The Marichev-Saigo-Maeda Fractional Calculus Operators Pertaining to the Generalized -Struve Function

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) $_{p} Ψ_{q} (z) =_{p} Ψ_{q} [\begin{matrix} {(a_{i}, ς_{i})}_{1, p} \\ {(b_{j}, τ_{j})}_{1, q} \end{matrix} | z] = \sum_{n = 0}^{\infty} \frac{\prod_{i = 1}^{p} Γ (a_{i} + n ς_{i})}{\prod_{j = 1}^{q} Γ (b_{j} + n τ_{j})} \frac{z^{n}}{n!}, z \in ℂ,$ _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) $_{p} Ψ_{q}^{k} (z) =_{p} Ψ_{q}^{k} [\begin{matrix} {(a_{i}, ς_{i})}_{1, p} \\ {(b_{j}, τ_{j})}_{1, q} \end{matrix} | z] = \sum_{n = 0}^{\infty} \frac{\prod_{i = 1}^{p} Γ_{k} (a_{i} + n ς_{i})}{\prod_{j = 1}^{q} Γ_{k} (b_{j} + n τ_{j})} \frac{z^{n}}{n!}, z \in ℂ,$ _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\kℤ⁻ for all n ∈ ℕ₀. The generalized k-gamma function [3] is defined as (1.3) $Γ_{k} (z) = \int_{0}^{\infty} e^{- \frac{t^{k}}{k}} t^{z - 1} dt; (ℜ (z) > 0; k \in ℝ^{+})$ {\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) $Γ_{k} (z) = lim_{n \to \infty} \frac{n! k^{n} {(nk)}^{\frac{z}{k} - 1}}{{(z)}_{n, k}}, k \in ℝ^{+}, z \in ℂ \ k ℤ^{-}$ {\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) $Γ_{k} (z) = k^{\frac{z}{k} - 1} Γ (\frac{z}{k}),$ {\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) ${(z)}_{n, k} = {\begin{matrix} 1 & if n = 0, \\ z (z + k) (z + 2 k) \dots (z + (n - 1) k) & if n \in ℕ . \end{matrix}}$ {(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) $(I_{0 +}^{ς, τ, γ} f) (x) = \frac{x^{- ς - τ}}{Γ (ς)} \int_{0}^{x} {(x - t)}^{ς - 1}_{2} F_{1} (ς + τ, - γ; ς; 1 - \frac{t}{x}) f (t) d t$ (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) $(I_{-}^{ς, τ, γ} f) (x) = \frac{1}{Γ (ς)} \int_{x}^{\infty} {\frac{{(t - x)}^{ς - 1}}{t^{ς + τ}}}_{2} F_{1} (ς + τ, - γ; ς; 1 - \frac{x}{t}) f (t) dt$ (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, ₂F₁(ς, τ; γ; z) is the Gauss hypergeometric function [11] defined for z ∈ ℂ, |z| < 1 and ς, τ ∈ ℂ, $γ \in ℂ \ ℤ_{0}^{-}$ \gamma \in \mathbb{C} \backslash \mathbb{Z}_0^ - by $_{2} F_{1} (ς, τ; γ; z) = \sum_{n = 0}^{\infty} \frac{{(ς)}_{n} {(τ)}_{n}}{{(γ)}_{n}} \frac{z^{n}}{n!},$ _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) $(D_{0 +}^{ς, τ, γ} f) (x) = {(\frac{d}{dx})}^{l} (I_{0 +}^{- ς + l, - τ - l, ς + γ - l} f) (x)$ (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) $(D_{-}^{ς, τ, γ} f) (x) = {(- \frac{d}{dx})}^{l} (I_{-}^{- ς + l, - τ - l, ς + γ} f) (x)$ (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 F₃ are defined as (1.11) $(I_{0 +}^{ς, ς^{'}, τ, τ^{'}, γ} f) (x) = \frac{x^{- ς}}{Γ (γ)} \int_{0}^{x} \frac{{(x - t)}^{γ - 1}}{t^{ς^{'}}} F_{3} (ς, ς^{'}, τ, τ, γ, 1 - \frac{t}{x}, 1 - \frac{x}{t}) f (t) dt$ (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) $(I_{-}^{ς, ς^{'}, τ, τ^{'}, γ} f) (x) = \frac{x^{- ς^{'}}}{Γ (γ)} \int_{x}^{\infty} \frac{{(t - x)}^{γ - 1}}{t^{ς}} F_{3} (ς, ς^{'}, τ, τ, γ, 1 - \frac{x}{t}, 1 - \frac{t}{x}) f (t) dt$ (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) $(D_{0 +}^{ς, ς^{'}, τ, τ^{'}, γ} f) (x) = {(\frac{d}{dx})}^{m} (I_{0 +}^{- ς^{'}, - ς, - τ^{'} + m, - τ, - γ + m} f) (x)$ (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) $(D_{-}^{ς, ς^{'}, τ, τ^{'}, γ} f) (x) = {(- \frac{d}{dx})}^{m} (I_{-}^{- ς^{'}, - ς, - τ^{'}, - τ + m, - γ + m} f) (x)$ (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) $F_{3} (ς, ς^{'}, τ, τ^{'}, γ; x, y) = \sum_{m, n, = 0}^{\infty} \frac{{(ς)}_{m} {(ς^{'})}_{n} {(τ)}_{m} {(τ^{'})}_{n}}{{(γ)}_{m + n}} \frac{x^{m} y^{n}}{m! n!}, max {| x |, | y |} < 1 .$ {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) $\begin{matrix} S_{ν, c}^{k} (t) = \sum_{n = 0}^{\infty} \frac{{(- c)}^{n}}{Γ_{k} (nk + ν + \frac{3 k}{2}) Γ (n + \frac{3}{2}) n!} {(\frac{t}{2})}^{2 n + \frac{ν}{k} + 1} \\ (k \in ℝ^{+}; c \in ℝ; ν > - 1) \end{matrix}$ \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) $H_{ν} (t) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{Γ (n + ν + \frac{3}{2}) Γ (n + \frac{3}{2}) n!} {(\frac{t}{2})}^{2 n + ν + 1}$ {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) $(I_{0 +}^{ς, ς^{'}, τ, τ^{'}, γ} t^{ρ - 1}) (x) = \frac{Γ (ρ) Γ (- ς^{'} + τ^{'} + ρ) Γ (- ς - ς^{'} - τ + γ + ρ)}{Γ (τ^{'} + ρ) Γ (- ς - ς^{'} + γ + ρ) Γ (- ς^{'} - τ + γ + ρ)} x^{- ς - ς^{'} + γ + ρ - 1}$ (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) $(I_{-}^{ς, ς^{'}, τ, τ^{'}, γ} t^{- ρ}) (x) = \frac{Γ (- τ + ρ) Γ (ς + ς^{'} - γ + ρ) Γ (ς + τ^{'} - γ + ρ)}{Γ (ρ) Γ (ς - τ + ρ) Γ (ς + ς^{'} + τ^{'} - γ + ρ)} x^{- ς - ς^{'} + γ - ρ}$ (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) $(D_{0 +}^{ς, ς^{'}, τ, τ^{'}, γ} t^{ρ - 1}) (x) = \frac{Γ (ρ) Γ (- τ + ς + ρ) Γ (ς + ς^{'} + τ^{'} - γ + ρ)}{Γ (- τ + ρ) Γ (ς + ς^{'} - γ + ρ) Γ (ς + τ^{'} - γ + ρ)} x^{ς + ς^{'} - γ + ρ - 1}$ (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) $(D_{-}^{ς, ς^{'}, τ, τ^{'}, γ} t^{- ρ}) (x) = \frac{Γ (τ^{'} + ρ) Γ (- ς - ς^{'} + γ + ρ) Γ (- ς^{'} - τ + γ + ρ)}{Γ (ρ) Γ (- ς^{'} + τ^{'} + ρ) Γ (- ς - ς^{'} - τ + γ + ρ)} x^{ς + ς^{'} - γ - ρ}$ (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 }}

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 $ℜ (γ) > 0, ℜ (\frac{λ}{k}) > max {0, ℜ (ς^{'} - τ^{'}), ℜ (ς + ς^{'} + τ - γ)}$ \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) $\begin{array}{l} (I_{0 +}^{ς, ς^{'}, τ, τ^{'}, γ} (t^{\frac{λ}{k} - 1} S_{ν, c}^{k} (t))) (x) = \frac{k^{γ + \frac{1}{2}} x^{- ς - ς^{'} + γ + \frac{λ}{k} + \frac{ν}{k}}}{2^{\frac{ν}{k} + 1}} \\ \times_{3} Ψ_{5}^{k} [\begin{matrix} (λ + ν + k, 2 k), & (- k ς^{'} + k τ^{'} + λ + ν + k, 2 k), \\ (k τ^{'} + λ + ν + k, 2 k), & (- k ς - k ς^{'} + k γ + λ + ν + k, 2 k), \end{matrix} \\ \begin{matrix} (- k ς - k ς^{'} - k τ + k γ + λ + ν + k, 2 k) \\ (- k ς^{'} - k τ + k γ + λ + ν + k, 2 k), & (ν + \frac{3 k}{2}, k), (\frac{3 k}{2}, k) \end{matrix} | \frac{- c x^{2} k}{4}] . \end{array}$ \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 $= \sum_{n = 0}^{\infty} \frac{{(- c)}^{n}}{Γ_{k} (nk + ν + \frac{3 k}{2}) Γ (n + \frac{3}{2}) n {! 2}^{2 n + \frac{ν}{k} + 1}} (I_{0 +}^{ς, ς^{'}, τ, τ^{'}, γ} {t^{\frac{λ}{k} + \frac{ν}{k} + 2 n}}) (x),$ = \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 $\begin{array}{l} = \sum_{n = 0}^{\infty} \frac{{(- c)}^{n} x^{- ς - ς^{'} + γ \frac{λ}{k} + \frac{ν}{k} + 2 n}}{Γ_{k} (nk + ν + \frac{3 k}{2}) Γ (n + \frac{3}{2}) n {! 2}^{2 n + \frac{ν}{k} + 1}} \frac{Γ (\frac{λ}{k} + \frac{ν}{k} + 2 n + 1)}{Γ (τ^{'} + \frac{λ}{k} + \frac{ν}{k} + 2 n + 1)} \\ \times \frac{Γ (- ς - ς^{'} - τ + γ + \frac{λ}{k} + \frac{ν}{k} + 2 n + 1) Γ (- ς^{'} + τ^{'} + \frac{λ}{k} + \frac{ν}{k} + 2 n + 1)}{Γ (- ς^{'} - τ + γ + \frac{λ}{k} + \frac{ν}{k} + 2 n + 1) Γ (- ς - ς^{'} + γ + \frac{λ}{k} \frac{ν}{k} + 2 n + 1)}, \end{array}$ \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 $\begin{array}{l} = \frac{x^{- ς - ς^{'} + γ + \frac{λ}{k} + \frac{ν}{k}}}{2^{\frac{ν}{k} + 1} k^{- γ - \frac{1}{2}}} \sum_{n = 0}^{\infty} \frac{Γ_{k} (λ + ν + k + 2 nk) Γ_{k} (- k ς^{'} + k τ^{'} + λ + ν + k + 2 nk)}{Γ_{k} (k τ^{'} + λ + ν + k + 2 nk) Γ_{k} (- k ς - k ς^{'} + k γ + λ + ν + k + 2 nk)} \\ \times \frac{Γ_{k} (- k ς - k ς^{'} - k τ + k γ + λ + ν + k + 2 nk)}{Γ_{k} (- k ς^{'} - k τ + k γ + λ + ν + k + 2 nk) Γ_{k} (nk + ν + \frac{3 k}{2}) Γ_{k} (\frac{3 k}{2} + nk) n!} {(\frac{- c x^{2} k}{2^{2}})}^{n} . \end{array}$ \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, $ℜ (γ) > 0, ℜ (\frac{λ}{k}) > max {ℜ (τ), ℜ (- ς - ς^{'} + γ), ℜ (- ς - τ^{'} + γ)}$ \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) $\begin{array}{l} (I_{-}^{ς, ς^{'}, τ, τ^{'}, γ} (t^{\frac{λ}{k} - 1} S_{ν, c}^{k} (t))) (x) = \frac{k^{γ + \frac{1}{2}} x^{- ς - ς^{'} + γ + \frac{λ}{k} + \frac{ν}{k}}}{2^{\frac{ν}{k} + 1}} \\ \times_{3} Ψ_{5}^{k} [\begin{matrix} (- k τ - λ - ν, - 2 k), & (k ς + k ς^{'} - k γ - nu, - 2 k), \\ (- λ - ν, - 2 k), & (k ς - k τ - λ - ν, - 2 k), \end{matrix} \\ \begin{matrix} (k ς + k τ^{'} - k γ - λ - ν, - 2 k) \\ (k ς + k ς^{'} + k τ^{'} - k γ - λ - ν, - 2 k), & (ν + \frac{3 k}{2}, k), (\frac{3 k}{2}, k) \end{matrix} | \frac{- c x^{2} k}{4}] . \end{array}$ \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 $= \sum_{n = 0}^{\infty} \frac{{(- c)}^{n}}{Γ_{k} (nk + ν + \frac{3 k}{2}) Γ (n + \frac{3}{2}) n {! 2}^{2 n + \frac{ν}{k} + 1}} (I_{-}^{ς, ς^{'}, τ, τ^{'}, γ} {t^{\frac{λ}{k} + \frac{ν}{k} + 2 n}}) (x)$ = \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 $\begin{array}{l} = \sum_{n = 0}^{\infty} \frac{{(- c)}^{n} x^{- ς - ς^{'} + γ + \frac{λ}{k} + \frac{ν}{k} + 2 n}}{Γ_{k} (nk + ν + \frac{3 k}{2}) Γ (n + \frac{3}{2}) n {! 2}^{2 n + \frac{ν}{k}} + 1} \frac{Γ (- τ - \frac{λ}{k} - \frac{ν}{k} - 2 n)}{Γ (- \frac{λ}{k} - \frac{ν}{k} - 2 n)} \\ \times \frac{Γ (ς + τ^{'} - γ - \frac{λ}{k} - \frac{ν}{k} - 2 n) Γ (ς + ς^{'} - γ - \frac{λ}{k} - \frac{ν}{k} - 2 n)}{Γ (ς + ς^{'} + τ^{'} - γ - \frac{λ}{k} - \frac{ν}{k} - 2 n) Γ (ς - τ - \frac{λ}{k} - \frac{ν}{k} - 2 n)} \\ = \sum_{n = 0}^{\infty} \frac{{(- c x^{2})}^{n} x^{- ς - ς^{'} + γ + \frac{λ}{k} + \frac{ν}{k}}}{Γ_{k} (nk + ν + \frac{3 k}{2}) n {! 2}^{2 n + \frac{ν}{k} + 1}} \frac{Γ (- τ - \frac{λ}{k} - \frac{ν}{k} - 2 n) Γ (ς + ς^{'} - γ - \frac{λ}{k} - \frac{ν}{k} - 2 n)}{Γ (n + \frac{3}{2}) Γ (- \frac{λ}{k} - \frac{ν}{k} - 2 n) Γ (ς - τ - \frac{λ}{k} - \frac{ν}{k} - 2 n)} \\ \times \frac{Γ (ς + τ^{'} - γ - \frac{λ}{k} - \frac{ν}{k} - 2 n)}{Γ (ς + ς^{'} + τ^{'} - γ - \frac{λ}{k} - \frac{ν}{k} - 2 n)} \\ = \frac{x^{- ς - ς^{'} + γ + \frac{λ}{k} + \frac{ν}{k}}}{2^{\frac{ν}{k} + 1} k^{- γ - \frac{1}{2}}} \sum_{n = 0}^{\infty} {(\frac{- ck x^{2}}{4})}^{n} \frac{1}{n!} \frac{Γ_{k} (- k τ - λ - ν - 2 nk)}{Γ_{k} (- λ - ν - 2 nk) Γ_{k} (k ς - k τ - λ - ν - 2 nk)} \\ \times \frac{Γ_{k} (k ς + k ς^{'} - k γ - λ - ν - 2 nk) Γ_{k} (k ς + k τ^{'} - k γ - λ - ν - 2 nk)}{Γ_{k} (k ς + k ς^{'} + k τ^{'} - k γ - λ - ν - 2 nk) Γ_{k} (nk + ν + \frac{3 k}{2}) Γ_{k} (\frac{3 k}{2} + nk)} \end{array}$ \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, $ℜ (γ) > 0, ℜ (\frac{λ}{k}) > max {ℜ (τ), ℜ (- ς - ς^{'} + γ), ℜ (- ς - τ^{'} + γ)}$ \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) $\begin{array}{l} (I_{-}^{ς, ς^{'}, τ, τ^{'}, γ} (t^{\frac{- λ}{k}} S_{ν, c}^{k} (t))) (x) = \frac{k^{γ - \frac{1}{2}} x^{- ς - ς^{'} + γ + \frac{ν}{k} - \frac{λ}{k} + 1}}{2^{\frac{ν}{k} + 1}} \\ \times_{3} Ψ_{5}^{k} [\begin{matrix} (- k τ + λ - ν, - 2 k), & (k ς + k ς^{'} - k γ + λ - nu - k, - 2 k), \\ (- λ - ν - k, - 2 k), & (k ς - k τ + λ - ν - k, - 2 k), \end{matrix} \\ \begin{matrix} (k ς + k τ^{'} - k γ + λ - ν - k, - 2 k) \\ (k ς + k ς^{'} + k τ^{'} - k γ + λ - ν - k, - 2 k), & (ν + \frac{3 k}{2}, k), (\frac{3 k}{2}, k) \end{matrix} | \frac{- c x^{2} k}{4}] . \end{array}$ \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 $= \sum_{n = 0}^{\infty} \frac{{(- c)}^{n}}{Γ_{k} (nk + ν + \frac{3 k}{2}) Γ (n + \frac{3}{2}) n {! 2}^{2 n + \frac{ν}{k} + 1}} (I_{-}^{ς, ς^{'}, τ, τ^{'}, γ} {t^{\frac{ν - λ}{k} + 2 n + 1}})$ = \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 $\begin{array}{l} = \frac{{(- c)}^{n} t^{- ς - ς^{'} + γ + \frac{ν}{k} - \frac{λ}{k} + 2 n + 1}}{Γ_{k} (nk + ν + \frac{3 k}{2}) Γ (n + \frac{3}{2}) n {! 2}^{2 n + \frac{ν}{k} + 1}} \frac{Γ (- τ + \frac{λ - ν}{k} - 2 n - 1)}{Γ (\frac{λ - nu}{k} - 2 n - 1)} \\ \times \frac{Γ (ς + ς^{'} - γ + \frac{λ - ν}{k} - 2 n - 1) Γ (ς + τ^{'} - γ + \frac{λ - ν}{k} - 2 n - 1)}{γ (ς - τ + \frac{λ - ν}{k} - 2 n - 1) Γ (ς + ς^{'} + τ^{'} - γ + \frac{λ - ν}{k} - 2 n - 1)} \end{array}$ \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 $\begin{array}{l} = \frac{x^{- ς - ς^{'} + γ + \frac{ν}{k} - \frac{λ}{k} + 1}}{k^{- γ + \frac{1}{2}} 2^{\frac{ν}{k} + 1}} \sum_{n = 0}^{\infty} \frac{{(- ck x^{2})}^{n}}{4^{n} n!} \frac{Γ_{k} (- k τ + λ - ν - k - 2 nk)}{Γ_{k} (λ - ν - k - 2 nk) Γ_{k} (k ς - k τ + λ - ν - k - 2 nk)} \\ \times \frac{Γ_{k} (k ς + k ς^{'} - k γ + λ - ν - k - 2 nk) Γ_{k} (k ς + k τ^{'} - k γ + λ - ν - k - 2 nk)}{Γ_{k} (k ς + k ς^{'} + k τ^{'} - k γ + λ - ν - k - 2 nk) Γ_{k} (nk + ν + \frac{3 k}{2}) Γ_{k} (nk + \frac{3 k}{2})} \end{array}$ \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 $ℜ (\frac{λ}{k}) > max {0, ℜ (- ς + τ), ℜ (- ς - ς^{'} - τ^{'} + γ)}$ \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) $\begin{array}{l} (D_{0 +}^{ς, ς^{'}, τ, τ^{'}, γ} (t^{\frac{λ}{k} - 1} S_{ν, c}^{k} (t))) (x) = \frac{k^{- γ + \frac{1}{2}} x^{ς + ς^{'} - γ + \frac{λ}{k} + \frac{ν}{k}}}{2^{\frac{ν}{k} + 1}} \\ \times_{3} Ψ_{5}^{k} [\begin{matrix} (λ + ν + k, 2 k), & (- k τ + k ς + λ + ν + k, 2 k), \\ (- k τ + λ + ν + k, 2 k), (k ς + k ς^{'} - k γ + λ + ν + k, 2 k), \end{matrix} \\ \begin{matrix} (k ς + k ς^{'} + k τ - k γ + λ + ν + k, 2 k) \\ (k ς + k τ^{'} - k γ + λ + ν + k, 2 k), & (ν + \frac{3 k}{2}, k), (\frac{3 k}{2}, k) \end{matrix} | \frac{- c x^{2} k}{4}] . \end{array}$ \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 $= \sum_{n = 0}^{\infty} \frac{{(- c)}^{n}}{Γ_{k} (nk + ν + \frac{3 k}{2}) Γ (n + \frac{3}{2}) n {! 2}^{\frac{ν}{k} + 2 n + 1}} (D_{0 +}^{ς, ς^{'}, τ, τ^{'}, γ} (t^{\frac{λ}{k} + \frac{ν}{k} + 2 n}))$ = \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 $\begin{array}{l} = \sum_{n = 0}^{\infty} \frac{{(- c)}^{n} Γ (\frac{λ}{k} + \frac{ν}{k} + 2 n + 1) Γ (- τ + ς + \frac{λ}{k} + \frac{ν}{k} + 2 n + 1)}{Γ_{k} (nk + ν + \frac{3 k}{2}) Γ (n + \frac{3}{2}) n {! 2}^{\frac{ν}{k} + 2 n + 1} Γ (- τ + \frac{λ}{k} + \frac{ν}{k} + 2 n + 1)} \\ \times \frac{Γ (ς + ς^{'} + τ^{'} - γ + \frac{λ}{k} + \frac{ν}{k} + 2 n + 1)}{Γ (ς + ς^{'} - γ + \frac{λ}{k} + \frac{ν}{k} + 2 n + 1) Γ (ς + τ^{'} - γ + \frac{λ}{k} + \frac{ν}{k} + 2 n + 1)} x^{ς + ς^{'} - γ + \frac{λ}{k} + \frac{ν}{k} + 2 n} \\ = \frac{x^{ς + ς^{'} - γ \frac{λ}{k} + \frac{ν}{k}}}{2^{\frac{ν}{k} + 1}} \sum_{n = 0}^{\infty} \frac{{(- c x^{2})}^{n}}{n {! 4}^{n} Γ_{k} (nk + ν + \frac{3 k}{2})} \frac{Γ (\frac{λ}{k} + \frac{ν}{k} + 2 n + 1)}{Γ (n + \frac{3}{2}) Γ (- τ + \frac{λ}{k} + \frac{ν}{k} + 2 n + 1)} \\ \times \frac{Γ (- τ + ς + \frac{λ}{k} + \frac{ν}{k} + 2 n + 1) Γ (ς + ς^{'} + τ^{'} - γ + \frac{λ}{k} + \frac{ν}{k} + 2 n + 1)}{Γ (ς + ς^{'} - γ + \frac{λ}{k} + \frac{ν}{k} + 2 n + 1) Γ (ς + τ^{'} - γ + \frac{λ}{k} + \frac{ν}{k} + 2 n + 1)} \\ = \frac{k^{- γ + \frac{1}{2}} x^{ς + ς^{'} - γ + \frac{λ}{k} + \frac{ν}{k}}}{2^{\frac{ν}{k} + 1}} \sum_{n = 0}^{\infty} \frac{{(- ck x^{2})}^{n}}{n {! 4}^{n}} \\ \times \frac{Γ_{k} (λ + ν + k + 2 nk) Γ_{k} (- k τ + k ς + λ + ν + k + 2 nk)}{Γ_{k} (nk + ν + \frac{3 k}{2}) Γ_{k} (nk + \frac{3 k}{2}) Γ_{k} (- k τ + λ + ν + k + 2 nk)} \\ \times \frac{Γ_{k} (k ς + k ς^{'} + k τ^{'} - k γ + λ + ν + k + 2 nk)}{Γ_{k} (k ς + k ς^{'} - k γ + λ + ν + k + 2 nk) Γ_{k} (k ς + k τ^{'} - k γ + λ + ν + k + 2 nk)} \end{array}$ \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 $ℜ (\frac{λ}{k}) > max {ℜ (- τ^{'}), ℜ (ς^{'} + τ - γ), ℜ (ς + ς^{'} - γ) + [ℜ (γ)] + 1}$ \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) $\begin{array}{l} (D_{-}^{ς, ς^{'}, τ, τ^{'}, γ} (t^{\frac{λ}{k} - 1} S_{ν, c}^{k} (t))) (x) = \frac{k^{- γ + \frac{1}{2}} x^{ς + ς^{'} - γ + \frac{λ}{k} + \frac{ν}{k}}}{2^{\frac{ν}{k} + 1}} \\ \times_{3} Ψ_{5}^{k} [\begin{matrix} (k τ^{'} - λ - ν, - 2 k), & (- k ς - k ς^{'} + k γ - λ - ν, - 2 k), \\ (- λ - ν, - 2 k), & (- k ς^{'} + k τ^{'} - λ - ν, - 2 k), \end{matrix} \\ \begin{matrix} (- k ς^{'} + k τ + k γ - λ - ν, - 2 k) \\ (- k ς - k ς^{'} - k τ + k γ - λ - ν, - 2 k), & (ν + \frac{3 k}{2}, k), (\frac{3 k}{2}, k) \end{matrix} | \frac{- c x^{2} k}{4}] . \end{array}$ \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 $= \sum_{n = 0}^{\infty} \frac{{(- c)}^{n}}{Γ_{k} (nk + ν + \frac{3 k}{2}) Γ (n + \frac{3}{2}) n {! 2}^{\frac{ν}{k} + 2 n + 1}} (D_{-}^{ς, ς^{'}, τ, τ^{'}, γ} (t^{\frac{λ}{k} + \frac{ν}{k} + 2 n}))$ = \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 $\begin{array}{l} = \sum_{n = 0}^{\infty} \frac{{(- c)}^{n} Γ (τ^{'} - \frac{λ}{k} - \frac{ν}{k} - 2 n)}{Γ_{k} (nk + ν + \frac{3 k}{2}) Γ (n + \frac{3}{2}) n {! 2}^{\frac{ν}{k} + 2 n + 1} Γ (- \frac{λ}{k} - \frac{ν}{k} - 2 n)} \\ \times \frac{Γ (- ς - ς^{'} + γ - \frac{λ}{k} - \frac{ν}{k} - 2 n) Γ (- ς^{'} - τ + γ - \frac{λ}{k} - \frac{ν}{k} - 2 n)}{Γ (- ς - ς^{'} - τ + γ - \frac{λ}{k} - \frac{ν}{k} - 2 n) Γ (- ς^{'} + τ^{'} - \frac{λ}{k} - \frac{ν}{k} - 2 n)} x^{ς + ς^{'} - γ + \frac{λ}{k} + \frac{ν}{k} + 2 n} \\ = \frac{x^{ς + ς^{'} - γ \frac{λ}{k} + \frac{ν}{k}}}{2^{\frac{ν}{k} + 1}} \sum_{n = 0}^{\infty} \frac{{(- c x^{2})}^{n}}{n {! 4}^{n} Γ_{k} (nk + ν + \frac{3 k}{2})} \frac{Γ (τ^{'} - \frac{λ}{k} - \frac{ν}{k} - 2 n)}{Γ (- \frac{λ}{k} - \frac{ν}{k} - 2 n)} \\ \times \frac{Γ (- ς - ς^{'} + γ - \frac{λ}{k} - \frac{ν}{k} - 2 n) Γ (- ς^{'} - τ + γ - \frac{λ}{k} - \frac{ν}{k} - 2 n)}{Γ (- ς^{'} + τ^{'} - \frac{λ}{k} - \frac{ν}{k} - 2 n) Γ (- ς - ς^{'} - τ + γ - \frac{λ}{k} - \frac{ν}{k} - 2 n)} \\ = \frac{k^{- γ + \frac{1}{2}} x^{ς + ς^{'} - γ + \frac{λ}{k} + \frac{ν}{k}}}{2^{\frac{ν}{k} + 1}} \sum_{n = 0}^{\infty} \frac{{(- ck x^{2})}^{n}}{n {! 4}^{n}} \\ \times \frac{Γ_{k} (k τ^{'} - λ - ν - 2 nk) Γ_{k} (- k ς - k ς^{'} + k γ - λ - ν - 2 nk)}{Γ_{k} (- λ - ν - 2 nk) Γ_{k} (- k ς^{'} + k τ^{'} - λ - ν - 2 nk)} \\ \times \frac{Γ_{k} (- k ς^{'} - k τ + k γ - λ - ν - 2 nk)}{Γ_{k} (- k ς - k ς^{'} - k τ + k γ - λ - ν - 2 nk) t Γ_{k} (ν + \frac{3 k}{2} + nk) Γ_{k} (\frac{3 k}{2} + nk)} \end{array}$ \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 $ℜ (\frac{λ}{k}) > max {ℜ (- τ^{'}), ℜ (ς^{'} + τ - γ), ℜ (ς + ς^{'} - γ) + [ℜ (γ)] + 1}$ \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) $\begin{array}{l} (D_{-}^{ς, ς^{'}, τ, τ^{'}, γ} (t^{- \frac{λ}{k}} S_{ν, c}^{k} (t))) (x) = \frac{k^{- γ + \frac{1}{2}} x^{ς + ς^{'} - γ + \frac{λ}{k} + \frac{ν}{k} + 1}}{2^{\frac{ν}{k} + 1}} \\ \times_{3} Ψ_{5}^{k} [\begin{matrix} (k τ^{'} + λ - ν - k, - 2 k), & (- k ς - k ς^{'} + k γ - λ - ν - k, - 2 k), \\ (- λ - ν - k, - 2 k), & (- k ς^{'} + k τ^{'} + λ - ν - k, - 2 k), \end{matrix} \\ \begin{matrix} (- k ς^{'} - k τ + k γ + λ - ν - k, - 2 k) \\ (- k ς - k ς^{'} - k τ + k γ + λ - ν - k, - 2 k), & (ν + \frac{3 k}{2}, k), (\frac{3 k}{2}, k) \end{matrix} | \frac{- c x^{2} k}{4}] . \end{array}$ \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 $= \sum_{n = 0}^{\infty} \frac{{(- c)}^{n}}{Γ_{k} (nk + ν + \frac{3 k}{2}) Γ (n + \frac{3}{2}) n {! 2}^{\frac{ν}{k} + 2 n + 1}} (D_{-}^{ς, ς^{'}, τ, τ^{'}, γ} (t^{\frac{ν}{k} - \frac{λ}{k} + 2 n + 1}))$ = \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 $\begin{array}{l} = \sum_{n = 0}^{\infty} \frac{{(- c)}^{n}}{Γ_{k} (nk + ν + \frac{3 k}{2}) Γ (n + \frac{3}{2}) n {! 2}^{\frac{ν}{k} + 2 n + 1}} \frac{Γ (τ^{'} + \frac{λ - ν}{k} - 2 n - 1)}{Γ (\frac{λ - ν}{k} - 2 n - 1)} \\ \times \frac{Γ (- ς - ς^{'} + γ + \frac{λ - ν}{k} - 2 n - 1) Γ (- ς^{'} - τ + γ + \frac{λ - ν}{k} - 2 n - 1)}{Γ (- ς^{'} + τ^{'} + \frac{λ - ν}{k} - 2 n - 1) Γ (- ς - ς^{'} - τ + γ + \frac{λ - ν}{k} - 2 n - 1)} x^{ς + ς^{'} - γ + \frac{ν}{k} - \frac{λ}{k} + 2 n + 1} \end{array}$ \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 $\begin{array}{l} = \frac{t^{ς + ς^{'} - γ + \frac{ν}{k} - \frac{λ}{k} + 1}}{2^{\frac{ν}{k} + 1}} \sum_{n = 0}^{\infty} \frac{{(- ck t^{2})}^{n}}{n {! 4}^{n} k^{γ - \frac{1}{2}}} \\ \times \frac{Γ_{k} (k τ^{'} + λ - ν - k - 2 nk)}{Γ_{k} (λ - ν - k - 2 nk) Γ_{k} (- k ς^{'} + k τ^{'} + λ - ν - k - 2 nk)} \\ \times \frac{Γ_{k} (- k ς - k ς^{'} + k γ + λ - ν - k - 2 nk) Γ_{k} (- k ς^{'} - k τ + k γ + λ - ν - k - 2 nk)}{Γ_{k} (- k ς - k ς^{'} - k τ + k γ + λ - ν - k - 2 nk) Γ_{k} (nk + ν + \frac{3 k}{2}) Γ_{k} (nk + \frac{3 k}{2})} \end{array}$ \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.

Concluding Remark

MSM fractional calculus operators have more advantage due to the generalize of Riemann-Liouville, Weyl, Erdélyi-Kober, and Saigo's fractional calculus operators; therefore, many authors are called as general operator. Now we are going to conclude of this paper by emphasizing that our leading results (Theorems 1 – 6) can be derived as the specific cases involving familiar fractional calculus operators as above said. On other hand, the k Struve function defined in (1.16) possesses the lead that a number of special functions occur to be the particular cases. Some of special cases respect to the integrals relating with k Struve function have been discovered in the earlier research works by various authors with not the same arguments.

eISSN:: 2444-8656
Lingua:: Inglese

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The Marichev-Saigo-Maeda Fractional Calculus Operators Pertaining to the Generalized K-Struve Function

Pubblicato online: 30 dic 2020

Pagine: 593 - 602

Ricevuto: 20 set 2019

Accettato: 11 nov 2019

DOI: https://doi.org/10.2478/amns.2020.2.00064

Parole chiaveGeneralized -Struve function, Marichev-Saigo-Maeda fractional calculus operators, Generalized -Wright function, Pochhammer symbol, -Gamma function

© 2020 Seema Kabra et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Parole chiave
Generalized -Struve function, Marichev-Saigo-Maeda fractional calculus operators, Generalized -Wright function, Pochhammer symbol, -Gamma function