( α + β + η +12 − λ k + v k ) n ( α + β + η − λ k + v k + 1) n × ( − c ) n (4 kx ) n n ! . By equation (13), we obtain the required given in (41). (cid:3)
Corollary 3.5.
Assume that α , η λ , v ∈ C and k > be such that R ( v ) > − , < R ( α ) < − R ( λ − v ) , and let λk − vk + α = 1 , , · · · then the followingresult holds: (cid:16) I α t λk − W kv,c ( t ) (cid:17) ( x ) = x λk − vk + α − (2 k ) vk Γ( − α − λk + vk + 1)Γ(1 − λk + vk )Γ( vk + 1)(42) × F − β +12 − λ k + v k , − α +22 − λ k + v k , | − c kx vk + 1 , − λ k + v k , − λ k + v k , . Corollary 3.6.
Assume that α , η , λ , v ∈ C and k > be such that R ( v ) > − , R ( α ) > , R ( λ + v ) < , R ( η )] and let λk − vk − η = 1 , , · · · ,then the following formula holds: (cid:16) K − η,α t λk − W kv,c ( t ) (cid:17) ( x ) = x λk − vk − (2 k ) vk Γ( η − λk + vk + 1)Γ(1 − λk + vk )Γ( vk + 1)(43) × F η +12 − λ k + v k , η +22 − λ k + v k | − c kx vk + 1 , α + η +12 − λk + vk , α + η +22 − λ k + v k . Corollary 3.5 and 3.6 follow from theorem 3.4 in respective cases β = − α and β = 0. References [1] R. D´aaz and E. Pariguan, On hypergeometric functions and Pochhammer k-symbol,Divulg. Mat. 15 (2007), no. 2, 179-192.[2] A. Erd´elyi,W. Magnus, F. Oberhettinger, and F.G. Tricomi, Higher TranscendentalFunctions,Vol.1, McGraw-Hill, NewYork, Toronto, London, 1953.[3] A. Erd´elyi, W. Magnus, F, Higher Transcendental Functions, Vol.2, McGraw-Hill,NewYork, Toronto, London, 1953.[4] K. S. Gehlot, Differential Equation of K -Bessels Function and its Properties, Nonl.Analysis and Differential Equations, 2 (2014),, no. 2, 61-67 ∗ , S. MUBEEN, AND M. ARSHAD [6] K. S. Gehlot and S. D. Purohit, Integral representations of the k-Bessel’s function,Honam Math. J. 38 (2016), no. 1, 17-23.[7] A.A. Kilbas, M. Saigo, and J.J. Trujillo, On the generalized Wright function, Fract.Calc. Appl. Anal. 54 (2002), pp. 437460.[8] A.A. Kilbas, N. Sebastian, Generalized fractional integration of Bessel function of thefirst kind , Integral Transforms and Special Functions,19 (2008),No. 12, 869883.[9] M. Saigo, A remark on integral operators involving the Gauss hypergeometric func-tions, Math. Rep. Kyushu Univ., 11 (1978), pp. 135143.[10] S.R. Mondal, Representation Formulae and Monotonicity of the Generalized k-BesselFunctions, arXiv:1611.07499 [math.CA], (2016).[11] S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives.Theory and Applications, Gordon and Breach, Yverdon, 1993.[12] E.M. Wright, The asymptotic expansion of the generalized hypergeometric functions,J. London. Math. Soc. 10 (1935), pp. 286293.[13] E.M. Wright, The asymptotic expansion of integral functions defined by Taylor Series,Philos. Trans. Roy. Soc. London A 238 (1940), pp. 423451.[14] E.M. Wright, The asymptotic expansion of the generalized hypergeometric functionII, Proc. London. Math. Soc. 46(2) (1935), pp. 389408.
Department of Mathematics, International Islamic University, City Islam-abad, Country Pakistan.
E-mail address : [email protected] Department of Mathematics,College of Arts and Science-Wadi Al dawaser,Prince Sattam bin Abdulaziz University, Saudi Arabia.
E-mail address : [email protected] Department of Mathematics, University of Sargodha, City Sargodha, Coun-try Pakistan.
E-mail address : [email protected] Department of Mathematics, International Islamic University, City Islam-abad, Country Pakistan.
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