Generalized fractional calculus of the multiindex Bessel function

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1 Available online at Math. Nat. Sci., , Research Article Journal Homepage: Generalized ractional calculus o the multiindex Bessel unction. L. Suthar a, S.. Purohit b,, R. K. Parmar c a epartment o Mathematics, Wollo University, essie, Ethiopia. b epartment o HEAS Mathematics, Rajasthan Technical University, Kota , Rajasthan, ndia. c epartment o Mathematics, Government College o Engineering and Technology, Bikaner , ndia. Communicated by O. K. Matthew Abstract The present paper is devoted to the study o the ractional calculus operators to obtain a number o key results or the generalized multiindex Bessel unction involving Saigo hypergeometric ractional integral and dierential operators in terms o generalized Wright unction. Various particular cases and consequences o our main ractional-calculus results as classical Riemann-Liouville and Erdélyi-Kober ractional integral and dierential ormulas are deduced. c 2017 All rights reserved. Keywords: Fractional calculus operators, multiindex Bessel unction, Wright unction MSC: 26A33, 33E12, 33C ntroduction Recently, the generalized multiindex Bessel unction is deined by Choi et al. 1 as ollows: J j m,γ β j m, k z n 0 γ k n m j 1 Γ j n + β j + 1 zn, m N, 1.1 n! where j, β j, γ C, j 1,, m, Rγ > 0, Rβ > 1, m j1 R j > max{0; Rk 1}, k > 0. The Pohhammer symbol is deined or γ C as ollows see 8, p.2 and p.5: { 1, n 0, γ n γγ γ + n + 1, n N, Γγ + n, γ C/Z 0, Γγ and Γ being the Gamma unction. Some important special cases o this unction are enumerated below: Corresponding author addresses: dlsuthar@gmail.com. L. Suthar, sunil_a_purohit@yahoo.com S.. Purohit, rakeshparmar27@gmail.com R. K. Parmar doi: /mns Received

2 . L. Suthar, S.. Purohit, R. K. Parmar, Math. Nat. Sci., , i we put k 0, m 1, 1 1, β 1 υ and replace z by z 2/ 4 in 1.1, we obtain J 1, γ υ, 0 z υ J υ z, z where J υ z is a well-known Bessel unction o the irst kind deined or complex z C, z 0 and υ C, Rυ > 1, by 3, see also 5 ii J j m,γ β j m z, k has the orm: J υ z k 0 J j m,γ β j m, k z 1 Γγ 1 ψ m 1 k Γυ + k + 1 z/2 υ+2k ; k! γ, 1 β 1 + 1, 1,..., β m + 1, m z. A detail account o Bessel unction, the reader may be reerred to the earlier extensive works by Erdélyi et al. 3 and Watson 9. The generalized Wright hypergeometric unction p ψ q z, or z C, complex a i, b j C, and i, β j R, i, β j 0; i 1, 2,, p; j 1, 2,, q is deined as ollows: ai, pψ q z p ψ i 1, p q z b j, β j 1, q k0 p i1 Γa i + i kz k q j1 Γb j + β j kk!. 1.2 Wright 10 introduced the generalized Wright unction 4 and proved several theorems on the asymptotic expansion o p ψ q z or instance, see or all values o the argument z, under the condition: q β j j1 p i > 1. i1 The generalized hypergeometric unction or complex a i, b j C and b j 0, 1, i 1, 2,, p;, j 1, 2,, q is given by the power series 2, Section 4.1 1: pf q a 1,, a p ; b 1,, b q ; z r0 a 1 r a p r z r b 1 r b q r r!, 1.3 where or convergence, we have z < 1 i p q + 1 and or any z i p q. The unction 1.3 is a special case o the generalized Wright unction 1.2 or 1 p β 1 β q 1 q j1 pf q a 1,, a p ; b 1,, b q ; z Γb j ai, 1 p i1 Γa i p ψ 1, p q z. b j, 1 1, q The object o this paper is to derive ractional integral and derivative o generalized multiindex Bessel unction 1.1 and the let and right sided operators o Saigo ractional calculus 6. The results derived in this paper are believed to be new. 2. Fractional calculus operators approach The Riemann-Liouville ractional integral and derivative operators are deined by see Samko, Kilbas and Marichev 7, Sect. 2 or > 0: x 1 x t dt, 2.1 Γ 0 x t 1

3 . L. Suthar, S.. Purohit, R. K. Parmar, Math. Nat. Sci., , x 1 Γ x t dt, 2.2 t x 1 d +1 x 1 1 Γ1 {} x, d +1 x 0 t dt, x t {} 2.3 d +1 x 1 1 Γ1 {} x, +1 d x t dt, t x {} where means the maximal integer not exceeding and {} is the ractional part o. A useul generalization o the Riemann-Liouville and Erdélyi-Kober ractional integral has been introduced by Saigo 6 in terms o Gauss hypergeometric unction. Let C and x R +, then the generalized ractional integration and ractional dierentiation operators associated with Gauss hypergeometric unction are deined as ollows: x x β Γ x 1 Γ x x 0 x 2.4 x t 1 2F 1 + β, γ; ; 1 t/x tdt, 2.5 t x 1 t β 2F 1 + β, γ; ; 1 x/t tdt, 2.6 β, +γ x R > 0; k R + 1, d k +k, β k, +γ k x, 2.7 x β, +γ x d k +k, β k, +γ x, 2.8 R > 0; k R + 1. Thereore, i we set β in 2.5, 2.6, 2.7, 2.8, reduce to 2.1, 2.2, 2.3, 2.4, o the let and right hand sided Riemann-Liouville ractional integral and derivative operators Let-sided ractional integration o generalized multiindex Bessel unction n this section, we establish image ormulas or the generalized multiindex Bessel unction involving let sided operators o Saigo ractional integral operators 2.1, in term o the generalized Wright unction. Theorem 2.1. Let m 1 be an integer, R µ j > 0, η j i 1,, m are arbitrary parameters and. be the Saigo let-sided ractional integral operator 2.5, then the ollowing result holds: t ρ 1 J µ j m, τ η j m, k a tυ x xρ β 1 Γτ The conditions or validity o 2.9 are i, R > 0 and a are any complex numbers; 3 ψ m+2 ρ β+γ, υ, ρ, υ, τ, k ρ β, υ, ρ++γ, υ, η j + 1, µ j m 1 ii ρ and υ are arbitrary, such that Rρ + υ n > 0 and Rρ + γ β + υ n > 0. a x υ. 2.9

4 . L. Suthar, S.. Purohit, R. K. Parmar, Math. Nat. Sci., , Proo. enote L.H.S. o Theorem 2.1 by 1, then 1 t ρ 1 J µ j m, τ η j m, k a tυ x, by virtue o 1.1 and 2.5, we have x β Γ x 0 x t 1 2F 1 + β, γ; ; 1 t/x t ρ 1 J µ j m,τ η j m,k a tυ dt, now, interchanging the order o integration and summation is permissible under the conditions stated with the theorem due to convergence o the integrals involved in the process and evaluating the inner integral by beta-unction, it gives xρ β 1 Γτ xρ β 1 Γτ which completes the proo o Theorem 2.1. Γρ β + γ + υ n Γρ + υ n Γτ + kn Γρ β + υ n Γρ + + γ + υ nγ η n0 j µ j n a tυ n n! ρ β+γ, υ, ρ, υ, τ, k 3ψ m+2 ax υ, ρ β, υ, ρ++γ, υ, η j + 1, µ j m 1 Corollary 2.2. Let m 1 be an integer, R µ j > 0, η j i 1,, m are arbitrary parameters and. be the Riemann-Lioville let-sided ractional integral operator 2.1, then the ollowing result holds: t ρ 1 J µ j m, τ η j m, k a tυ x xρ+ 1 Γτ The conditions or validity o 2.10 are i, R > 0 and a are any complex numbers; ii ρ and υ are arbitrary, such that Rρ + υn > 0. 2 ψ m+1 ρ, υ, τ, k ρ β, υ, η j + 1, µ j m Right-sided ractional integration o generalized multiindex Bessel unction a x υ n this section, we establish image ormulas or the generalized multiindex Bessel unction involving right sided operators o Saigo ractional integral operators 2.6, in term o the generalized Wright unction. Theorem 2.3. Let m 1 be an integer, Rµ j > 0, η j i 1,, m are arbitrary parameters and the Saigo s right-sided ractional integral operator 2.6, then the ollowing result holds: t ρ J µ j m, τ η j m, k a t x xρ β Γτ 3 ψ m+2 The conditions or validity o 2.11 are i, R > 0 and a are any complex numbers; β ρ, υ, γ ρ, υ, τ, k ρ, υ, +β+γ ρ, υ, η j +1, µ j m 1 ii ρ and υ are arbitrary such that Rβ ρ + υn > 0 and Rγ ρ + υ n > 0. Proo. enote L.H.S. o Theorem 2.3 by 2, then 2,β,γ t ρ J µ j m, τ η j m, k a t x,. be a x using the deinition o generalized multiindex Bessel unction 1.1, ractional integral ormula 2.6 and proceeding similarly to the proo o Theorem 2.1, we obtain

5 . L. Suthar, S.. Purohit, R. K. Parmar, Math. Nat. Sci., , xρ β Γτ n0 xρ β Γτ 3 ψ m+2 which completes the proo o Theorem 2.3. Γβ ρ + υ n Γγ ρ + υ n Γτ + kn Γ ρ + υ n Γ + β + γ ρ + υ nγ η j µ j n a t n n! β ρ, υ, γ ρ, υ, τ, k a x, ρ, υ, +β+γ ρ, υ, η j 1,,µ j m 1 Corollary 2.4. Let m 1 be an integer, Rµ j > 0, η j i 1,, m are arbitrary parameters and. be the Riemann Liouville right-sided ractional integral operator 2.2, then the ollowing result holds: t ρ J µ j m, τ η j m, k a t x xρ+ Γτ 2 ψ m+1 The conditions or validity o 2.12 are i, R > 0 and a are any complex numbers; ii ρ and υ are arbitrary such that R ρ + υn > 0. ρ, υ, τ, k ρ, υ, η j + 1, µ j m 1 a x Let-sided ractional dierentiation o generalized multiindex Bessel unction n this section, we establish image ormulas or the generalized multiindex Bessel unction involving let sided o Saigo ractional dierentiation operators 2.7, in term o the generalized Wright unction. Theorem 2.5. Let m 1 be an integer, Rµ j > 0, η j i 1,, m are arbitrary parameters and. be the Saigo let-sided ractional dierentiation operator 2.7, then the ollowing result holds: t ρ 1 J µ j m, τ η j m, k a tυ x xρ+β 1 ρ++β+γ, υ, ρ, υ, τ, k 3ψ m+2 ax υ Γτ The conditions or validity o 2.13 are i, R > 0 and a are any complex numbers; ρ+β, υ, ρ+γ, υ, η j +1, µ j m 1 ii ρ and υ are arbitrary such that Rρ + υ n > 0 and Rρ + + β + γ + υn > 0. Proo. enote L.H.S. o Theorem 2.5 by 3, then 3 t ρ 1 J µ j m, τ η j m, k a tυ x, by virtue o 1.1 and 2.7, we have d k d k x +β Γ + k +k, β k, +γ k x t ρ 1 J µ j m, τ η j m, k a tυ dt, 0 t ρ 1 J µ j m, τ η j m, k a tυ, x t +k 1 2F 1 β, γ + k; + k; 1 t/x now, interchanging the order o integration and summation is permissible under the conditions xρ+β 1 Γτ xρ+β 1 Γτ which completes the proo o Theorem 2.5. Γρ + + β + γ + υ n Γρ + υ n Γτ + kn Γρ + β + υ n Γρ + γ + υ nγ η n0 j µ j n a tυ n n! ρ++β+γ, υ, ρ, υ, τ,k 3ψ m+2 ax υ, ρ+β, υ, ρ+γ, υ, η j + 1, µ j m 1

6 . L. Suthar, S.. Purohit, R. K. Parmar, Math. Nat. Sci., , Corollary 2.6. Let m 1 be an integer, Rµ j > 0, η j i 1,, m are arbitrary parameters and,. be the Riemann Liouville let-sided ractional dierentiation operator 2.3, then the ollowing result holds: t ρ 1 J µ j m, τ η j m, k a tυ x xρ 1 ρ, υ, τ, k 2ψ m+1 ax υ Γτ The conditions or validity o 2.14 are i, R > 0 and a are any complex numbers; ii ρ and υ are arbitrary such that Rρ + υn > 0. ρ,υ,η j + 1,µ j m Right-sided ractional dierentiation o generalized multiindex Bessel unction n this section, we establish image ormulas or the generalized multiindex Bessel unction involving right sided operators o Saigo ractional dierentiation operators 2.8, in term o the generalized Wright unction. Theorem 2.7. Let m 1 be an integer, R µ j > 0, η j i 1,, m are arbitrary parameters and. be the Saigo let-sided ractional dierentiation operator 2.8, then the ollowing result holds: t ρ J µ j m, τ η j m, k a t x x+β+ρ β ρ, υ, γ ρ, υ, τ, k 3ψ m+2 ax Γτ The conditions or validity o 2.15 are i, R > 0 and a are any complex numbers; ρ, υ, γ β, ρ υ, η j + 1, µ j m 1 ii ρ and υ are arbitrary, such that Rγ ρ + υn > 0 and R β ρ + υn > 0. Proo. enote L.H.S. o Theorem 2.7 by 4, then 4 t ρ J µ j m, τ η j m, k a t x, by virtue o 1.1 and 2.8, we have d k d +k, β k, +γ k x +β Γ + k t ρ J µ j m,τ η j m, k a t dt, x t ρ J µ j m, τ η j m, k a t, t x +k 1 t +β 2F 1 β, γ ; + k; 1 x/t now, interchanging the order o integration and summation is permissible under the conditions x+β+ρ Γτ x+β+ρ Γτ which completes the proo o Theorem 2.7. Γ β ρ + υ n Γγ ρ + υ n Γτ + kn Γ ρ + υ n Γγ β ρ + υ nγ η n0 j µ j n a t n n! β ρ, υ, γ ρ, υ, τ, k 3ψ m+2 ax, ρ, υ, γ β, ρ υ, η j + 1, µ j m 1 Corollary 2.8. Let m 1 be an integer, Rµ j > 0, η j i 1,, m are arbitrary parameters and,. be the Riemann Liouville let-sided ractional dierentiation operator 2.8, then the ollowing result holds: t ρ J µ j m, τ η j m, k a t x xρ Γτ 2 ψ γ ρ, υ, τ, k m+1 ax The conditions or validity o 2.16 are γ+, ρ υ, η j + 1, µ j m 1

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