1 Introduction and Preliminaries
The Wright function play an important role in the partial differential equation of fractional order which is familiar and extensively treated in papers by a number of authors including Gorenflo et al. [6 ].
For ς i , τ j ∈ ℝ\{0} and a i ,b j ∈ ℂ,i = (1̅, p ); j = (1̅,q ) the generalized form of Wright function is defined by Wright ([13 ,14 ,15 ,16 ,17 ]) as following:
(1.1) ![]()
p Ψ q ( z ) = p Ψ q [ ( a i , ς i ) 1 , p ( b j , τ j ) 1 , q | z ] = ∑ n = 0 ∞ ∏ i = 1 p Γ ( a i + n ς i ) ∏ j = 1 q Γ ( b j + n τ j ) z n n ! , z ∈ ℂ , _p{\Psi _q}(z{) = _p}{\Psi _q}\left[ {\matrix{ {{{({a_i},{\varsigma _i})}_{1,p}}} \cr {{{({b_j},{\tau _j})}_{1,q}}} \cr } |z} \right] = \sum\limits_{n = 0}^\infty {{\prod\nolimits_{i = 1}^p \Gamma ({a_i} + n{\varsigma _i})} \over {\prod\nolimits_{j = 1}^q \Gamma ({b_j} + n{\tau _j})}}{{{z^n}} \over {n!}},z \in \mathbb{C},
where Γ (z ) is the well-known Euler gamma function [4 ].The condition for existence of (1.1) with its depiction in terms of Mellin-Barnes integral and the H-function were obtained by Kilbas et al. [10 ].
The generalized form of the above Wright function (1.1) was given by Gehlot and Prajapati [5 ], named as generalized K-Wright function which is defined as
(1.2) ![]()
p Ψ q k ( z ) = p Ψ q k [ ( a i , ς i ) 1 , p ( b j , τ j ) 1 , q | z ] = ∑ n = 0 ∞ ∏ i = 1 p Γ k ( a i + n ς i ) ∏ j = 1 q Γ k ( b j + n τ j ) z n n ! , z ∈ ℂ , _p\Psi _q^k(z{) = _p}\Psi _q^k\left[ {\matrix{ {{{({a_i},{\varsigma _i})}_{1,p}}} \cr {{{({b_j},{\tau _j})}_{1,q}}} \cr } |z} \right] = \sum\limits_{n = 0}^\infty {{\prod\nolimits_{i = 1}^p {\Gamma _k}({a_i} + n{\varsigma _i})} \over {\prod\nolimits_{j = 1}^q {\Gamma _k}({b_j} + n{\tau _j})}}{{{z^n}} \over {n!}},z \in \mathbb{C},
where k ∈ ℝ+ and (a i + nς i ), (b j + bτ j ) ∈ ℂ\k ℤ− for all n ∈ ℕ0 . The generalized k -gamma function [3 ] is defined as
(1.3) ![]()
Γ k ( z ) = ∫ 0 ∞ e − t k k t z − 1 dt ; ( ℜ ( z ) > 0 ; k ∈ ℝ + ) {\Gamma _k}(z) = \int_0^\infty {e^{ - {{{t^k}} \over k}}}{t^{z - 1}}dt;\;\;(\Re (z) > 0;\;k \in {\mathbb{R}^ + })
and
(1.4) ![]()
Γ k ( z ) = lim n → ∞ n ! k n ( nk ) z k − 1 ( z ) n , k , k ∈ ℝ + , z ∈ ℂ \ k ℤ − {\Gamma _k}(z) = \mathop {\lim }\limits_{n \to \infty } {{n!{k^n}{{(nk)}^{{z \over k} - 1}}} \over {{{(z)}_{n,k}}}},\;\;k \in {\mathbb{R}^ + },\;z \in \backslash k{\mathbb{Z}^ - }
Also
(1.5) ![]()
Γ k ( z ) = k z k − 1 Γ ( z k ) , {\Gamma _k}(z) = {k^{{z \over k} - 1}}\Gamma \left( {{z \over k}} \right),
where (z )n,k is the k -Pochammer symbol introduced by Diaz and Pariguan [3 ] defined for complex z ∈ ℂ and k ∈ ℝ as
(1.6) ![]()
( z ) n , k = { 1 if n = 0 , z ( z + k ) ( z + 2 k ) … ( z + ( n − 1 ) k ) if n ∈ ℕ . } {(z)_{n,k}} = \left\{ {\matrix{ 1 & {if\;\;n = 0,} \cr {z(z + k)(z + 2k) \ldots (z + (n - 1)k)} & {if\;\;n \in \mathbb{N}.} \cr } } \right\}
On taking k = 1, then the generalized K-Wright function (1.2) diminishes to the generalized Wright function (1.1) .
1.1 Saigo fractional calculus operators
Saigo [18 ] defined the fractional integral and differential operators with the Gauss hyergeometric function as kernel, which are remarkable generalizations of the Riemann-Liouville (R-L) and Erdélyi-Kober fractional calculus operators (see; [11 ]).
For ς , τ , γ ∈ ℂ and x ∈ ℝ+ with ℜ(ς ) > 0, the left-hand and the right-hand sided generalized fractional integral operators connected with Gauss hypergeometric function are defined as below:
(1.7) ![]()
( I 0 + ς , τ , γ f ) ( x ) = x − ς − τ Γ ( ς ) ∫ 0 x ( x − t ) ς − 1 2 F 1 ( ς + τ , − γ ; ς ; 1 − t x ) f ( t ) d t (I_{0 + }^{\varsigma ,\tau ,\gamma }f)(x) = {{{x^{ - \varsigma - \tau }}} \over {\Gamma (\varsigma )}}\int_0^x {(x - t)^{\varsigma - 1}}_2{F_1}(\varsigma + \tau , - \gamma ;\varsigma ;1 - {t \over x})f(t)dt
and
(1.8) ![]()
( I − ς , τ , γ f ) ( x ) = 1 Γ ( ς ) ∫ x ∞ ( t − x ) ς − 1 t ς + τ 2 F 1 ( ς + τ , − γ ; ς ; 1 − x t ) f ( t ) dt (I_ - ^{\varsigma ,\tau ,\gamma }f)(x) = {1 \over {\Gamma (\varsigma )}}\int_x^\infty {{{{{(t - x)}^{\varsigma - 1}}} \over {{t^{\varsigma + \tau }}}}{_2}}{F_1}(\varsigma + \tau , - \gamma ;\varsigma ;1 - {x \over t})f(t)dt
respectively. Here, 2 F 1 (ς, τ ; γ ; z ) is the Gauss hypergeometric function [11 ] defined for z ∈ ℂ, |z | < 1 and ς , τ ∈ ℂ,
γ ∈ ℂ \ ℤ 0 − \gamma \in \mathbb{C} \backslash \mathbb{Z}_0^ -
by
![]()
2 F 1 ( ς , τ ; γ ; z ) = ∑ n = 0 ∞ ( ς ) n ( τ ) n ( γ ) n z n n ! , _2{F_1}(\varsigma ,\tau ;\gamma ;z) = \sum\limits_{n = 0}^\infty {{{{(\varsigma )}_n}{{(\tau )}_n}} \over {{{(\gamma )}_n}}}{{{z^n}} \over {n!}},
where (z )n = (z )n, 1 . The corresponding fractional differential operators are
(1.9) ![]()
( D 0 + ς , τ , γ f ) ( x ) = ( d dx ) l ( I 0 + − ς + l , − τ − l , ς + γ − l f ) ( x ) (D_{0 + }^{\varsigma ,\tau ,\gamma }f)(x) = {\left( {{d \over {dx}}} \right)^l}(I_{0 + }^{ - \varsigma + l, - \tau - l,\varsigma + \gamma - l}f)(x)
and
(1.10) ![]()
( D − ς , τ , γ f ) ( x ) = ( − d dx ) l ( I − − ς + l , − τ − l , ς + γ f ) ( x ) (D_ - ^{\varsigma ,\tau ,\gamma }f)(x) = {\left( { - {d \over {dx}}} \right)^l}(I_ - ^{ - \varsigma + l, - \tau - l,\varsigma + \gamma }f)(x)
where l = [ℜ (ς )] + 1 and [ℜ (ς )] is the integer part of ℜ(ς ). Substituting τ = −ς and τ = 0 in equation (1.7) – (1.10) , we get the corresponding R-L and Erdélyi-Kober fractional operators, respectively.
1.2 Marichev-Saigo-Maeda fractional operators
Marichev [13 ] was introduced and studied fractional calculus operators which are the generalization of the Saigo operators, later generalized by Saigo and Maeda [19 ]. For ς , ς ′, τ , τ ′, γ ∈ ℂ and x ∈ ℝ+ with ℜ(γ ) > 0, the left-hand and right-hand sided MSM fractional integral and derivative operators associated with third Appell function F 3 are defined as
(1.11) ![]()
( I 0 + ς , ς ′ , τ , τ ′ , γ f ) ( x ) = x − ς Γ ( γ ) ∫ 0 x ( x − t ) γ − 1 t ς ′ F 3 ( ς , ς ′ , τ , τ , γ , 1 − t x , 1 − x t ) f ( t ) dt (I_{0 + }^{\varsigma ,\varsigma ',\tau ,\tau ',\gamma }f)(x) = {{{x^{ - \varsigma }}} \over {\Gamma (\gamma )}}\int_0^x {{{{(x - t)}^{\gamma - 1}}} \over {{t^{\varsigma '}}}}{F_3}(\varsigma ,\varsigma ',\tau ,\tau ,\gamma ,1 - {t \over x},1 - {x \over t})f(t)dt
and
(1.12) ![]()
( I − ς , ς ′ , τ , τ ′ , γ f ) ( x ) = x − ς ′ Γ ( γ ) ∫ x ∞ ( t − x ) γ − 1 t ς F 3 ( ς , ς ′ , τ , τ , γ , 1 − x t , 1 − t x ) f ( t ) dt (I_ - ^{\varsigma ,\varsigma ',\tau ,\tau ',\gamma }f)(x) = {{{x^{ - \varsigma '}}} \over {\Gamma (\gamma )}}\int_x^\infty {{{{(t - x)}^{\gamma - 1}}} \over {{t^\varsigma }}}{F_3}(\varsigma ,\varsigma ',\tau ,\tau ,\gamma ,1 - {x \over t},1 - {t \over x})f(t)dt (1.13) ![]()
( D 0 + ς , ς ′ , τ , τ ′ , γ f ) ( x ) = ( d dx ) m ( I 0 + − ς ′ , − ς , − τ ′ + m , − τ , − γ + m f ) ( x ) (D_{0 + }^{\varsigma ,\varsigma ',\tau ,\tau ',\gamma }f)(x) = {\left( {{d \over {dx}}} \right)^m}(I_{0 + }^{ - \varsigma ', - \varsigma , - \tau ' + m, - \tau , - \gamma + m}f)(x)
and
(1.14) ![]()
( D − ς , ς ′ , τ , τ ′ , γ f ) ( x ) = ( − d dx ) m ( I − − ς ′ , − ς , − τ ′ , − τ + m , − γ + m f ) ( x ) (D_ - ^{\varsigma ,\varsigma ',\tau ,\tau ',\gamma }f)(x) = {\left( { - {d \over {dx}}} \right)^m}(I_ - ^{ - \varsigma ', - \varsigma , - \tau ', - \tau + m, - \gamma + m}f)(x)
respectively, where m = [ℜ(γ )] + 1 and the third Appell function [17 ], is defined by
(1.15) ![]()
F 3 ( ς , ς ′ , τ , τ ′ , γ ; x , y ) = ∑ m , n , = 0 ∞ ( ς ) m ( ς ′ ) n ( τ ) m ( τ ′ ) n ( γ ) m + n x m y n m ! n ! , max { | x | , | y | } < 1 . {F_3}(\varsigma ,\varsigma ',\tau ,\tau ',\gamma ;x,y) = \sum\limits_{m,n, = 0}^\infty {{{{(\varsigma )}_m}{{(\varsigma ')}_n}{{(\tau )}_m}{{(\tau ')}_n}} \over {{{(\gamma )}_{m + n}}}}{{{x^m}{y^n}} \over {m!n!}},\;\;\;\max \{ |x|,|y|\} < 1.
1.3 Generalized k -Struve function
The generalized k -Struve function was defined by Nisar et al. [14 ] as
(1.16) ![]()
S ν , c k ( t ) = ∑ n = 0 ∞ ( − c ) n Γ k ( nk + ν + 3 k 2 ) Γ ( n + 3 2 ) n ! ( t 2 ) 2 n + ν k + 1 ( k ∈ ℝ + ; c ∈ ℝ ; ν > − 1 ) \matrix{ {S_{\nu ,c}^k(t) = \sum\limits_{n = 0}^\infty {{{{( - c)}^n}} \over {{\Gamma _k}(nk + \nu + {{3k} \over 2})\Gamma (n + {3 \over 2})n!}}{{\left( {{t \over 2}} \right)}^{2n + {\nu \over k} + 1}}} \cr {(k \in {\mathbb{R}^ + };c \in \mathbb{R};\nu > - 1)} \cr }
taking k→ 1 and c = 1; (1.15) reduces to yield the well-known Struve function of order ν is defined by [1 ] as
(1.17) ![]()
H ν ( t ) = ∑ n = 0 ∞ ( − 1 ) n Γ ( n + ν + 3 2 ) Γ ( n + 3 2 ) n ! ( t 2 ) 2 n + ν + 1 {H_\nu }(t) = \sum\limits_{n = 0}^\infty {{{{( - 1)}^n}} \over {\Gamma (n + \nu + {3 \over 2})\Gamma (n + {3 \over 2})n!}}{\left( {{t \over 2}} \right)^{2n + \nu + 1}}
For more details about Struve functions, their generalizations and properties, the esteemed reader is invited to consider references [2 , 7 , 8 , 14 , 15 , 20 ,21 ,22 ].
The following MSM integral operators are required here [19 , p. 394] to obtain the MSM fractional integration of generalized k -Struve function.
Lemma 1
Let ς , ς ′, τ , τ ′, γ , ρ ∈ ℂ such that ℜ(ς ) > 0
(i) ℜ(ρ ) > 0 max{0,ℜ(ς ′ − τ ′),ℜ(ς + ς ′ + τ − γ )}, then
(1.18) ![]()
( I 0 + ς , ς ′ , τ , τ ′ , γ t ρ − 1 ) ( x ) = Γ ( ρ ) Γ ( − ς ′ + τ ′ + ρ ) Γ ( − ς − ς ′ − τ + γ + ρ ) Γ ( τ ′ + ρ ) Γ ( − ς − ς ′ + γ + ρ ) Γ ( − ς ′ − τ + γ + ρ ) x − ς − ς ′ + γ + ρ − 1 (I_{0 + }^{\varsigma ,\varsigma ',\tau ,\tau ',\gamma }{t^{\rho - 1}})(x) = {{\Gamma (\rho )\Gamma ( - \varsigma ' + \tau ' + \rho )\Gamma ( - \varsigma - \varsigma ' - \tau + \gamma + \rho )} \over {\Gamma (\tau ' + \rho )\Gamma ( - \varsigma - \varsigma ' + \gamma + \rho )\Gamma ( - \varsigma ' - \tau + \gamma + \rho )}}{x^{ - \varsigma - \varsigma ' + \gamma + \rho - 1}}
(ii) If ℜ(ρ ) > max{ℜ(τ ),ℜ(−ς − ς ′ + γ ),ℜ(−ς − τ ′ + γ }, then
(1.19) ![]()
( I − ς , ς ′ , τ , τ ′ , γ t − ρ ) ( x ) = Γ ( − τ + ρ ) Γ ( ς + ς ′ − γ + ρ ) Γ ( ς + τ ′ − γ + ρ ) Γ ( ρ ) Γ ( ς − τ + ρ ) Γ ( ς + ς ′ + τ ′ − γ + ρ ) x − ς − ς ′ + γ − ρ (I_ - ^{\varsigma ,\varsigma ',\tau ,\tau ',\gamma }{t^{ - \rho }})(x) = {{\Gamma ( - \tau + \rho )\Gamma (\varsigma + \varsigma ' - \gamma + \rho )\Gamma (\varsigma + \tau ' - \gamma + \rho )} \over {\Gamma (\rho )\Gamma (\varsigma - \tau + \rho )\Gamma (\varsigma + \varsigma ' + \tau ' - \gamma + \rho )}}{x^{ - \varsigma - \varsigma ' + \gamma - \rho }}
Further, to obtain the MSM fractional differentiation of the generalized k -Struve function, following results will be used from [9 ] as below:
Lemma 2
Let ς, ς ′, τ, τ ′, γ, ρ ∈ ℂ, such that ℜ(ς ) > 0;
(i) If ℜ(ρ ) > max{0,ℜ(−ς + τ ),ℜ(−ς − ς ′ − τ ′ + γ )}
(1.20) ![]()
( D 0 + ς , ς ′ , τ , τ ′ , γ t ρ − 1 ) ( x ) = Γ ( ρ ) Γ ( − τ + ς + ρ ) Γ ( ς + ς ′ + τ ′ − γ + ρ ) Γ ( − τ + ρ ) Γ ( ς + ς ′ − γ + ρ ) Γ ( ς + τ ′ − γ + ρ ) x ς + ς ′ − γ + ρ − 1 (D_{0 + }^{\varsigma ,\varsigma ',\tau ,\tau ',\gamma }{t^{\rho - 1}})(x) = {{\Gamma (\rho )\Gamma ( - \tau + \varsigma + \rho )\Gamma (\varsigma + \varsigma ' + \tau ' - \gamma + \rho )} \over {\Gamma ( - \tau + \rho )\Gamma (\varsigma + \varsigma ' - \gamma + \rho )\Gamma (\varsigma + \tau ' - \gamma + \rho )}}{x^{\varsigma + \varsigma ' - \gamma + \rho - 1}}
(ii) If ℜ(ρ ) > max{ℜ(−τ ′),ℜ(ς ′ + τ − γ ),ℜ(ς + ς ′ − γ ) + [ℜ(γ )] + 1}, then
(1.21) ![]()
( D − ς , ς ′ , τ , τ ′ , γ t − ρ ) ( x ) = Γ ( τ ′ + ρ ) Γ ( − ς − ς ′ + γ + ρ ) Γ ( − ς ′ − τ + γ + ρ ) Γ ( ρ ) Γ ( − ς ′ + τ ′ + ρ ) Γ ( − ς − ς ′ − τ + γ + ρ ) x ς + ς ′ − γ − ρ (D_ - ^{\varsigma ,\varsigma ',\tau ,\tau ',\gamma }{t^{ - \rho }})(x) = {{\Gamma (\tau ' + \rho )\Gamma ( - \varsigma - \varsigma ' + \gamma + \rho )\Gamma ( - \varsigma ' - \tau + \gamma + \rho )} \over {\Gamma (\rho )\Gamma ( - \varsigma ' + \tau ' + \rho )\Gamma ( - \varsigma - \varsigma ' - \tau + \gamma + \rho )}}{x^{\varsigma + \varsigma ' - \gamma - \rho }}
2 Fractional Calculus Approach
In this section, the following six theorems for k -Struve function concerning to MSM fractional integral and differential operators are established here as main results.
Theorem 1
Let ς, ς ′, τ, tau ′, γ, ρ ∈ ℂ and k ∈ ℝ+ be such that
ℜ ( γ ) > 0 , ℜ ( λ k ) > max { 0 , ℜ ( ς ′ − τ ′ ) , ℜ ( ς + ς ′ + τ − γ ) } \Re (\gamma ) > 0,\Re \left( {{\lambda \over k}} \right) > \max \{ 0,\Re (\varsigma ' - \tau '),\Re (\varsigma + \varsigma ' + \tau - \gamma )\}
. Also let c ∈ ℝ; ν > −1, then for t > 0
(2.1) ![]()
( I 0 + ς , ς ′ , τ , τ ′ , γ ( t λ k − 1 S ν , c k ( t ) ) ) ( x ) = k γ + 1 2 x − ς − ς ′ + γ + λ k + ν k 2 ν k + 1 × 3 Ψ 5 k [ ( λ + ν + k , 2 k ) , ( − k ς ′ + k τ ′ + λ + ν + k , 2 k ) , ( k τ ′ + λ + ν + k , 2 k ) , ( − k ς − k ς ′ + k γ + λ + ν + k , 2 k ) , ( − k ς − k ς ′ − k τ + k γ + λ + ν + k , 2 k ) ( − k ς ′ − k τ + k γ + λ + ν + k , 2 k ) , ( ν + 3 k 2 , k ) , ( 3 k 2 , k ) | − c x 2 k 4 ] . \matrix{ {\left( {I_{0 + }^{\varsigma ,\varsigma ',\tau ,\tau ',\gamma }\left( {{t^{{\lambda \over k} - 1}}S_{\nu ,c}^k(t)} \right)} \right)(x) = {{{k^{\gamma + {1 \over 2}}}{x^{ - \varsigma - \varsigma ' + \gamma + {\lambda \over k} + {\nu \over k}}}} \over {{2^{{\nu \over k} + 1}}}}} \hfill \cr {{ \times _3}\Psi _5^k\left[ {\matrix{ {(\lambda + \nu + k,2k),} & {( - k\varsigma ' + k\tau ' + \lambda + \nu + k,2k),} \cr {(k\tau ' + \lambda + \nu + k,2k),} & {( - k\varsigma - k\varsigma ' + k\gamma + \lambda + \nu + k,2k),} \cr } } \right.} \hfill \cr {\left. {\matrix{ {( - k\varsigma - k\varsigma ' - k\tau + k\gamma + \lambda + \nu + k,2k)} & {} \cr {( - k\varsigma ' - k\tau + k\gamma + \lambda + \nu + k,2k),} & {(\nu + {{3k} \over 2},k),({{3k} \over 2},k)}\cr } \;\;|{{ - c{x^2}k} \over 4}} \right].} \hfill \cr }
Proof
On using (1.16) and taking the left-hand sided MSM fractional integral operator inside the summation, the left-hand side of (2.1) becomes
![]()
= ∑ n = 0 ∞ ( − c ) n Γ k ( nk + ν + 3 k 2 ) Γ ( n + 3 2 ) n ! 2 2 n + ν k + 1 ( I 0 + ς , ς ′ , τ , τ ′ , γ { t λ k + ν k + 2 n } ) ( x ) , = \sum\limits_{n = 0}^\infty {{{{( - c)}^n}} \over {{\Gamma _k}(nk + \nu + {{3k} \over 2})\Gamma (n + {3 \over 2})n{{!2}^{2n + {\nu \over k} + 1}}}}\left( {I_{0 + }^{\varsigma ,\varsigma ',\tau ,\tau ',\gamma }\{ {t^{{\lambda \over k} + {\nu \over k} + 2n}}\} } \right)(x),
Making use of (1.18) , we obtain
![]()
= ∑ n = 0 ∞ ( − c ) n x − ς − ς ′ + γ λ k + ν k + 2 n Γ k ( nk + ν + 3 k 2 ) Γ ( n + 3 2 ) n ! 2 2 n + ν k + 1 Γ ( λ k + ν k + 2 n + 1 ) Γ ( τ ′ + λ k + ν k + 2 n + 1 ) × Γ ( − ς − ς ′ − τ + γ + λ k + ν k + 2 n + 1 ) Γ ( − ς ′ + τ ′ + λ k + ν k + 2 n + 1 ) Γ ( − ς ′ − τ + γ + λ k + ν k + 2 n + 1 ) Γ ( − ς − ς ′ + γ + λ k ν k + 2 n + 1 ) , \matrix{ { = \sum\limits_{n = 0}^\infty {{{{( - c)}^n}{x^{ - \varsigma - \varsigma ' + \gamma {\lambda \over k} + {\nu \over k} + 2n}}} \over {{\Gamma _k}(nk + \nu + {{3k} \over 2})\Gamma (n + {3 \over 2})n{{!2}^{2n + {\nu \over k} + 1}}}}{{\Gamma ({\lambda \over k} + {\nu \over k} + 2n + 1)} \over {\Gamma (\tau ' + {\lambda \over k} + {\nu \over k} + 2n + 1)}}} \hfill \cr { \times {{\Gamma ( - \varsigma - \varsigma ' - \tau + \gamma + {\lambda \over k} + {\nu \over k} + 2n + 1)\Gamma ( - \varsigma ' + \tau ' + {\lambda \over k} + {\nu \over k} + 2n + 1)} \over {\Gamma ( - \varsigma ' - \tau + \gamma + {\lambda \over k} + {\nu \over k} + 2n + 1)\Gamma ( - \varsigma - \varsigma ' + \gamma + {\lambda \over k}{\nu \over k} + 2n + 1)}},} \hfill \cr }
Now, using equation (1.5) on above term, then we get
![]()
= x − ς − ς ′ + γ + λ k + ν k 2 ν k + 1 k − γ − 1 2 ∑ n = 0 ∞ Γ k ( λ + ν + k + 2 nk ) Γ k ( − k ς ′ + k τ ′ + λ + ν + k + 2 nk ) Γ k ( k τ ′ + λ + ν + k + 2 nk ) Γ k ( − k ς − k ς ′ + k γ + λ + ν + k + 2 nk ) × Γ k ( − k ς − k ς ′ − k τ + k γ + λ + ν + k + 2 nk ) Γ k ( − k ς ′ − k τ + k γ + λ + ν + k + 2 nk ) Γ k ( nk + ν + 3 k 2 ) Γ k ( 3 k 2 + nk ) n ! ( − c x 2 k 2 2 ) n . \matrix{ { = {{{x^{ - \varsigma - \varsigma ' + \gamma + {\lambda \over k} + {\nu \over k}}}} \over {{2^{{\nu \over k} + 1}}{k^{ - \gamma - {1 \over 2}}}}}\sum\limits_{n = 0}^\infty {{{\Gamma _k}(\lambda + \nu + k + 2nk){\Gamma _k}( - k\varsigma ' + k\tau ' + \lambda + \nu + k + 2nk)} \over {{\Gamma _k}(k\tau ' + \lambda + \nu + k + 2nk){\Gamma _k}( - k\varsigma - k\varsigma ' + k\gamma + \lambda + \nu + k + 2nk)}}} \hfill \cr { \times {{{\Gamma _k}( - k\varsigma - k\varsigma ' - k\tau + k\gamma + \lambda + \nu + k + 2nk)} \over {{\Gamma _k}( - k\varsigma ' - k\tau + k\gamma + \lambda + \nu + k + 2nk){\Gamma _k}(nk + \nu + {{3k} \over 2}){\Gamma _k}({{3k} \over 2} + nk)n!}}{{\left( {{{ - c{x^2}k} \over {{2^2}}}} \right)}^n}.} \hfill \cr }
Using the definition of (1.2) in the above term, we arrive at the result (2.1) .
Next theorem gives the right-hand MSM fractional integration of
S ν , c k ( . ) S_{\nu ,c}^k(.)
.
Theorem 2
Let ς, ς ′, τ, τ ′, γ, ρ ∈ ℂ and k ∈ ℝ+ be such that ℜ(γ ) > 0,
ℜ ( γ ) > 0 , ℜ ( λ k ) > max { ℜ ( τ ) , ℜ ( − ς − ς ′ + γ ) , ℜ ( − ς − τ ′ + γ ) } \Re (\gamma ) > 0,\Re \left( {{\lambda \over k}} \right) > \max \{ \Re (\tau ),\Re ( - \varsigma - \varsigma ' + \gamma ),\Re ( - \varsigma - \tau ' + \gamma )\}
. Also let c ∈ ℝ; ν > −1, then for t > 0
(2.2) ![]()
( I − ς , ς ′ , τ , τ ′ , γ ( t λ k − 1 S ν , c k ( t ) ) ) ( x ) = k γ + 1 2 x − ς − ς ′ + γ + λ k + ν k 2 ν k + 1 × 3 Ψ 5 k [ ( − k τ − λ − ν , − 2 k ) , ( k ς + k ς ′ − k γ − nu , − 2 k ) , ( − λ − ν , − 2 k ) , ( k ς − k τ − λ − ν , − 2 k ) , ( k ς + k τ ′ − k γ − λ − ν , − 2 k ) ( k ς + k ς ′ + k τ ′ − k γ − λ − ν , − 2 k ) , ( ν + 3 k 2 , k ) , ( 3 k 2 , k ) | − c x 2 k 4 ] . \matrix{ {\left( {I_ - ^{\varsigma ,\varsigma ',\tau ,\tau ',\gamma }\left( {{t^{{\lambda \over k} - 1}}S_{\nu ,c}^k(t)} \right)} \right)(x) = {{{k^{\gamma + {1 \over 2}}}{x^{ - \varsigma - \varsigma ' + \gamma + {\lambda \over k} + {\nu \over k}}}} \over {{2^{{\nu \over k} + 1}}}}} \hfill \cr {{ \times _3}\Psi _5^k\left[ {\matrix{ {( - k\tau - \lambda - \nu , - 2k),} & {(k\varsigma + k\varsigma ' - k\gamma - nu, - 2k),} \cr {( - \lambda - \nu , - 2k),} & {(k\varsigma - k\tau - \lambda - \nu , - 2k),} \cr } } \right.} \hfill \cr {\left. {\matrix{ {(k\varsigma + k\tau ' - k\gamma - \lambda - \nu , - 2k)} & {} \cr {(k\varsigma + k\varsigma ' + k\tau ' - k\gamma - \lambda - \nu , - 2k),} & {(\nu + {{3k} \over 2},k),({{3k} \over 2},k)} \cr } \;\;|{{ - c{x^2}k} \over 4}} \right].} \hfill \cr }
Proof
On using (1.16) and taking the right-hand sided MSM fractional integral operator inside the summation, the left hand side of (2.2) becomes
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= ∑ n = 0 ∞ ( − c ) n Γ k ( nk + ν + 3 k 2 ) Γ ( n + 3 2 ) n ! 2 2 n + ν k + 1 ( I − ς , ς ′ , τ , τ ′ , γ { t λ k + ν k + 2 n } ) ( x ) = \sum\limits_{n = 0}^\infty {{{{( - c)}^n}} \over {{\Gamma _k}(nk + \nu + {{3k} \over 2})\Gamma (n + {3 \over 2})n{{!2}^{2n + {\nu \over k} + 1}}}}\left( {I_ - ^{\varsigma ,\varsigma ',\tau ,\tau ',\gamma }\{ {t^{{\lambda \over k} + {\nu \over k} + 2n}}\} } \right)(x)
On using (1.19) , we get
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= ∑ n = 0 ∞ ( − c ) n x − ς − ς ′ + γ + λ k + ν k + 2 n Γ k ( nk + ν + 3 k 2 ) Γ ( n + 3 2 ) n ! 2 2 n + ν k + 1 Γ ( − τ − λ k − ν k − 2 n ) Γ ( − λ k − ν k − 2 n ) × Γ ( ς + τ ′ − γ − λ k − ν k − 2 n ) Γ ( ς + ς ′ − γ − λ k − ν k − 2 n ) Γ ( ς + ς ′ + τ ′ − γ − λ k − ν k − 2 n ) Γ ( ς − τ − λ k − ν k − 2 n ) = ∑ n = 0 ∞ ( − c x 2 ) n x − ς − ς ′ + γ + λ k + ν k Γ k ( nk + ν + 3 k 2 ) n ! 2 2 n + ν k + 1 Γ ( − τ − λ k − ν k − 2 n ) Γ ( ς + ς ′ − γ − λ k − ν k − 2 n ) Γ ( n + 3 2 ) Γ ( − λ k − ν k − 2 n ) Γ ( ς − τ − λ k − ν k − 2 n ) × Γ ( ς + τ ′ − γ − λ k − ν k − 2 n ) Γ ( ς + ς ′ + τ ′ − γ − λ k − ν k − 2 n ) = x − ς − ς ′ + γ + λ k + ν k 2 ν k + 1 k − γ − 1 2 ∑ n = 0 ∞ ( − ck x 2 4 ) n 1 n ! Γ k ( − k τ − λ − ν − 2 nk ) Γ k ( − λ − ν − 2 nk ) Γ k ( k ς − k τ − λ − ν − 2 nk ) × Γ k ( k ς + k ς ′ − k γ − λ − ν − 2 nk ) Γ k ( k ς + k τ ′ − k γ − λ − ν − 2 nk ) Γ k ( k ς + k ς ′ + k τ ′ − k γ − λ − ν − 2 nk ) Γ k ( nk + ν + 3 k 2 ) Γ k ( 3 k 2 + nk ) \matrix{ { = \sum\limits_{n = 0}^\infty {{{{( - c)}^n}{x^{ - \varsigma - \varsigma ' + \gamma + {\lambda \over k} + {\nu \over k} + 2n}}} \over {{\Gamma _k}(nk + \nu + {{3k} \over 2})\Gamma (n + {3 \over 2})n{{!2}^{2n + {\nu \over k}}} + 1}}{{\Gamma ( - \tau - {\lambda \over k} - {\nu \over k} - 2n)} \over {\Gamma ( - {\lambda \over k} - {\nu \over k} - 2n)}}} \hfill \cr { \times {{\Gamma (\varsigma + \tau ' - \gamma - {\lambda \over k} - {\nu \over k} - 2n)\Gamma (\varsigma + \varsigma ' - \gamma - {\lambda \over k} - {\nu \over k} - 2n)} \over {\Gamma (\varsigma + \varsigma ' + \tau ' - \gamma - {\lambda \over k} - {\nu \over k} - 2n)\Gamma (\varsigma - \tau - {\lambda \over k} - {\nu \over k} - 2n)}}} \hfill \cr { = \sum\limits_{n = 0}^\infty {{{{( - c{x^2})}^n}{x^{ - \varsigma - \varsigma ' + \gamma + {\lambda \over k} + {\nu \over k}}}} \over {{\Gamma _k}(nk + \nu + {{3k} \over 2})n{{!2}^{2n + {\nu \over k} + 1}}}}{{\Gamma ( - \tau - {\lambda \over k} - {\nu \over k} - 2n)\Gamma (\varsigma + \varsigma ' - \gamma - {\lambda \over k} - {\nu \over k} - 2n)} \over {\Gamma (n + {3 \over 2})\Gamma ( - {\lambda \over k} - {\nu \over k} - 2n)\Gamma (\varsigma - \tau - {\lambda \over k} - {\nu \over k} - 2n)}}} \hfill \cr { \times {{\Gamma (\varsigma + \tau ' - \gamma - {\lambda \over k} - {\nu \over k} - 2n)} \over {\Gamma (\varsigma + \varsigma ' + \tau ' - \gamma - {\lambda \over k} - {\nu \over k} - 2n)}}} \hfill \cr { = {{{x^{ - \varsigma - \varsigma ' + \gamma + {\lambda \over k} + {\nu \over k}}}} \over {{2^{{\nu \over k} + 1}}{k^{ - \gamma - {1 \over 2}}}}}\sum\limits_{n = 0}^\infty {{\left( {{{ - ck{x^2}} \over 4}} \right)}^n}{1 \over {n!}}{{{\Gamma _k}( - k\tau - \lambda - \nu - 2nk)} \over {{\Gamma _k}( - \lambda - \nu - 2nk){\Gamma _k}(k\varsigma - k\tau - \lambda - \nu - 2nk)}}} \hfill \cr { \times {{{\Gamma _k}(k\varsigma + k\varsigma ' - k\gamma - \lambda - \nu - 2nk){\Gamma _k}(k\varsigma + k\tau ' - k\gamma - \lambda - \nu - 2nk)} \over {{\Gamma _k}(k\varsigma + k\varsigma ' + k\tau ' - k\gamma - \lambda - \nu - 2nk){\Gamma _k}(nk + \nu + {{3k} \over 2}){\Gamma _k}({{3k} \over 2} + nk)}}} \hfill \cr }
and the result follows on making use of (1.5) and definition of generalized k -Wright function.
Theorem 3
Let ς, ς ′, τ, τ ′, γ, ρ ∈ ℂ and k ∈ ℝ+ be such that ℜ(γ ) > 0,
ℜ ( γ ) > 0 , ℜ ( λ k ) > max { ℜ ( τ ) , ℜ ( − ς − ς ′ + γ ) , ℜ ( − ς − τ ′ + γ ) } \Re (\gamma ) > 0,\Re ({\lambda \over k}) > \max \{ \Re (\tau ),\Re ( - \varsigma - \varsigma ' + \gamma ),\Re ( - \varsigma - \tau ' + \gamma )\}
. Also let c ∈ ℝ; ν > −1, then for t > 0
(2.3) ![]()
( I − ς , ς ′ , τ , τ ′ , γ ( t − λ k S ν , c k ( t ) ) ) ( x ) = k γ − 1 2 x − ς − ς ′ + γ + ν k − λ k + 1 2 ν k + 1 × 3 Ψ 5 k [ ( − k τ + λ − ν , − 2 k ) , ( k ς + k ς ′ − k γ + λ − nu − k , − 2 k ) , ( − λ − ν − k , − 2 k ) , ( k ς − k τ + λ − ν − k , − 2 k ) , ( k ς + k τ ′ − k γ + λ − ν − k , − 2 k ) ( k ς + k ς ′ + k τ ′ − k γ + λ − ν − k , − 2 k ) , ( ν + 3 k 2 , k ) , ( 3 k 2 , k ) | − c x 2 k 4 ] . \matrix{ {\left( {I_ - ^{\varsigma ,\varsigma ',\tau ,\tau ',\gamma }\left( {{t^{{{ - \lambda } \over k}}}S_{\nu ,c}^k(t)} \right)} \right)(x) = {{{k^{\gamma - {1 \over 2}}}{x^{ - \varsigma - \varsigma ' + \gamma + {\nu \over k} - {\lambda \over k} + 1}}} \over {{2^{{\nu \over k} + 1}}}}} \hfill \cr {{ \times _3}\Psi _5^k\left[ {\matrix{ {( - k\tau + \lambda - \nu , - 2k),} & {(k\varsigma + k\varsigma ' - k\gamma + \lambda - nu - k, - 2k),} \cr {( - \lambda - \nu - k, - 2k),} & {(k\varsigma - k\tau + \lambda - \nu - k, - 2k),} \cr } } \right.} \hfill \cr {\left. {\matrix{ {(k\varsigma + k\tau ' - k\gamma + \lambda - \nu - k, - 2k)} & {} \cr {(k\varsigma + k\varsigma ' + k\tau ' - k\gamma + \lambda - \nu - k, - 2k),} & {(\nu + {{3k} \over 2},k),({{3k} \over 2},k)} \cr } \;\;|{{ - c{x^2}k} \over 4}} \right].} \hfill \cr }
Proof
On using (1.16) and taking the right-hand sided MSM fractional integral operator inside the summation, the left hand side of (2.3) becomes
![]()
= ∑ n = 0 ∞ ( − c ) n Γ k ( nk + ν + 3 k 2 ) Γ ( n + 3 2 ) n ! 2 2 n + ν k + 1 ( I − ς , ς ′ , τ , τ ′ , γ { t ν − λ k + 2 n + 1 } ) = \sum\limits_{n = 0}^\infty {{{{( - c)}^n}} \over {{\Gamma _k}(nk + \nu + {{3k} \over 2})\Gamma (n + {3 \over 2})n{{!2}^{2n + {\nu \over k} + 1}}}}\left( {I_ - ^{\varsigma ,\varsigma ',\tau ,\tau ',\gamma }\{ {t^{{{\nu - \lambda } \over k} + 2n + 1}}\} } \right)
On using (1.19) , we obtain
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= ( − c ) n t − ς − ς ′ + γ + ν k − λ k + 2 n + 1 Γ k ( nk + ν + 3 k 2 ) Γ ( n + 3 2 ) n ! 2 2 n + ν k + 1 Γ ( − τ + λ − ν k − 2 n − 1 ) Γ ( λ − nu k − 2 n − 1 ) × Γ ( ς + ς ′ − γ + λ − ν k − 2 n − 1 ) Γ ( ς + τ ′ − γ + λ − ν k − 2 n − 1 ) γ ( ς − τ + λ − ν k − 2 n − 1 ) Γ ( ς + ς ′ + τ ′ − γ + λ − ν k − 2 n − 1 ) \matrix{ { = {{{{( - c)}^n}{t^{ - \varsigma - \varsigma ' + \gamma + {\nu \over k} - {\lambda \over k} + 2n + 1}}} \over {{\Gamma _k}(nk + \nu + {{3k} \over 2})\Gamma (n + {3 \over 2})n{{!2}^{2n + {\nu \over k} + 1}}}}{{\Gamma ( - \tau + {{\lambda - \nu } \over k} - 2n - 1)} \over {\Gamma ({{\lambda - nu} \over k} - 2n - 1)}}} \hfill \cr { \times {{\Gamma (\varsigma + \varsigma ' - \gamma + {{\lambda - \nu } \over k} - 2n - 1)\Gamma (\varsigma + \tau ' - \gamma + {{\lambda - \nu } \over k} - 2n - 1)} \over {\gamma (\varsigma - \tau + {{\lambda - \nu } \over k} - 2n - 1)\Gamma (\varsigma + \varsigma ' + \tau ' - \gamma + {{\lambda - \nu } \over k} - 2n - 1)}}} \hfill \cr }
Making use of (1.5) , we get
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= x − ς − ς ′ + γ + ν k − λ k + 1 k − γ + 1 2 2 ν k + 1 ∑ n = 0 ∞ ( − ck x 2 ) n 4 n n ! Γ k ( − k τ + λ − ν − k − 2 nk ) Γ k ( λ − ν − k − 2 nk ) Γ k ( k ς − k τ + λ − ν − k − 2 nk ) × Γ k ( k ς + k ς ′ − k γ + λ − ν − k − 2 nk ) Γ k ( k ς + k τ ′ − k γ + λ − ν − k − 2 nk ) Γ k ( k ς + k ς ′ + k τ ′ − k γ + λ − ν − k − 2 nk ) Γ k ( nk + ν + 3 k 2 ) Γ k ( nk + 3 k 2 ) \matrix{ { = {{{x^{ - \varsigma - \varsigma ' + \gamma + {\nu \over k} - {\lambda \over k} + 1}}} \over {{k^{ - \gamma + {1 \over 2}}}{2^{{\nu \over k} + 1}}}}\sum\limits_{n = 0}^\infty {{{{( - ck{x^2})}^n}} \over {{4^n}n!}}{{{\Gamma _k}( - k\tau + \lambda - \nu - k - 2nk)} \over {{\Gamma _k}(\lambda - \nu - k - 2nk){\Gamma _k}(k\varsigma - k\tau + \lambda - \nu - k - 2nk)}}} \hfill \cr { \times {{{\Gamma _k}(k\varsigma + k\varsigma ' - k\gamma + \lambda - \nu - k - 2nk){\Gamma _k}(k\varsigma + k\tau ' - k\gamma + \lambda - \nu - k - 2nk)} \over {{\Gamma _k}(k\varsigma + k\varsigma ' + k\tau ' - k\gamma + \lambda - \nu - k - 2nk){\Gamma _k}(nk + \nu + {{3k} \over 2}){\Gamma _k}(nk + {{3k} \over 2})}}} \hfill \cr }
This on expressing in terms of k -Wright function
p Ψ q k _p\Psi _q^k
using (1.2) leads to the right-hand side of (2.3) . This completes the proof of theorem.
The next theorem obtains the left-hand sided MSM fractional differentiation of k -Struve function.
Theorem 4
Let ς, ς ′, τ, τ ′, γ, ρ ∈ ℂ and k ∈ ℝ+ be such that
ℜ ( λ k ) > max { 0 , ℜ ( − ς + τ ) , ℜ ( − ς − ς ′ − τ ′ + γ ) } \Re ({\lambda \over k}) > \max \{ 0,\Re ( - \varsigma + \tau ),\Re ( - \varsigma - \varsigma ' - \tau ' + \gamma )\}
. Also let c ∈ ℝ; ν > −1, then for t > 0
(2.4) ![]()
( D 0 + ς , ς ′ , τ , τ ′ , γ ( t λ k − 1 S ν , c k ( t ) ) ) ( x ) = k − γ + 1 2 x ς + ς ′ − γ + λ k + ν k 2 ν k + 1 × 3 Ψ 5 k [ ( λ + ν + k , 2 k ) , ( − k τ + k ς + λ + ν + k , 2 k ) , ( − k τ + λ + ν + k , 2 k ) , ( k ς + k ς ′ − k γ + λ + ν + k , 2 k ) , ( k ς + k ς ′ + k τ − k γ + λ + ν + k , 2 k ) ( k ς + k τ ′ − k γ + λ + ν + k , 2 k ) , ( ν + 3 k 2 , k ) , ( 3 k 2 , k ) | − c x 2 k 4 ] . \matrix{ {\left( {D_{0 + }^{\varsigma ,\varsigma ',\tau ,\tau ',\gamma }\left( {{t^{{\lambda \over k} - 1}}S_{\nu ,c}^k(t)} \right)} \right)(x) = {{{k^{ - \gamma + {1 \over 2}}}{x^{\varsigma + \varsigma ' - \gamma + {\lambda \over k} + {\nu \over k}}}} \over {{2^{{\nu \over k} + 1}}}}} \hfill \cr {{ \times _3}\Psi _5^k\left[ {\matrix{ {(\lambda + \nu + k,2k),} & {( - k\tau + k\varsigma + \lambda + \nu + k,2k),} \cr {( - k\tau + \lambda + \nu + k,2k),(k\varsigma + k\varsigma ' - k\gamma + \lambda + \nu + k,2k),} \cr } } \right.} \hfill \cr {\left. {\matrix{ {(k\varsigma + k\varsigma ' + k\tau - k\gamma + \lambda + \nu + k,2k)} & {} \cr {(k\varsigma + k\tau ' - k\gamma + \lambda + \nu + k,2k),} & {(\nu + {{3k} \over 2},k),({{3k} \over 2},k)} \cr } \;\;|{{ - c{x^2}k} \over 4}} \right].} \hfill \cr }
Proof
On using (1.16) and taking the left-hand sided MSM fractional derivative inside the summation, the left-hand side of (2.4) becomes
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= ∑ n = 0 ∞ ( − c ) n Γ k ( nk + ν + 3 k 2 ) Γ ( n + 3 2 ) n ! 2 ν k + 2 n + 1 ( D 0 + ς , ς ′ , τ , τ ′ , γ ( t λ k + ν k + 2 n ) ) = \sum\limits_{n = 0}^\infty {{{{( - c)}^n}} \over {{\Gamma _k}(nk + \nu + {{3k} \over 2})\Gamma (n + {3 \over 2})n{{!2}^{{\nu \over k} + 2n + 1}}}}\left( {D_{0 + }^{\varsigma ,\varsigma ',\tau ,\tau ',\gamma }\left( {{t^{{\lambda \over k} + {\nu \over k} + 2n}}} \right)} \right)
Using (1.20) in above term, we obtain
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= ∑ n = 0 ∞ ( − c ) n Γ ( λ k + ν k + 2 n + 1 ) Γ ( − τ + ς + λ k + ν k + 2 n + 1 ) Γ k ( nk + ν + 3 k 2 ) Γ ( n + 3 2 ) n ! 2 ν k + 2 n + 1 Γ ( − τ + λ k + ν k + 2 n + 1 ) × Γ ( ς + ς ′ + τ ′ − γ + λ k + ν k + 2 n + 1 ) Γ ( ς + ς ′ − γ + λ k + ν k + 2 n + 1 ) Γ ( ς + τ ′ − γ + λ k + ν k + 2 n + 1 ) x ς + ς ′ − γ + λ k + ν k + 2 n = x ς + ς ′ − γ λ k + ν k 2 ν k + 1 ∑ n = 0 ∞ ( − c x 2 ) n n ! 4 n Γ k ( nk + ν + 3 k 2 ) Γ ( λ k + ν k + 2 n + 1 ) Γ ( n + 3 2 ) Γ ( − τ + λ k + ν k + 2 n + 1 ) × Γ ( − τ + ς + λ k + ν k + 2 n + 1 ) Γ ( ς + ς ′ + τ ′ − γ + λ k + ν k + 2 n + 1 ) Γ ( ς + ς ′ − γ + λ k + ν k + 2 n + 1 ) Γ ( ς + τ ′ − γ + λ k + ν k + 2 n + 1 ) = k − γ + 1 2 x ς + ς ′ − γ + λ k + ν k 2 ν k + 1 ∑ n = 0 ∞ ( − ck x 2 ) n n ! 4 n × Γ k ( λ + ν + k + 2 nk ) Γ k ( − k τ + k ς + λ + ν + k + 2 nk ) Γ k ( nk + ν + 3 k 2 ) Γ k ( nk + 3 k 2 ) Γ k ( − k τ + λ + ν + k + 2 nk ) × Γ k ( k ς + k ς ′ + k τ ′ − k γ + λ + ν + k + 2 nk ) Γ k ( k ς + k ς ′ − k γ + λ + ν + k + 2 nk ) Γ k ( k ς + k τ ′ − k γ + λ + ν + k + 2 nk ) \matrix{ { = \sum\limits_{n = 0}^\infty {{{{( - c)}^n}\Gamma ({\lambda \over k} + {\nu \over k} + 2n + 1)\Gamma ( - \tau + \varsigma + {\lambda \over k} + {\nu \over k} + 2n + 1)} \over {{\Gamma _k}(nk + \nu + {{3k} \over 2})\Gamma (n + {3 \over 2})n{{!2}^{{\nu \over k} + 2n + 1}}\Gamma ( - \tau + {\lambda \over k} + {\nu \over k} + 2n + 1)}}} \hfill \cr { \times {{\Gamma (\varsigma + \varsigma ' + \tau ' - \gamma + {\lambda \over k} + {\nu \over k} + 2n + 1)} \over {\Gamma (\varsigma + \varsigma ' - \gamma + {\lambda \over k} + {\nu \over k} + 2n + 1)\Gamma (\varsigma + \tau ' - \gamma + {\lambda \over k} + {\nu \over k} + 2n + 1)}}{x^{\varsigma + \varsigma ' - \gamma + {\lambda \over k} + {\nu \over k} + 2n}}} \hfill \cr { = {{{x^{\varsigma + \varsigma ' - \gamma {\lambda \over k} + {\nu \over k}}}} \over {{2^{{\nu \over k} + 1}}}}\sum\limits_{n = 0}^\infty {{{{( - c{x^2})}^n}} \over {n{{!4}^n}{\Gamma _k}(nk + \nu + {{3k} \over 2})}}{{\Gamma ({\lambda \over k} + {\nu \over k} + 2n + 1)} \over {\Gamma (n + {3 \over 2})\Gamma ( - \tau + {\lambda \over k} + {\nu \over k} + 2n + 1)}}} \hfill \cr { \times {{\Gamma ( - \tau + \varsigma + {\lambda \over k} + {\nu \over k} + 2n + 1)\Gamma (\varsigma + \varsigma ' + \tau ' - \gamma + {\lambda \over k} + {\nu \over k} + 2n + 1)} \over {\Gamma (\varsigma + \varsigma ' - \gamma + {\lambda \over k} + {\nu \over k} + 2n + 1)\Gamma (\varsigma + \tau ' - \gamma + {\lambda \over k} + {\nu \over k} + 2n + 1)}}} \hfill \cr { = {{{k^{ - \gamma + {1 \over 2}}}{x^{\varsigma + \varsigma ' - \gamma + {\lambda \over k} + {\nu \over k}}}} \over {{2^{{\nu \over k} + 1}}}}\sum\limits_{n = 0}^\infty {{{{( - ck{x^2})}^n}} \over {n{{!4}^n}}}} \hfill \cr { \times {{{\Gamma _k}(\lambda + \nu + k + 2nk){\Gamma _k}( - k\tau + k\varsigma + \lambda + \nu + k + 2nk)} \over {{\Gamma _k}(nk + \nu + {{3k} \over 2}){\Gamma _k}(nk + {{3k} \over 2}){\Gamma _k}( - k\tau + \lambda + \nu + k + 2nk)}}} \hfill \cr { \times {{{\Gamma _k}(k\varsigma + k\varsigma ' + k\tau ' - k\gamma + \lambda + \nu + k + 2nk)} \over {{\Gamma _k}(k\varsigma + k\varsigma ' - k\gamma + \lambda + \nu + k + 2nk){\Gamma _k}(k\varsigma + k\tau ' - k\gamma + \lambda + \nu + k + 2nk)}}} \hfill \cr }
In above term, we use equation (1.5) , and the result follows by using (1.2) , then we arrive at (2.4) .
The next theorem gives the right-hand sided MSM fractional derivative of k -Struve function.
Theorem 5
Let ς, ς ′, τ, τ ′, γ, ρ ∈ ℂ and k ∈ ℝ+ be such that
ℜ ( λ k ) > max { ℜ ( − τ ′ ) , ℜ ( ς ′ + τ − γ ) , ℜ ( ς + ς ′ − γ ) + [ ℜ ( γ ) ] + 1 } \Re ({\lambda \over k}) > \max \{ \Re ( - \tau '),\Re (\varsigma ' + \tau - \gamma ),\Re (\varsigma + \varsigma ' - \gamma ) + [\Re (\gamma )] + 1\}
. Also let c ∈ ℝ; ν > −1, then for t > 0
(2.5) ![]()
( D − ς , ς ′ , τ , τ ′ , γ ( t λ k − 1 S ν , c k ( t ) ) ) ( x ) = k − γ + 1 2 x ς + ς ′ − γ + λ k + ν k 2 ν k + 1 × 3 Ψ 5 k [ ( k τ ′ − λ − ν , − 2 k ) , ( − k ς − k ς ′ + k γ − λ − ν , − 2 k ) , ( − λ − ν , − 2 k ) , ( − k ς ′ + k τ ′ − λ − ν , − 2 k ) , ( − k ς ′ + k τ + k γ − λ − ν , − 2 k ) ( − k ς − k ς ′ − k τ + k γ − λ − ν , − 2 k ) , ( ν + 3 k 2 , k ) , ( 3 k 2 , k ) | − c x 2 k 4 ] . \matrix{ {\left( {D_ - ^{\varsigma ,\varsigma ',\tau ,\tau ',\gamma }\left( {{t^{{\lambda \over k} - 1}}S_{\nu ,c}^k(t)} \right)} \right)(x) = {{{k^{ - \gamma + {1 \over 2}}}{x^{\varsigma + \varsigma ' - \gamma + {\lambda \over k} + {\nu \over k}}}} \over {{2^{{\nu \over k} + 1}}}}} \hfill \cr {{ \times _3}\Psi _5^k\left[ {\matrix{ {(k\tau ' - \lambda - \nu , - 2k),} & {( - k\varsigma - k\varsigma ' + k\gamma - \lambda - \nu , - 2k),} \cr {( - \lambda - \nu , - 2k),} & {( - k\varsigma ' + k\tau ' - \lambda - \nu , - 2k),} \cr } } \right.} \hfill \cr {\left. {\matrix{ {( - k\varsigma ' + k\tau + k\gamma - \lambda - \nu , - 2k)} & {} \cr {( - k\varsigma - k\varsigma ' - k\tau + k\gamma - \lambda - \nu , - 2k),} & {(\nu + {{3k} \over 2},k),({{3k} \over 2},k)} \cr } \;\;|{{ - c{x^2}k} \over 4}} \right].} \hfill \cr }
Proof
On using (1.16) and taking the left-hand sided MSM fractional derivative inside the summation, the left-hand side of (2.5) becomes
![]()
= ∑ n = 0 ∞ ( − c ) n Γ k ( nk + ν + 3 k 2 ) Γ ( n + 3 2 ) n ! 2 ν k + 2 n + 1 ( D − ς , ς ′ , τ , τ ′ , γ ( t λ k + ν k + 2 n ) ) = \sum\limits_{n = 0}^\infty {{{{( - c)}^n}} \over {{\Gamma _k}(nk + \nu + {{3k} \over 2})\Gamma (n + {3 \over 2})n{{!2}^{{\nu \over k} + 2n + 1}}}}\left( {D_ - ^{\varsigma ,\varsigma ',\tau ,\tau ',\gamma }\left( {{t^{{\lambda \over k} + {\nu \over k} + 2n}}} \right)} \right)
Using (1.21) in above term, we obtain
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= ∑ n = 0 ∞ ( − c ) n Γ ( τ ′ − λ k − ν k − 2 n ) Γ k ( nk + ν + 3 k 2 ) Γ ( n + 3 2 ) n ! 2 ν k + 2 n + 1 Γ ( − λ k − ν k − 2 n ) × Γ ( − ς − ς ′ + γ − λ k − ν k − 2 n ) Γ ( − ς ′ − τ + γ − λ k − ν k − 2 n ) Γ ( − ς − ς ′ − τ + γ − λ k − ν k − 2 n ) Γ ( − ς ′ + τ ′ − λ k − ν k − 2 n ) x ς + ς ′ − γ + λ k + ν k + 2 n = x ς + ς ′ − γ λ k + ν k 2 ν k + 1 ∑ n = 0 ∞ ( − c x 2 ) n n ! 4 n Γ k ( nk + ν + 3 k 2 ) Γ ( τ ′ − λ k − ν k − 2 n ) Γ ( − λ k − ν k − 2 n ) × Γ ( − ς − ς ′ + γ − λ k − ν k − 2 n ) Γ ( − ς ′ − τ + γ − λ k − ν k − 2 n ) Γ ( − ς ′ + τ ′ − λ k − ν k − 2 n ) Γ ( − ς − ς ′ − τ + γ − λ k − ν k − 2 n ) = k − γ + 1 2 x ς + ς ′ − γ + λ k + ν k 2 ν k + 1 ∑ n = 0 ∞ ( − ck x 2 ) n n ! 4 n × Γ k ( k τ ′ − λ − ν − 2 nk ) Γ k ( − k ς − k ς ′ + k γ − λ − ν − 2 nk ) Γ k ( − λ − ν − 2 nk ) Γ k ( − k ς ′ + k τ ′ − λ − ν − 2 nk ) × Γ k ( − k ς ′ − k τ + k γ − λ − ν − 2 nk ) Γ k ( − k ς − k ς ′ − k τ + k γ − λ − ν − 2 nk ) t Γ k ( ν + 3 k 2 + nk ) Γ k ( 3 k 2 + nk ) \matrix{ { = \sum\limits_{n = 0}^\infty {{{{( - c)}^n}\Gamma (\tau ' - {\lambda \over k} - {\nu \over k} - 2n)} \over {{\Gamma _k}(nk + \nu + {{3k} \over 2})\Gamma (n + {3 \over 2})n{{!2}^{{\nu \over k} + 2n + 1}}\Gamma ( - {\lambda \over k} - {\nu \over k} - 2n)}}} \hfill \cr { \times {{\Gamma ( - \varsigma - \varsigma ' + \gamma - {\lambda \over k} - {\nu \over k} - 2n)\Gamma ( - \varsigma ' - \tau + \gamma - {\lambda \over k} - {\nu \over k} - 2n)} \over {\Gamma ( - \varsigma - \varsigma ' - \tau + \gamma - {\lambda \over k} - {\nu \over k} - 2n)\Gamma ( - \varsigma ' + \tau ' - {\lambda \over k} - {\nu \over k} - 2n)}}{x^{\varsigma + \varsigma ' - \gamma + {\lambda \over k} + {\nu \over k} + 2n}}} \hfill \cr { = {{{x^{\varsigma + \varsigma ' - \gamma {\lambda \over k} + {\nu \over k}}}} \over {{2^{{\nu \over k} + 1}}}}\sum\limits_{n = 0}^\infty {{{{( - c{x^2})}^n}} \over {n{{!4}^n}{\Gamma _k}(nk + \nu + {{3k} \over 2})}}{{\Gamma (\tau ' - {\lambda \over k} - {\nu \over k} - 2n)} \over {\Gamma ( - {\lambda \over k} - {\nu \over k} - 2n)}}} \hfill \cr { \times {{\Gamma ( - \varsigma - \varsigma ' + \gamma - {\lambda \over k} - {\nu \over k} - 2n)\Gamma ( - \varsigma ' - \tau + \gamma - {\lambda \over k} - {\nu \over k} - 2n)} \over {\Gamma ( - \varsigma ' + \tau ' - {\lambda \over k} - {\nu \over k} - 2n)\Gamma ( - \varsigma - \varsigma ' - \tau + \gamma - {\lambda \over k} - {\nu \over k} - 2n)}}} \hfill \cr { = {{{k^{ - \gamma + {1 \over 2}}}{x^{\varsigma + \varsigma ' - \gamma + {\lambda \over k} + {\nu \over k}}}} \over {{2^{{\nu \over k} + 1}}}}\sum\limits_{n = 0}^\infty {{{{( - ck{x^2})}^n}} \over {n{{!4}^n}}}} \hfill \cr { \times {{{\Gamma _k}(k\tau ' - \lambda - \nu - 2nk){\Gamma _k}( - k\varsigma - k\varsigma ' + k\gamma - \lambda - \nu - 2nk)} \over {{\Gamma _k}( - \lambda - \nu - 2nk){\Gamma _k}( - k\varsigma ' + k\tau ' - \lambda - \nu - 2nk)}}} \hfill \cr { \times {{{\Gamma _k}( - k\varsigma ' - k\tau + k\gamma - \lambda - \nu - 2nk)} \over {{\Gamma _k}( - k\varsigma - k\varsigma ' - k\tau + k\gamma - \lambda - \nu - 2nk)t{\Gamma _k}(\nu + {{3k} \over 2} + nk){\Gamma _k}({{3k} \over 2} + nk)}}} \hfill \cr }
Thus, in accordance with (1.2) , we get the required result (2.5) .
Theorem 6
Let ς, ς ′, τ, τ ′, γ, ρ ∈ ℂ and k ∈ ℝ+ be such that
ℜ ( λ k ) > max { ℜ ( − τ ′ ) , ℜ ( ς ′ + τ − γ ) , ℜ ( ς + ς ′ − γ ) + [ ℜ ( γ ) ] + 1 } \Re ({\lambda \over k}) > \max \{ \Re ( - \tau '),\Re (\varsigma ' + \tau - \gamma ),\Re (\varsigma + \varsigma ' - \gamma ) + [\Re (\gamma )] + 1\}
. Also let c ∈ ℝ; ν > −1, then for t > 0
(2.6) ![]()
( D − ς , ς ′ , τ , τ ′ , γ ( t − λ k S ν , c k ( t ) ) ) ( x ) = k − γ + 1 2 x ς + ς ′ − γ + λ k + ν k + 1 2 ν k + 1 × 3 Ψ 5 k [ ( k τ ′ + λ − ν − k , − 2 k ) , ( − k ς − k ς ′ + k γ − λ − ν − k , − 2 k ) , ( − λ − ν − k , − 2 k ) , ( − k ς ′ + k τ ′ + λ − ν − k , − 2 k ) , ( − k ς ′ − k τ + k γ + λ − ν − k , − 2 k ) ( − k ς − k ς ′ − k τ + k γ + λ − ν − k , − 2 k ) , ( ν + 3 k 2 , k ) , ( 3 k 2 , k ) | − c x 2 k 4 ] . \matrix{ {\left( {D_ - ^{\varsigma ,\varsigma ',\tau ,\tau ',\gamma }\left( {{t^{ - {\lambda \over k}}}S_{\nu ,c}^k(t)} \right)} \right)(x) = {{{k^{ - \gamma + {1 \over 2}}}{x^{\varsigma + \varsigma ' - \gamma + {\lambda \over k} + {\nu \over k} + 1}}} \over {{2^{{\nu \over k} + 1}}}}} \hfill \cr {{ \times _3}\Psi _5^k\left[ {\matrix{ {(k\tau ' + \lambda - \nu - k, - 2k),} & {( - k\varsigma - k\varsigma ' + k\gamma - \lambda - \nu - k, - 2k),} \cr {( - \lambda - \nu - k, - 2k),} & {( - k\varsigma ' + k\tau ' + \lambda - \nu - k, - 2k),} \cr } } \right.} \hfill \cr {\left. {\matrix{ {( - k\varsigma ' - k\tau + k\gamma + \lambda - \nu - k, - 2k)} & {} \cr {( - k\varsigma - k\varsigma ' - k\tau + k\gamma + \lambda - \nu - k, - 2k),} & {(\nu + {{3k} \over 2},k),({{3k} \over 2},k)} \cr } \;\;|{{ - c{x^2}k} \over 4}} \right].} \hfill \cr }
Proof
On using (1.16) and taking the right-hand sided MSM fractional derivative inside the summation, the left-hand side of (2.6) becomes
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= ∑ n = 0 ∞ ( − c ) n Γ k ( nk + ν + 3 k 2 ) Γ ( n + 3 2 ) n ! 2 ν k + 2 n + 1 ( D − ς , ς ′ , τ , τ ′ , γ ( t ν k − λ k + 2 n + 1 ) ) = \sum\limits_{n = 0}^\infty {{{{( - c)}^n}} \over {{\Gamma _k}(nk + \nu + {{3k} \over 2})\Gamma (n + {3 \over 2})n{{!2}^{{\nu \over k} + 2n + 1}}}}\left( {D_ - ^{\varsigma ,\varsigma ',\tau ,\tau ',\gamma }\left( {{t^{{\nu \over k} - {\lambda \over k} + 2n + 1}}} \right)} \right)
Using (1.21) , we have
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= ∑ n = 0 ∞ ( − c ) n Γ k ( nk + ν + 3 k 2 ) Γ ( n + 3 2 ) n ! 2 ν k + 2 n + 1 Γ ( τ ′ + λ − ν k − 2 n − 1 ) Γ ( λ − ν k − 2 n − 1 ) × Γ ( − ς − ς ′ + γ + λ − ν k − 2 n − 1 ) Γ ( − ς ′ − τ + γ + λ − ν k − 2 n − 1 ) Γ ( − ς ′ + τ ′ + λ − ν k − 2 n − 1 ) Γ ( − ς − ς ′ − τ + γ + λ − ν k − 2 n − 1 ) x ς + ς ′ − γ + ν k − λ k + 2 n + 1 \matrix{ { = \sum\limits_{n = 0}^\infty {{{{( - c)}^n}} \over {{\Gamma _k}(nk + \nu + {{3k} \over 2})\Gamma (n + {3 \over 2})n{{!2}^{{\nu \over k} + 2n + 1}}}}{{\Gamma (\tau ' + {{\lambda - \nu } \over k} - 2n - 1)} \over {\Gamma ({{\lambda - \nu } \over k} - 2n - 1)}}} \hfill \cr { \times {{\Gamma ( - \varsigma - \varsigma ' + \gamma + {{\lambda - \nu } \over k} - 2n - 1)\Gamma ( - \varsigma ' - \tau + \gamma + {{\lambda - \nu } \over k} - 2n - 1)} \over {\Gamma ( - \varsigma ' + \tau ' + {{\lambda - \nu } \over k} - 2n - 1)\Gamma ( - \varsigma - \varsigma ' - \tau + \gamma + {{\lambda - \nu } \over k} - 2n - 1)}}{x^{\varsigma + \varsigma ' - \gamma + {\nu \over k} - {\lambda \over k} + 2n + 1}}} \hfill \cr }
Making use of (1.5) , we obtain
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= t ς + ς ′ − γ + ν k − λ k + 1 2 ν k + 1 ∑ n = 0 ∞ ( − ck t 2 ) n n ! 4 n k γ − 1 2 × Γ k ( k τ ′ + λ − ν − k − 2 nk ) Γ k ( λ − ν − k − 2 nk ) Γ k ( − k ς ′ + k τ ′ + λ − ν − k − 2 nk ) × Γ k ( − k ς − k ς ′ + k γ + λ − ν − k − 2 nk ) Γ k ( − k ς ′ − k τ + k γ + λ − ν − k − 2 nk ) Γ k ( − k ς − k ς ′ − k τ + k γ + λ − ν − k − 2 nk ) Γ k ( nk + ν + 3 k 2 ) Γ k ( nk + 3 k 2 ) \matrix{ { = {{{t^{\varsigma + \varsigma ' - \gamma + {\nu \over k} - {\lambda \over k} + 1}}} \over {{2^{{\nu \over k} + 1}}}}\sum\limits_{n = 0}^\infty {{{{( - ck{t^2})}^n}} \over {n{{!4}^n}{k^{\gamma - {1 \over 2}}}}}} \hfill \cr { \times {{{\Gamma _k}(k\tau ' + \lambda - \nu - k - 2nk)} \over {{\Gamma _k}(\lambda - \nu - k - 2nk){\Gamma _k}( - k\varsigma ' + k\tau ' + \lambda - \nu - k - 2nk)}}} \hfill \cr { \times {{{\Gamma _k}( - k\varsigma - k\varsigma ' + k\gamma + \lambda - \nu - k - 2nk){\Gamma _k}( - k\varsigma ' - k\tau + k\gamma + \lambda - \nu - k - 2nk)} \over {{\Gamma _k}( - k\varsigma - k\varsigma ' - k\tau + k\gamma + \lambda - \nu - k - 2nk){\Gamma _k}(nk + \nu + {{3k} \over 2}){\Gamma _k}(nk + {{3k} \over 2})}}} \hfill \cr }
This on expressing in terms of k -Wright function
p Ψ q k _p\Psi _q^k
using (1.2) leads to the right-hand side of (2.6) . This completes the proof.