FRACTIONAL INTEGRATION OF THE PRODUCT OF BESSEL FUNCTIONS OF THE FIRST KIND. Abstract

FRACTIONAL INTEGRATION OF THE PRODUCT OF BESSEL FUNCTIONS OF THE FIRST KIND Anaoly A. Kilbas,1, Nicy Sebasian Dedicaed o 75h birhday of Prof. A.M. Mahai Absrac Two inegral ransforms involving he Gauss-hypergeomeric funcion in he kernels are considered. They generalize he classical Riemann-Liouville and Erdélyi-Kober fracional inegral operaors. Formulas for composiions of such generalized fracional inegrals wih he produc of Bessel funcions of he firs kind are proved. Special cases for he produc of cosine and sine funcions are given. The resuls are esablished in erms of generalized Lauricella funcion due o Srivasava and Daous. Corresponding asserions for he Riemann-Liouville and Erdélyi-Kober fracional inegrals are presened. Mahemaical Subec Classificaion 010: 6A33, 33C10, 33C0, 33C50, 33C60, 6A09 Key Words and Phrases: fracional inegrals, Bessel funcion of he firs kind, generalized hypergeomeric series, generalized Lauricella series in several variables, cosine and sine rigonomeric funcions 1. Inroducion This paper deals wih wo inegral ransforms defined for x > 0 and complex α, β, η C (R(α) > 0) by c 010, FCAA Diogenes Co. (Bulgaria). All righs reserved.

160 A.A. Kilbas, N. Sebasian and (I α,β,η xαβ 0+ f)(x) = Γ(α) x 0 ( (x ) α1 F 1 α + β, η; α; 1 ) f()d x (1.1) (I α,β,η f)(x) = 1 ( ( x) α1 αβ F 1 α + β, η; α; 1 x ) f()d. Γ(α) x (1.) Here Γ(α) is he Euler gamma funcion 1, Secion 1, R(α) denoes he real par of α, and F 1 (a, b; c; z) is he Gauss hypergeomeric funcion defined for complex a, b, c, C, c 0, 1,, by he hypergeomeric series 1,.1() F 1 (a, b; c; z) = k=0 (a) k (b) k (c) k z k k!, (1.3) where (z) k is he Pochhammer symbol defined for z C and k N 0 = N {0}, N = {1,,...} by (z) 0 = 1, (z) k = z(z + 1)...(z + k 1) (k N). (1.) The series in (1.3) is absoluely convergen for z < 1 and z = 1(z 1), R(c a b) > 0. (1.5) Operaors (1.1) and (1.) were inroduced by Saigo 5, and heir properies were invesigaed by many auhors; see bibliography and a shor survey of resuls in 3, Secion 7.1, For Secions 7.7 and 7.8. When β = α, (1.1) and (1.) coincide wih he classical lef and righ-hand sided Riemann- Liouville fracional inegrals of order α C, R(α) > 0, 6, Secion 5.1: (I α,α,η 0+ f)(x) = (I0+f)(x) α 1 Γ(α) (I α,α,η f)(x) = (I α f)(x) 1 Γ(α) x o x (x ) α1 f()d (x > 0), (1.6) ( x) α1 f()d (x > 0). (1.7) If β = 0, (1.1) and (1.) are he so-called Erdélyi-Kober fracional inegrals defined for complex α, η C (R(α) > 0) by 6, Secion 18.1: (I α,0,η 0+ f)(x) = (I+ η,αf)(x) xαη Γ(α) x o (x ) α1 η f()d (x > 0), (1.8)

FRACTIONAL INTEGRATION OF THE PRODUCT... 161 (I α,0,η f)(x) = (K η,αf)(x) xη Γ(α) x ( x) α1 αη f()d (x > 0). (1.9) We invesigae composiions of inegral ransforms (1.1) and (1.) wih he produc of Bessel funcion of he firs kind, J ν (z), which is defined for complex z C (z 0) and ν C (R(ν) > 1) by, 7.() J ν (z) = k=0 (1) k ( z )ν+k Γ(ν + k + 1)k!. (1.10) We prove ha such composiions are expressed in erms of he generalized Lauricella funcion due o Srivasava and Daous 7, which is defined by = F A: B ; ;B (n) C: D ; ;D (n) = F A: B ; ;B (n) (a): θ,, θ (n), (b ): φ ; ; (b) (n) : φ (n) ; C: D ; ;D (n) (c): ψ,, ψ (n), (d ): δ ; ; (d) (n) : δ (n) ; z 1,, z n k 1,,k n =0 A (a ) k1 θ + +knθ(n) C (c ) k1 ψ + +k nψ (n) he coefficiens { B z 1. z n B(n) (b ) k1 φ D(n) (d ) k1 δ D θ (m) ( = 1,..., A); φ (m) ( = 1,..., B (m) ) ψ (m) ( = 1,..., C); δ (m) (b (n) ) knφ (n) (d (n) ) kn δ (n) z k 1 1 k 1! zkn n k n!, (1.11) ( = 1,..., D (m) ); m {1,..., n} (1.1) are real and posiive, and (a) abbreviaes he array of A parameers a 1,..., a A, (b (m) ) abbreviaes he array of B (m) parameers b (m) } ( = 1,..., B (m) ); m {1,..., n}, wih similar inerpreaions for (c) and (d (m) ) (m = 1,..., n). (z) a is a generalizaion of he Pochhammer symbol (1.): (z) a = Γ(z + a) Γ(a) The muliple series (1.11) converges absoluely eiher (z, a C). (1.13) (i) i > 0 (i = 1,..., n), z 1,..., z n C,

16 A.A. Kilbas, N. Sebasian or (ii) i = 0 (i = 1,..., n), z 1,..., z n C, z i < ϱ i (i = 1,..., n), and divergen when i < 0 (i = 1,..., n); excep for he rivial case z 1 = z n = 0, where wih i 1 + E i = (µ i ) 1+ C D (i) ψ (i) D (i) + δ (i) ϱ i = δ (i) A θ (i) B (i) φ (i) (i = 1,..., n), (1.1) min {E i} (i = 1,..., n), (1.15) µ 1,...,µ n >0 B (i) φ (i) ( C n i=1 ( A n i=1 µ i ψ (i) µ i θ (i) ) ψ (i) ) θ (i) D (i) B (i) (δ (i) ) δ(i) (φ (i) )φ(i). (1.16) For more deails see 7. Special cases of (1.11) are esablished in erms of generalized hypergeomeric funcion of one and wo variables respecively, for he sake of compleeness we define hese funcions here. A generalized hypergeomeric funcion p F q (z) is defined for complex a i, b C, b 0, 1,... (i = 1,,... p; = 1,,... q) by he generalized hypergeomeric series 1,.1(1) pf q (a 1,..., a p ; b 1,..., b q ; z) = k=0 (a 1 ) k... (a p ) k z k (b 1 ) k... (b q ) k k!. (1.17) This series is absoluely convergen for all values of z C if p q; and i is an enire funcion of z. We define a generalizaion of he Kampé de Férie funcion by means of he double hypergeomeric series 7 F p:q;k l:m;n (ap):(bq);(c k); (α l ):(β m );(γ n ); x, y= r,s=0 { p (a ) r+s }{ q (b ) r }{ k (c ) s } { l (α ) r+s }{ m (β ) r }{ n x r y s (γ ) s } r! s!. (1.18)

FRACTIONAL INTEGRATION OF THE PRODUCT... 163 The above double series is absoluely convergen for all values of x and y, if p + q < l + m + 1 and p + k < l + n + 1. Also, if p + q = l + m + 1 and p + k = l + n + 1, we mus have any one of he following ses of condiions: p l, max{ x, y } < 1; p > l, x 1/(pl) + y 1/(pl) < 1. The paper is organized as follows. Formulas for composiions of inegral ransforms (1.1) and (1.) wih he produc of Bessel funcions (1.10) are proved in erms of generalized Lauricella funcion (1.11) in Secion and 3, respecively. The corresponding resuls for he Riemann-Liouville and Erdélyi-Kober fracional inegrals (1.6), (1.7) and (1.8), (1.9) are also presened in Secions and 3. Special cases giving composiions of fracional inegrals wih he produc of cosine and sine funcions are considered in Secions.. Lef-sided fracional inegraion of Bessel funcions Our resuls in Secions and 3 are based on he preliminary asserions giving composiion formulas of generalized fracional inegrals (1.1) and (1.) wih a power funcion. Lemma 1. (, Lemmas 1-) Le α, β, η C. (a) If R(α) > 0 and R(σ) > max 0, R(β η), hen (I α,β,η 0+ σ1 )(x) = Γ(σ)Γ(σ + η β) Γ(σ β)γ(σ + α + η) xσβ1. (.1) (b) If R(α) > 0 and R(σ) < 1 + min R(β), R(η), hen (I α,β,η σ1 )(x) = Γ(β σ + 1)Γ(η σ + 1) Γ(1 σ)γ(α + β + η σ + 1) xσβ1. (.) The generalized lef-sided fracional inegraion (1.1) of he produc of Bessel funcions(1.10) is given by he following resul. Theorem 1. Le n N, α, β, η, σ, ν C and a, ρ R + ( = 1,..., n) be such ha n R(α) > 0, R(ν ) > 1, R(σ + ρ ν ) > max0, R(β η). (.3)

16 A.A. Kilbas, N. Sebasian Then here holds he formula 0+ σ1 n J ν (a ρ ) (x) n = x σβ1 ( a x ρ ) ν Γ(u)Γ(v) Γ(ν + 1) Γ(w)Γ(z) F :0,...,0 u:ρ 1,...,ρ n,v:ρ 1,...,ρ n :1,...,1 w:ρ 1,...,ρ n,z:ρ 1,...,ρ n:ν 1 +1:1,...,ν ; n+1:1: a 1 xρ 1,..., a nx ρn (.) where u = σ + n ρ ν, v = σ + η β + n ρ ν, w = σ β + n ρ ν, z = σ + α + η + n ρ ν and F :0,...,0 :1,...,1 is given by (1.11). P r o o f. Firs of all we noe ha i in (1.1) is given by i = 1+n > 0 (i = 1,..., n N), and herefore F :0,...,0 :1,...,1 in he righ hand side of (.) is defined. Now we prove (.). Applying equaion (1.10), Using (1.1) and (1.11) and changing he orders of inegraion and summaion, we find = 0+ = σ1 k 1,...,k n=0 0+ k 1 =0 σ1 n J ν (a ρ ) (x) (1) k 1 ( a 1 ρ 1 ) ν 1+k 1 Γ(ν 1 + k 1 + 1) k 1! (1) k n ( a n ρ n ) ν n+k n (x) Γ(ν n + k n + 1) k n! k n =0 (1) k 1 ( a 1 ) ν 1+k 1 Γ(ν 1 + 1)(ν 1 + 1) k1 k 1! (1) kn ( an ) νn+kn Γ(ν n + 1)(ν n + 1) kn k n! (I α,β,η 0+ {σ+ν 1ρ 1 + +ν nρ n+ρ 1 k 1 + +ρ nk n1 })(x). By (.3), for any k N 0 ( = 1,..., n) R(σ + n ρ ν + n ρ k ) R(σ + n ρ ν ) > max0, R(β η). Applying Lemma 1(a) and using (.1) wih σ replaced by σ + n ρ ν + n ρ k ( = 1,..., n), we obain 0+ σ1 n J ν (a ρ ) (x),

FRACTIONAL INTEGRATION OF THE PRODUCT... 165 = k 1,...,k n=0 (1) k 1 ( a 1 ) ν 1+k 1 Γ(ν 1 + 1)(ν 1 + 1) k1 k 1! (1) kn ( an ) νn+kn Γ(ν n + 1)(ν n + 1) kn k n! Γ(σ + n (ν ρ + ρ k ))Γ(σ + η β + n (ν ρ + ρ k )) Γ(σ β + n (ν ρ + ρ k ))Γ(σ + α + η + n (ν ρ + ρ k )) n = x σβ1 ( a x ρ ) ν Γ(ν +1) k 1,...,k n=0 x σβ1+ n (ν ρ +ρ k ) Γ(σ + n ρ ν )Γ(σ + η β + n ρ ν ) Γ(σ β + n ρ ν )Γ(σ + α + η + n ρ ν ) (σ+ n ρ ν ) ρ1 k 1 + +ρ nk n (σ+ηβ+ n ρ ν ) ρ1 k 1 + +ρ nk n (σβ+ n ρ ν ) ρ1 k 1 + +ρ nk n (σ+α+η+ n ρ ν ) ρ1 k 1 + +ρ nk n 1 ( a (ν 1 + 1) k1 (ν n + 1) kn k 1! 1 xρ 1 ) k 1 ( a n xρn ) kn This, in accordance wih Equaion (1.11), gives he resul in (.). This complee he proof of he heorem. Corollary 1.1. Le α, σ, ν C and a, ρ R + ( = 1,..., n) be such ha R(α) > 0, R(ν ) > 1 and R(σ + n ρ ν ) > 0. Then n I 0+ α σ1 J ν (a ρ ) (x) n = x σ+α1 ( a x ρ ) ν Γ(σ + n ρ ν ) Γ(ν + 1) Γ(σ + α + n ρ ν ) F 1:0,...,0 σ+ n ρ ν :ρ 1,...,ρ n : 1:1,...,1 σ+α+ n ρ ν :ρ 1,...,ρ n:ν 1 +1:1,...,ν ; a 1 xρ1 n+1:1: k n!.,..., a nx ρn (.5) Corollary 1.. Le α, η, σ, ν C and a, ρ R + ( = 1,..., n) be such ha R(α) > 0, R(ν ) > 1 and R(σ + n ρ ν ) > R(η). Then n I η,α + σ1 J ν (a ρ ) (x).

166 A.A. Kilbas, N. Sebasian n = x σ1 ( a x ρ ) ν Γ(σ + η + n ρ ν ) Γ(ν + 1) Γ(σ + α + η + n ρ ν ) F 1:0,...,0 σ+η+ n ρ ν :ρ 1,...,ρ n : 1:1,...,1 σ+α+η+ n ρ ν :ρ 1,...,ρ n:ν 1 +1:1,...,ν ; a 1 xρ1 n+1:1:,..., a nx ρn. (.6) Corollary 1.3. Le α, β, σ, ν 1, ν C and a 1, a, ρ 1, ρ R + be such ha R(α) > 0, R(ν 1 ) > 1, R(ν ) > 1 and R(σ + ρ 1 ν 1 + ρ ν ) > max0, R(β η). Then ( I α,β,η 0+ σ1 J ν1 ()J ν () ) (x) F :0,0 a a+1 :1,1, :1,1, b b+1 :1,1, :1,1 c = xc1 ν 1+ν Γ(a)Γ(b) Γ(c)Γ(d)Γ(ν 1 + 1)Γ(ν + 1) :1,1, c+1 :1,1, d :1,1: :1,1, d+1 :1,1:ν 1+1:1,...,ν n +1:1: ; x, x, (.7) where a = σ + ν 1 + ν, b = σ + η β + ν 1 + ν, c = σ β + ν 1 + ν, d = σ + α + η + ν 1 + ν and F :0,0 :1,1 is defined in (1.18). Corollaries 1.1 and 1. follow from Theorem 1 in respecive cases β = α and β = 0, if we ake (1.6) and (1.8) ino accoun. Corollary 1.3 follows from Theorem 1, if we pu n =, a 1 = 1, a = 1, ρ 1 = 1, ρ = 1, use (1.11) and ake ino accoun he relaion (z) k = k ( z ) k ( ) z + 1 where (z) k is he Pochhammer symbol (1.). k (z C, k N 0 ), (.8) Remark 1. When n = 1, a 1 = 1, ρ 1 = 1, ν 1 = ν equaion (.) is reduced o ( I α,β,η 0+ σ1 J ν () ) (x) = xσ+νβ1 Γ(σ + ν)γ(σ + ν + η β) ν Γ(σ + ν β)γ(σ + ν + α + η)γ(ν + 1) σ+ν F, σ+ν+1, σ+ν+ηβ, σ+ν+ηβ+1 5 ν+1, σ+νβ, σ+νβ+1, σ+ν+α+η, σ+ν+α+η+1 This resul was proved in, Theorem 3. ; x. (.9)

FRACTIONAL INTEGRATION OF THE PRODUCT... 167 3. Righ-sided fracional inegraion of Bessel funcions The following resul yields generalized righ-hand sided fracional inegraion (1.) of he produc of Bessel funcions. Theorem. Le α, β, η, σ, ν C and a, ρ R + ( = 1,..., n) be such ha n R(α) > 0, R(ν ) > 1, R(σ ρ ν ) < 1 + minr(β), R(η). (3.1) Then here holds he formula σ1 n J ν ( a ρ ) (x) n = x σβ1 ( a x ρ ) ν Γ(p)Γ(q) Γ(ν + 1) Γ(r)Γ(s) F :0,...,0 p:ρ 1,...,ρ n,q:ρ 1,...,ρ n : :1,...,1 r:ρ 1,...,ρ n,s:ρ 1,...,ρ n:ν 1 +1:1,...,ν ; a 1 n+1:1: x ρ,..., a n 1 x ρ, n (3.) where p = 1 + β σ + n ρ ν, q = 1 + η σ + n ρ ν, r = 1 σ + n ρ ν, s = α + β + η σ + n ρ ν + 1 and F :0,...,0 :1,...,1 is given by (1.11). P r o o f. Firs of all we noe ha i in (1.1) is given by i = 1+n > 0 (i = 1,..., n N), and herefore F :0,...,0 :1,...,1 in he righ hand side of (3.) is defined. Now we prove (3.). Using Equaions (1.) and (1.10) and changing he orders of inegraion and summaion, we have n ( σ1 a ) J ν ρ (x) = (I α,β,η = σ1 k 1,...,k n =0 k 1 =0 (1) k 1 ( a 1 ρ 1 )ν 1+k 1 Γ(ν 1 + k 1 + 1) k 1! k n=0 (1) k n ( an ρ n ) νn+kn Γ(ν n + k n + 1) k n! (x) (1) k 1 ( a 1 ) ν 1+k 1 Γ(ν 1 + 1)(ν 1 + 1) k1 k 1! (1) kn ( an ) νn+kn Γ(ν n + 1)(ν n + 1) kn k n!

168 A.A. Kilbas, N. Sebasian (I α,β,η { σν 1ρ 1 ν nρ nρ 1 k 1 ρ nk n1 })(x). By (3.1), for any k N 0 ( = 1,..., n) R(σ n ρ ν n ρ k ) R(σ n ρ ν ) < 1 + minr(β), R(η). Applying Lemma 1(b) and using (.) wih σ replaced by σ n ρ ν n ρ k ( = 1,..., n), we obain n ( σ1 a ) J ν ρ (x) = k 1,...,k n =0 (1) k 1 ( a 1 ) ν 1+k 1 Γ(ν 1 + 1)(ν 1 + 1) k1 k 1! (1) kn ( an ) νn+kn Γ(ν n + 1)(ν n + 1) kn k n! Γ(β σ + 1 + n (ν ρ + ρ k ))Γ(η σ + 1 + n (ν ρ + ρ k )) Γ(1σ+ n (ν ρ + ρ k ))Γ(1+α+β+ησ+ n (ν ρ + ρ k )) n = x σβ1 ( a x ρ ) ν Γ(ν + 1) x σβ1 n (ν ρ +ρ k ) Γ(p)Γ(q) Γ(r)Γ(s) k 1,...,k n =0 (p) ρ1 k 1 + +ρ n k n (q) ρ1 k 1 + +ρ n k n (r) ρ1 k 1 + +ρ n k n (s) ρ1 k 1 + +ρ n k n a 1 x ρ 1 )k 1 1 ( (ν 1 + 1) k1 (ν n + 1) kn k 1! By equaion (1.11), his yields he resul in (3.). ( a n x ρn )k n. k n! Corollary.1. Le α, σ, ν C and a, ρ R + ( = 1,..., n) be such ha R(ν ) > 1, and 0 < R(α) < 1 R(σ n ρ ν ). Then n ( I α σ1 a ) J ν ρ (x) ( n a ) ν = x σ+α1 x ρ Γ(1 σ α + n ρ ν ) Γ(ν + 1) Γ(1 σ + n ρ ν ) F 1:0,...,0 1σα+ n ρ ν :ρ 1,...,ρ n : 1:1,...,1 1σ+ n ρ ν :ρ 1,...,ρ n:ν 1 +1:1,...,ν ; a 1 n+1:1: x ρ,..., 1 a n x ρ n. (3.3)

FRACTIONAL INTEGRATION OF THE PRODUCT... 169 Corollary.. Le α, η, σ, ν C and a, ρ R + ( = 1,..., n) be such ha R(α) > 0, R(ν ) > 1, and R(σ n ρ ν ) < 1 + R(η). Then n ( K η,α σ1 a ) J ν ρ (x) n = x σ1 ( a x ρ ) ν Γ(1 + η σ + n ρ ν ) Γ(ν + 1) Γ(1 + η + α σ + n ρ ν ) F 1:0,...,0 1+ησ+ n ρ ν :ρ 1,...,ρ n : 1:1,...,1 1+α+ησ+ n ρ ν :ρ 1,...,ρ n:ν 1 +1:1,...,ν ; a 1 n+1:1: x ρ,..., 1 a n x ρ n (3.). Corollary.3. Le α, β, η, σ, ν 1, ν C, a 1, a and ρ 1, ρ R + be such ha R(α) > 0, R(ν 1 ) > 1, R(ν ) > 1, R(σ ρ 1 ν 1 ρ ν ) < 1+minR(β), R(η) and R(β σ +ν 1 +ν +1) > 0, R(η σ +ν 1 +ν +1) > 0. Then ( I α,β,η σ1 J ν1 ( 1 ) J ν ( 1 )) (x) = xσν 1ν β1 ν 1+ν Γ(c)Γ(f) Γ(g)Γ(h)Γ(ν 1 + 1)Γ(ν + 1) F :0,0 e e+1 :1,1, :1,1, f f+1 :1,1, :1,1 g :1,1, g+1 :1,1, h ; 1 :1,1:ν 1+1:1,...,ν n +1:1: x, 1 x, (3.5) :1,1: h+1 :1,1, where e = β σ + ν 1 + ν + 1, f = η σ + ν 1 + ν + 1, g = 1 σ + ν 1 + ν, h = α + β + η σ + ν 1 + ν + 1 and F :0,0 :1,1 is defined in (1.18). According o (1.7) and (1.9), Corollaries.1 and. follow from Theorem in respecive cases β = α and β = 0. Corollary 1.3 follows from Theorem 1, if we pu n =, a 1 = 1, a = 1, ρ 1 = 1, ρ = 1 and ake (.8) ino accoun. Remark. When n = 1, a 1 = 1, ρ 1 = 1, ν 1 = ν, equaion (3.9) is reduced o ( ( )) I α,β,η 1 σ1 J ν (x) = xσνβ1 ν Γ(β σ + ν + 1)Γ(η σ + ν + 1) Γ(1 σ + ν)γ(α + β + η σ + ν + 1)Γ(ν + 1)

170 A.A. Kilbas, N. Sebasian βσ+ν+1, F βσ+ν+, ησ+ν+1, ησ+ν+ 5 ν+1, 1σ+ν, σ+ν, α+β+ησ+ν+1, α+β+ησ+ν+ This formula was proved in, Theorem. ; 1 x. (3.6). Fracional inegraion of cosine and sine funcions For ν = 1 and ν = 1, he Bessel funcion J ν(z) in (1.10) coincides ( ) 1 wih cosine- and sine-funcions, apar from he muliplier πz : ( ) 1 ( ) 1 J 1 (z) = cos(z), J 1 (z) = sin(z). (.1) πz πz Seing ν 1 = = ν n = 1 and ρ 1 = = ρ n = 1, from Theorem 1 and Corollaries 1.1 and 1. we deduce he following resuls: Theorem 3. Le α, β, η, σ C, a R +, = 1,..., n be such ha R(α) > 0, R(σ) > 0, R(σ + η β) > 0, R(σ) > max0, R(β η) Then here holds he formula 0+ σ1 n cos(a ) (x) = x σβ1 Γ(σ)Γ(σ + η β) Γ(σ β)γ(σ + α + η) F :0,...,0 σ:,...,,σ+ηβ:,...,: :1,...,1 ; σβ:,...,,α+η+σ:,...,: 1 :1,..., 1 :1: a 1 x,..., nx a. (.) Corollary 3.1. Le α, σ C and a R + ( = 1,..., n) be such ha R(α) > 0 and R(σ) > 0. Then n I 0+ α σ1 cos(a ) (x) = x σ+α1 Γ(σ) Γ(σ + α) F 1:0,...,0 σ:,...,: 1:1,...,1 ; σ+α:,...,: 1 :1,..., 1 :1: a 1 x,..., nx a. (.3)

FRACTIONAL INTEGRATION OF THE PRODUCT... 171 Corollary 3.. Le α, η, σ C and a R + ( = 1,..., n) be such ha R(α) > 0 and R(σ) > R(η). Then n I η,α + σ1 cos(a ) (x) = x σ1 Γ(σ + η) Γ(σ + α + η) F 1:0,...,0 σ+η:,...,: 1:1,...,1 ; σ+α+η:,...,: 1 :1,..., 1 :1: a 1 x,..., nx a. (.) Theorem. Le α, β, η, σ C and a R + ( = 1,..., n) be such ha Then R(α) > 0, R(σ) > 0, R(σ + η β) > 0, R(σ) > max0, R(β η). = π n n 0+ n a σn1 n sin(a ) (x) x σβ1 Γ(σ)Γ(σ + η β) Γ(σ β)γ(σ + α + η) F :0,...,0 σ:,...,,σ+ηβ:,...,: :1,...,1 ; σβ:,...,,α+η+σ:,...,: 3 :1,..., 3 :1: a 1 x,..., nx a. (.5) Corollary.1. Le α, σ C and a R + ( = 1,..., n) be such ha R(α) > 0 and R(σ) > 0. Then = π n n F 1:0,...,0 1:1,...,1 n I 0+ α σn1 sin(a ) (x) n a x σβ1 Γ(σ) Γ(σ + α) σ:,...,: ; σ+α:,...,,: 3 :1,..., 3 :1: a 1 x,..., nx a. (.6)

17 A.A. Kilbas, N. Sebasian Corollary.. Le α, η, σ C and a R + ( = 1,..., n) be such ha R(α) > 0 and R(σ) > R(η). Then n I η,α + σn1 = π n n n a sin(a ) (x) x σ1 Γ(σ + η) Γ(σ + α + η) F 1:0,...,0 σ+η:,...,: 1:1,...,1 ; α+η+σ:,...,: 3 :1,..., 3 :1: a 1 x,..., nx a. (.7) Similarly, seing ν 1 = = ν n = 1 and ρ 1 = = ρ n = 1 and aking (.1) ino accoun, from Theorem and Corollaries.1 and., we obain he following resuls: Theorem 5. Le α, β, η, σ C and a R + ( = 1,..., n) be such ha Then R(α) > 0, R(β σ) > 0, R(η σ) > 0, R(σ) < minr(β), R(η). σ n cos ( a ) (x) = x σβ Γ(β σ)γ(η σ) Γ(σ)Γ(α + β + η σ) F :0,...,0 βσ:,...,,ησ:,...,: :1,...,1 ; a 1 σ:...,,α+β+ησ:,...,: 1 :1,..., 1 :1: x,..., a n x. (.8) Corollary 5.1. Le α, σ C and a R + ( = 1,..., n) be such ha 0 < R(α) < R(σ). Then n ( I α σ a ) cos (x) σ+α Γ(α σ) = x F 1:0,...,0 ασ:,...,: 1:1,...,1 Γ(σ) ; a 1 σ:,...,: 1 :1,..., 1 :1: x,..., a n x. (.9)

FRACTIONAL INTEGRATION OF THE PRODUCT... 173 Corollary 5.. Le α, η, σ C and a R + ( = 1,..., n) be such ha R(α) > 0 and R(σ) < R(η). Then n ( K η,α σ a ) cos (x) = x σ Γ(η σ) Γ(α + η σ) F 1:0,...,0 ησ:,...,: 1:1,...,1 ; a 1 α+ησ:,...,: 1 :1,..., 1 :1: x,..., a n x. (.10) Theorem 6. Le α, β, η, σ C and a R + ( = 1,..., n) be such ha R(α) > 0, R(β σ > 1, R(η σ) > 1, R(σ) < 1 + minr(β), R(η). Then here holds he formula = π n n n a σ+n1 n sin ( a ) (x) x σβ1 Γ(β σ + 1)Γ(η σ + 1) Γ(1 σ)γ(α + β + η σ + 1) F :0,...,0 βσ:,...,,ησ+1:,...,: :1,...,1 ; a 1 1σ:,...,,α+β+ησ+1:,...,: 3 :1,..., 3 :1: x,..., a n x. (.11) Corollary 6.1. Le α, σ C and a R + ( = 1,..., n) be such ha 0 < R(α) < 1 R(σ). Then n ( I α σ+n1 a ) sin (x) = π n n n a x σ+α1 Γ(1 α σ) Γ(1 σ)

17 A.A. Kilbas, N. Sebasian F 1:0,...,0 1σα:,...,: 1:1,...,1 ; a 1 1σ:,...,: 3 :1,..., 3 :1: x,..., a n x. (.1) Corollary 6.. Le α, η, σ C and a R + ( = 1,..., n) be such ha R(α) > 0 and R(σ) < 1 + R(η). Then n ( K η,α σ+n1 a ) sin (x) = π n n n a x σ1 Γ(η σ + 1) Γ(α + η σ + 1) F 1:0,...,0 ησ+1:,...,: 1:1,...,1 ; a 1 α+ησ+1:,...,: 3 :1,..., 3 :1: x,..., a n x. (.13) Remark 3. When n = 1, a 1 = 1, hen all he resuls in Secion coincide wih ha proved in, Secions 5 and 6. Acknowledgemens The auhors would like o hank he Deparmen of Science and Technology, Governmen of India, New Delhi, for he financial assisance for his work under proec-number SR/S/MS:87/05, and he Cenre for Mahemaical Sciences for providing all faciliies. The firs co-auhor (A.A.K.) was suppored, in par, by he Belarusian Fundamenal Research Fund (Proec F08MC-08) and by Naional Science Fund - Minisry of Educaion, Youh and Science, Bulgaria (Proec D ID 0/5/009 Inegral Transform Mehods, Special Funcions and Applicaions ).

FRACTIONAL INTEGRATION OF THE PRODUCT... 175 References 1 A. Erdélyi, W. Magnus, F. Oberheinger, F.G. Tricomi, Higher Transcendenal Funcions, Vol. I. McGraw-Hill, New York - Torono - London (1953). A. Erdélyi, W. Magnus, F. Oberheinger, F.G. Tricomi, Higher Transcendenal Funcions, Vol. II. McGraw-Hill, New York - Torono - London (1953). 3 A.A. Kilbas, M. Saigo, H-Transforms. Theory and Applicaions. Chapman and Hall/CRC, Boca Raon, Florida (00). A.A. Kilbas, N. Sebasian, Generalized fracional inegraion of Bessel funcion of he firs kind. Inegral Transforms Spec. Func. 19, No 1 (008), 869-883. 5 M. Saigo, A remark on inegral operaors involving he Gauss hypergeomeric funcions. Mah. Rep. College of General Edu. Kyushu Universiy 11 (1978), 135-13. 6 S.G. Samko, A.A. Kilbas, O.I. Marichev, Fracional Inegrals and Derivaives. Theory and Applicaions. Gordon and Breach Sci. Publ., London - New York (1993). 7 H.M. Srivasava, M.C. Daous, A noe on he convergence of Kampé de Férie s double hypergeomeric series. Mah. Nachr. 53 (197), 151-159.,1 Dep. Mah. & Mechanics, Belarusian Sae Universiy Minsk 0030, BELARUS (Corresponding auhor) Cenre for Mahemaical Sciences Pala Campus Arunapuram P.O. Palai, Kerala 686 57, INDIA e-mail: nicyseb@yahoo.com; www.cmsinl.org Received: December 8, 009