New results for generalized Hurwitz-Lerch Zeta functions using Laplace transform

Yağcı, Oğuz; Şahin, Recep; Nisar, Kottakkaran Sooppy

Introduction

The Hurwitz-Lerch Zeta function is defined by Srivastava and Choi [1, 2]: (1) $\begin{matrix} Φ (z, r, α) = \sum_{n = 0}^{\infty} \frac{z^{n}}{{(n + α)}^{r}}, \\ (α \in ℂ \ ℤ_{0}; r \in ℂ when | z | < 1; ℜ (r) > 1 when | z | = 1) . \end{matrix}$ \matrix{{\Phi \left( {z,r,\alpha } \right) = \sum\limits_{n = 0}^\infty \frac{{{z^n}}}{{{{\left( {n + \alpha } \right)}^r}}},}\\{(\alpha \in \mathbb{C}\backslash {\mathbb{Z}_0};\;r \in \mathbb{C}\,{\rm{when}}\,\left| z \right| < 1;\;\Re (r) > 1\,{\rm{when}}\,\left| z \right| = 1).}}

In 1997, Goyal and Laddha [3] defined and studied an extension of (1) one variable in the following equation: (2) $\begin{matrix} Φ_{ε}^{⋆} (z, r, α) = \sum_{n = 0}^{\infty} {(ε)}_{n} \frac{z^{n}}{{(n + α)}^{r} n!}, \\ (ε \in ℂ; α \in \ ℤ_{0}; | x | < 1) . \end{matrix}$ \matrix{{\Phi_\varepsilon^\star \left( {z,r,\alpha } \right) = \sum\limits_{n = 0}^\infty {{(\varepsilon )}_n}{\kern 1pt} \frac{{{z^n}}}{{{{\left( {n + \alpha } \right)}^r}{\kern 1pt} n!}},}\\{(\varepsilon \in \mathbb{C};\alpha \in \backslash {\mathbb{Z}_0};|x| < 1).}}

Afterwards, many researchers studied many different generalizations and extensions of the Hurwitz-Lerch Zeta function one variable. The readers who are interested can refer to these earlier publications for further researches and applications (Lin and Srivastava [4], Lin et al. [5], Choi et al. [6], Gupta et al. [7], Garg et al. [8], Srivastava et al. [9], Daman and Pathan [10], Srivastava et al. [11], and Srivastava [12]).

Later on, in 2017, Choi and Parmar [13] introduced and examined a generalization of (1) two variables in the following equation: (3) $\begin{matrix} Φ_{m, n, n^{'}; u} (v, y, r, p) = \sum_{a, b = 0}^{\infty} \frac{{(m)}_{a + b} {(n)}_{a} {(n^{'})}_{b}}{{(u)}_{a + b} a! b!} \frac{v^{a} y^{b}}{{(a + b + p)}^{r}} \\ (u, p \in ℂ \ ℤ_{0}; m, n, n^{'} \in ℂ; r, v, y \in ℂ when | v | < 1 and | y | < 1; \\ ℜ (r - u - m - n - n^{'}) > 1 when | v | = 1 and | y | = 1) . \end{matrix}$ \matrix{{{\Phi_{\mathfrak{m},\mathfrak{n},{\mathfrak{n}^{'}};\mathfrak{u}}}\left( {\mathfrak{v},\mathfrak{y},\mathfrak{r},\mathfrak{p}} \right) = \sum\limits_{\mathfrak{a},\mathfrak{b} = 0}^\infty \frac{{{{\left( \mathfrak{m} \right)}_{\mathfrak{a} + \mathfrak{b}}}{{\left( \mathfrak{n} \right)}_\mathfrak{a}}{{\left( {{\mathfrak{n}^{'}}} \right)}_\mathfrak{b}}}}{{{{\left( \mathfrak{u} \right)}_{\mathfrak{a} + \mathfrak{b}}}\mathfrak{a}!\;\mathfrak{b}!}}\;\frac{{{\mathfrak{v}^\mathfrak{a}}{\mathfrak{y}^\mathfrak{b}}}}{{{{\left( {\mathfrak{a} + \mathfrak{b} + \mathfrak{p}} \right)}^\mathfrak{r}}}}}\\{(\mathfrak{u},\mathfrak{p} \in \mathbb{C}\backslash {\mathbb{Z}_0};\;\mathfrak{m},\mathfrak{n},{\mathfrak{n}^{'}} \in \mathbb{C};\;\mathfrak{r},\mathfrak{v},\mathfrak{y} \in \mathbb{C}\,{\rm{when}}\,\left| \mathfrak{v} \right| < 1\,{\rm{and}}\,\left| \mathfrak{y} \right| < 1;}\\{\Re (\mathfrak{r} - \mathfrak{u} - \mathfrak{m} - \mathfrak{n} - {\mathfrak{n}^{'}}) > 1\,{\rm{when}}\,\left| \mathfrak{v} \right| = 1\,{\rm{and}}\,\left| \mathfrak{y} \right| = 1).}} A large number of generalizations and extensions of the Hurwitz-Lerch Zeta function two variables have been studied by Choi et al. [14], and Srivastava et al. [15].

In 2011, Srivastava et al. [16] introduced and studied the following extension of the generalized Hurwitz-Lerch Zeta function: (4) $\begin{matrix} Φ_{λ, μ; ω}^{(σ, ρ, κ)} (z, s, a) = \sum_{n = 0}^{\infty} \frac{{(λ)}_{σ n} {(μ)}_{ρ n}}{{(ω)}_{κ n} n!} \frac{z^{n}}{{(n + a)}^{s}}, \\ (λ, μ \in ℂ; a, ω \in ℂ \ ℤ_{0}^{-}; σ, ρ, κ \in ℝ^{+}; κ - σ - ρ > - 1 when s, z \in ℂ; \\ κ - σ - ρ = - 1 and s \in ℂ when | z | < δ^{⋆} = σ^{- σ} ρ^{- ρ} κ^{κ}; \\ κ - σ - ρ = - 1 and ℜ (s + ω - λ - μ) > 1 when | z | = δ^{⋆}) . \end{matrix}$ \matrix{{\Phi_{\lambda ,{\kern 1pt} \mu ;{\kern 1pt} \omega }^{(\sigma ,{\kern 1pt} \rho ,{\kern 1pt} \kappa )}(z,s,a) = {\kern 1pt} \sum\limits_{n = 0}^\infty \frac{{{{(\lambda )}_{\sigma n}}{\kern 1pt} {{(\mu )}_{\rho n}}}}{{{{(\omega )}_{\kappa n}}{\kern 1pt} n!}}\frac{{{z^n}}}{{{{(n + a)}^s}}},}\\{(\lambda ,{\kern 1pt} \mu {\kern 1pt} \in {\kern 1pt} \mathbb{C};{\kern 1pt} a,{\kern 1pt} \omega {\kern 1pt} \in {\kern 1pt} \mathbb{C}\backslash \mathbb{Z}_0^ - ;{\kern 1pt} \sigma ,{\kern 1pt} \rho ,{\kern 1pt} \kappa {\kern 1pt} \in {\kern 1pt} {\mathbb{R}^ + };{\kern 1pt} \kappa - \sigma - \rho > - 1{\kern 1pt} \,{\rm{when}}{\kern 1pt} \,s,{\kern 1pt} z{\kern 1pt} \in {\kern 1pt} \mathbb{C};}\\{{\kern 1pt} \kappa - \sigma - \rho = - 1{\kern 1pt} \,{\rm{and}}\,{\kern 1pt} s{\kern 1pt} \in {\kern 1pt} \mathbb{C}{\kern 1pt} \,{\rm{when}}\,{\kern 1pt} |z| < {\kern 1pt} {\delta^ \star } = {\kern 1pt} {\sigma^{ - \sigma }}{\rho^{ - \rho }}{\kappa^\kappa };}\\{{\kern 1pt} \kappa - \sigma - \rho = - 1{\kern 1pt} \,{\rm{and}}\,{\kern 1pt} \Re (s + \omega - \lambda - \mu ) > 1{\kern 1pt} \,{\rm{when}}\,{\kern 1pt} |z| = {\kern 1pt} {\delta^ \star }).}}

The main goal of this work is to demonstrate the generalized FKEs involving generalized Hurwitz-Lerch Zeta function (4). Here, we consider the Laplace transform methodology to get the results.

The rest of the paper is organized as follows. The fractional Kinetic equations are given in Section 2. The numerical results and graphics are described in Section 3. The conclusion is introduced by Section 4.

Fractional Kinetic equations

In [17], experts designated the fractional differential equation for the rate of change of reaction. The destruction rate and the production rate are given by (5) $\frac{d z}{d y} = - d (z_{y}) + p (z_{y}),$ \frac{{d\mathfrak{z}}}{{d\mathfrak{y}}} = {\kern 1pt} - d({\mathfrak{z}_\mathfrak{y}}) + p({\mathfrak{z}_\mathfrak{y}}), where 𝔷= 𝔷(𝔶) the rate of the reaction, 𝔡 = 𝔡(𝔷) the rate of destruction, p = p(𝔷) the rate of production and 𝔷_𝔶 denotes the function defined by 𝔷_𝔶(𝔶^*) = 𝔷(𝔶−𝔶^*), 𝔶^* > 0.

The special case of equation (5) for spatial fluctuations and inhomogeneities in 𝔷(𝔶) the quantities are neglected, that is the equation (6) $\frac{d z}{d y} = - c_{i} z_{i} (y)$ \frac{{d\mathfrak{z}}}{{d\mathfrak{y}}} = - {c_i}{\kern 1pt} {\mathfrak{z}_i}(\mathfrak{y}) with the initial condition that 𝔷_i(𝔶 = 0) = 𝔷₀ is the number of density of the species i at time 𝔶 = 0 and c_i > 0. If we shift the index i and integrate the standard Kinetic equation (6), we have (7) $z (y) - z_{0} = - c_{0} D_{y}^{- 1} z (y)$ \mathfrak{z}(\mathfrak{y}) - {\mathfrak{z}_0} = - c{{\kern 1pt}_0}\mathcal{D}_\mathfrak{y}^{ - 1}{\kern 1pt} \mathfrak{z}(\mathfrak{y}) where $_{0} D_{y}^{- 1}$ {_0\mathcal{D}_\mathfrak{y}^{ - 1}} is the special case of the Riemann-Liouville integral operator $_{0} D_{y}^{- x}$ {_0\mathcal{D}_\mathfrak{y}^{ - \mathfrak{x}}} given as Srivastava and Saxena [18], (8) $\begin{matrix} _{0} D_{y}^{- x} f (y) = \frac{1}{Γ (x)} \int_{0}^{y} {(y - s)}^{x - 1} f (s) d s, \\ (y > 0, ℜ (x) > 0) . \end{matrix}$ \matrix{{_0\mathcal{D}_\mathfrak{y}^{ - \mathfrak{x}}{\kern 1pt} f(\mathfrak{y}) = {\kern 1pt} \frac{1}{{\Gamma (\mathfrak{x})}}{\kern 1pt} \int_0^\mathfrak{y} {\kern 1pt} {{(\mathfrak{y} - s)}^{\mathfrak{x} - 1}}{\kern 1pt} f(s)ds,}\\{(\mathfrak{y} > 0,{\kern 1pt} \Re (\mathfrak{x}) > 0).}}

The fractional generalisation of the standard Kinetic equation (7) is given by Haubold and Mathai [17] as follows: (9) $z (y) - z_{0} = - c^{x}_{0} D_{y}^{- 1} z (y)$ \mathfrak{z}(\mathfrak{y}) - {\mathfrak{z}_0} = - {c^\mathfrak{x}}{{\kern 1pt}_0}\mathcal{D}_\mathfrak{y}^{ - 1}{\kern 1pt} \mathfrak{z}(\mathfrak{y}) and obtained the solution of (6) as follows: (10) $z (y) = z_{0} \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{Γ (x k + 1)} {(c y)}^{x k} .$ \mathfrak{z}(\mathfrak{y}) = {\mathfrak{z}_0}{\kern 1pt} \sum\limits_{k = 0}^\infty {\kern 1pt} \frac{{{{( - 1)}^k}}}{{\Gamma (\mathfrak{x}k + 1)}}{(c\mathfrak{y})^{\mathfrak{x}k}}.

Furthermore, Saxena and Kalla [19] considered the following fractional Kinetic equation: (11) $z (y) - z_{0} f (y) = - c^{x}_{0} D_{y}^{- 1} z (y) (ℜ (x) > 0),$ \mathfrak{z}(\mathfrak{y}) - {\mathfrak{z}_0}{\kern 1pt} f(\mathfrak{y}) = - {c^\mathfrak{x}}{{\kern 1pt}_0}\mathcal{D}_\mathfrak{y}^{ - 1}{\kern 1pt} \mathfrak{z}(\mathfrak{y}){\kern 1pt} {\kern 1pt} {\kern 1pt} (\Re (\mathfrak{x}) > 0), where 𝔷(𝔶) denotes the number density of a given species at time 𝔶, 𝔷₀ = 𝔷(0) is the number of density of that species at time 𝔶 = 0, c is a constant and f ∈ L(0, ∞).

By applying the Laplace transform (Erdélyi et al. [20], Srivastava and Karlsson [21], and Srivastava and Manocha [22]) to the equation (11), we obtain (12) $\begin{matrix} L {z (y); p} = z_{0} \frac{F (p)}{1 + c^{x} p^{- x}} = z_{0} (\sum_{n = 0}^{\infty} {(- c^{x})}^{n} p^{- x n}) F (p), \\ (n \in z_{0}, | \frac{c}{p} | < 1) . \end{matrix}$ \matrix{{\mathfrak{L}{\kern 1pt} \{ \mathfrak{z}(\mathfrak{y});p\} = {\mathfrak{z}_0}{\kern 1pt} \frac{{F(p)}}{{1 + {c^\mathfrak{x}}{\kern 1pt} {p^{ - \mathfrak{x}}}}} = {\kern 1pt} {\mathfrak{z}_0}{\kern 1pt} (\sum\limits_{n = 0}^\infty {\kern 1pt} {{( - {c^\mathfrak{x}})}^n}{p^{ - \mathfrak{x}n}}){\kern 1pt} F(p),}\\{(n \in {\kern 1pt} {\mathfrak{z}_0},|\frac{c}{p}| < 1).}}

The extension and generalisation of FKEs involving many fractional operators were found in (Saxena et al. [23], Saxena et al. [24], Chouhan and Sarwat [25], Chouhan et al. [26], Agarwal et al. [27], Agarwal et al. [28], Agarwal and Nisar [29], Baleanu et al. [30], Nisar et al. [31], Nisar [32], Şahin and Yağcı [33], Yağcı and Şahin [34], Akel et al. [35], and Hidan et al. [36]).

2.1

Solution of generalized FKEs involving generalized Hurwitz-Lerch Zeta function

Here, we present the solution of the generalised FKEs by considering generalized Hurwitz-Lerch Zeta function (4).

Theorem 1

If 𝔡 > 0, 𝔵 > 0; λ, μ, δ ∈ ℂ, and 𝔡 ≠ δ be such that $a, ω \in ℂ \ ℤ_{0}^{-}$ a,{\kern 1pt} \omega {\kern 1pt} \in {\kern 1pt} \mathbb{C}\backslash \mathbb{Z}_0^ - ; σ, ρ, κ∈ ℝ⁺, then the solution of the following fractional equation (13) $z (y) - z_{0} Φ_{λ, μ; ω}^{(σ, ρ, κ)} (d^{x} y^{x}, s, a) = - δ^{x}_{0} D_{y}^{- x}$ \mathfrak{z}(\mathfrak{y}) - {\mathfrak{z}_0}{\kern 1pt} \Phi_{\lambda ,{\kern 1pt} \mu ;{\kern 1pt} \omega }^{(\sigma ,{\kern 1pt} \rho ,{\kern 1pt} \kappa )}({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}},s,a) = {\kern 1pt} - {\delta^\mathfrak{x}}{{\kern 1pt}_0}\mathcal{D}_\mathfrak{y}^{ - \mathfrak{x}} is given by (14) $z (y) = z_{0} \sum_{n = 0}^{\infty} \frac{{(λ)}_{σ n} {(μ)}_{ρ n} Γ (x n + 1) {(d^{x} y^{x})}^{n}}{{(ω)}_{κ n} n! {(n + a)}^{s}} E_{x, x n + 1} (- δ^{x} y^{x}),$ \mathfrak{z}(\mathfrak{y}) = {\mathfrak{z}_0}{\kern 1pt} {\kern 1pt} \sum\limits_{n = 0}^\infty \frac{{{{(\lambda )}_{\sigma n}}{\kern 1pt} {{(\mu )}_{\rho n}}{\kern 1pt} \Gamma (\mathfrak{x}n + 1){{({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}})}^n}}}{{{{(\omega )}_{\kappa n}}{\kern 1pt} n!{\kern 1pt} {{(n + a)}^s}}}{\kern 1pt} {E_{\mathfrak{x},\mathfrak{x}n + 1}}( - {\delta^\mathfrak{x}}{\kern 1pt} {\mathfrak{y}^\mathfrak{x}}), where ℰ_{𝔵,𝔵𝔫+1}(.) is the Mittag-Leffler function (Mittag and Leffler [37]).

Proof

The Laplace transform of the Riemann-Liouville fractional integral operator is given by Srivastava and Saxena [18], and Erdélyi et al. [20]: (15) $L {_{0} D_{y}^{- x} f (y); s} = s^{- x} F (s)$ \mathfrak{L}{\kern 1pt} {\{_0}\mathcal{D}_\mathfrak{y}^{ - \mathfrak{x}}f(\mathfrak{y});s\} = {\kern 1pt} {s^{ - \mathfrak{x}}}{\kern 1pt} F(s) where F(p) is given in Erdélyi et al. [20], Srivastava and Karlsson [21], and Srivastava and Manocha [22]. Now, applying the Laplace transform to both sides of (13), we obtain (16) $\begin{array}{l} L {z (y); p} = z_{0} L {Φ_{λ, μ; ω}^{(σ, ρ, κ)} (d^{x} y^{x}, s, a); p} - δ^{x} L {_{0} D_{y}^{- x} z (y); p} \\ z (p) = z_{0} {\int_{0}^{\infty} e^{- p y} \sum_{n = 0}^{\infty} \frac{{(λ)}_{σ n} {(μ)}_{ρ n} {(d^{x} y^{x})}^{n}}{{(ω)}_{κ n} n! {(n + a)}^{s}} d y} - δ^{x} p^{- x} z (p) \\ z (p) + δ^{x} p^{- x} z (p) \\ = z_{0} \sum_{n = 0}^{\infty} \frac{{(λ)}_{σ n} {(μ)}_{ρ n} d^{x n}}{{(ω)}_{κ n} n! {(n + a)}^{s}} \int_{0}^{\infty} e^{- p y} y^{x n} d y \\ = z_{0} \sum_{n = 0}^{\infty} \frac{{(λ)}_{σ n} {(μ)}_{ρ n} d^{x n}}{{(ω)}_{κ n} n! {(n + a)}^{s}} \frac{Γ (x n + 1)}{p^{x n + 1}} \\ z (p) = z_{0} \sum_{n = 0}^{\infty} \frac{{(λ)}_{σ n} z μ)_{ρ n} Γ (x n + 1) d^{x n}}{{(ω)}_{κ n} n! {(n + a)}^{s}} p^{- (x n + 1)} \sum_{r = 0}^{\infty} [- {(\frac{p}{δ})}^{- x}]^{r} . \end{array}$ \matrix{\mathfrak{L}\{ \mathfrak{z}(\mathfrak{y});p\} = {\mathfrak{z}_0}\mathfrak{L}\{ \Phi_{\lambda ,{\kern 1pt} \mu ;{\kern 1pt} \omega }^{(\sigma ,{\kern 1pt} \rho ,{\kern 1pt} \kappa )}({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}},s,a);p\} - {\delta^\mathfrak{x}}\mathfrak{L}{\{_0}\mathcal{D}_\mathfrak{y}^{ - \mathfrak{x}}\mathfrak{z}(\mathfrak{y});p\} \\\mathfrak{z}(p) = {\mathfrak{z}_0}\{ \int_0^\infty {{e^{ - p\mathfrak{y}}}\sum\limits_{n = 0}^\infty {\frac{{{{(\lambda )}_{\sigma n}}{{(\mu )}_{\rho n}}{{({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}})}^n}}}{{{{(\omega )}_{\kappa n}}n!{{(n + a)}^s}}}} d\mathfrak{y}} \} - {\delta^\mathfrak{x}}{p^{ - \mathfrak{x}}}\mathfrak{z}(p)\\\mathfrak{z}(p) + {\delta^\mathfrak{x}}{p^{ - \mathfrak{x}}}\mathfrak{z}(p)\\\,\,\,\,\,\,\,\,\, = {\mathfrak{z}_0}\sum\limits_{n = 0}^\infty {\frac{{{{(\lambda )}_{\sigma n}}{{(\mu )}_{\rho n}}{\mathfrak{d}^{\mathfrak{x}n}}}}{{{{(\omega )}_{\kappa n}}n!{{(n + a)}^s}}}} \int_0^\infty {{e^{ - p\mathfrak{y}}}{\mathfrak{y}^{\mathfrak{x}n}}d\mathfrak{y}} \\\,\,\,\,\,\,\,\,\, = {\mathfrak{z}_0}\sum\limits_{n = 0}^\infty {\frac{{{{(\lambda )}_{\sigma n}}{{(\mu )}_{\rho n}}{\mathfrak{d}^{\mathfrak{x}n}}}}{{{{(\omega )}_{\kappa n}}n!{{(n + a)}^s}}}} \frac{{\Gamma (\mathfrak{x}n + 1)}}{{{p^{\mathfrak{x}n + 1}}}}\\\mathfrak{z}(p) = {\mathfrak{z}_0}\sum\limits_{n = 0}^\infty {\frac{{{{(\lambda )}_{\sigma n}}z\mu {)_{\rho n}}\Gamma (\mathfrak{x}n + 1){\mathfrak{d}^{\mathfrak{x}n}}}}{{{{(\omega )}_{\kappa n}}n!{{(n + a)}^s}}}} {p^{ - (\mathfrak{x}n + 1)}}\sum\limits_{r = 0}^\infty {[ - {{(\frac{p}{\delta })}^{ - \mathfrak{x}}}} {]^r}.}

The inverse Laplace transform of (16) is given by Erdélyi et al. [20] (17) $L^{- 1} {p^{- x}; y} = \frac{t^{x - 1}}{Γ (x)}, (ℜ (x) > 0),$ {\mathfrak{L}^{ - 1}}\{ {p^{ - \mathfrak{x}}};\mathfrak{y}\} = \frac{{{t^{\mathfrak{x} - 1}}}}{{\Gamma (\mathfrak{x})}},{\kern 1pt} {\kern 1pt} {\kern 1pt} (\Re (\mathfrak{x}) > 0), we get (18) $\begin{matrix} L^{- 1} {z (p)} = z_{0} \sum_{n = 0}^{\infty} \frac{{(λ)}_{σ n} {(μ)}_{ρ n} Γ (x n + 1) d^{x n}}{{(ω)}_{κ n} n! {(n + a)}^{s}} \\ \times L^{- 1} {p^{- (x n + 1)} \sum_{r = 0}^{\infty} {[- {(\frac{p}{δ})}^{- x}]}^{r}} \\ z (y) = \sum_{n = 0}^{\infty} \frac{{(λ)}_{σ n} {(μ)}_{ρ n} Γ (x n + 1) {(d^{x} y^{x})}^{n}}{{(ω)}_{κ n} n! {(n + a)}^{s}} \sum_{r = 0}^{\infty} {(- 1)}^{r} δ^{x r} \frac{y^{x r}}{Γ (x n + x r + 1)} . \end{matrix}$ \matrix{{{L^{ - 1}}\{ \mathfrak{z}(p)\} = {\kern 1pt} {\mathfrak{z}_0}{\kern 1pt} \sum\limits_{n = 0}^\infty \frac{{{{(\lambda )}_{\sigma n}}{\kern 1pt} {{(\mu )}_{\rho n}}{\kern 1pt} \Gamma (\mathfrak{x}n + 1){\kern 1pt} {\mathfrak{d}^{\mathfrak{x}n}}}}{{{{(\omega )}_{\kappa n}}{\kern 1pt} n!{\kern 1pt} {{(n + a)}^s}}}}\\{ \times {\kern 1pt} {L^{ - 1}}\left\{ {{p^{ - (\mathfrak{x}n + 1)}}\sum\limits_{r = 0}^\infty {{\left[ { - {{(\frac{p}{\delta })}^{ - \mathfrak{x}}}} \right]}^r}} \right\}}\\{\mathfrak{z}(\mathfrak{y}) = {\kern 1pt} \sum\limits_{n = 0}^\infty \frac{{{{(\lambda )}_{\sigma n}}{\kern 1pt} {{(\mu )}_{\rho n}}{\kern 1pt} \Gamma (\mathfrak{x}n + 1){{({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}})}^n}}}{{{{(\omega )}_{\kappa n}}{\kern 1pt} n!{\kern 1pt} {{(n + a)}^s}}}\sum\limits_{r = 0}^\infty {\kern 1pt} {{( - 1)}^r}{\delta^{\mathfrak{x}r}}{\kern 1pt} \frac{{{\mathfrak{y}^{\mathfrak{x}r}}}}{{\Gamma (\mathfrak{x}n + \mathfrak{x}r + 1)}}.}}

So, we can yield the required result (14).

Theorem 2

If 𝔡 > 0, 𝔵 > 0; λ, μ ∈ ℂ be such that $a, ω \in ℂ \ ℤ_{0}^{-}$ {\kern 1pt} a,{\kern 1pt} \omega {\kern 1pt} \in {\kern 1pt} \mathbb{C}\backslash \mathbb{Z}_0^ - ; σ, ρ, κ ∈ ℝ⁺, then the solution of the following fractional equation (19) $z (y) - z_{0} Φ_{λ, μ; ω}^{(σ, ρ, κ)} (d^{x} y^{x}, s, a) = - d^{x}_{0} D_{y}^{- x}$ \mathfrak{z}(\mathfrak{y}) - {\mathfrak{z}_0}{\kern 1pt} \Phi_{\lambda ,{\kern 1pt} \mu ;{\kern 1pt} \omega }^{(\sigma ,{\kern 1pt} \rho ,{\kern 1pt} \kappa )}({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}},s,a) = {\kern 1pt} - {\mathfrak{d}^\mathfrak{x}}{{\kern 1pt}_0}\mathcal{D}_\mathfrak{y}^{ - \mathfrak{x}} is given by (20) $z (y) = z_{0} \sum_{n = 0}^{\infty} \frac{{(λ)}_{σ n} {(μ)}_{ρ n} Γ (x n + 1) {(d^{x} y^{x})}^{n}}{{(ω)}_{κ n} n! {(n + a)}^{s}} ℰ_{x, x n + 1} (- d^{x} y^{x}),$ \mathfrak{z}(\mathfrak{y}) = {\mathfrak{z}_0}{\kern 1pt} {\kern 1pt} \sum\limits_{n = 0}^\infty \frac{{{{(\lambda )}_{\sigma n}}{\kern 1pt} {{(\mu )}_{\rho n}}{\kern 1pt} \Gamma (\mathfrak{x}n + 1){{({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}})}^n}}}{{{{(\omega )}_{\kappa n}}{\kern 1pt} n!{\kern 1pt} {{(n + a)}^s}}}{\kern 1pt} {\mathcal{E}_{\mathfrak{x},\mathfrak{x}n + 1}}( - {\mathfrak{d}^\mathfrak{x}}{\kern 1pt} {\mathfrak{y}^\mathfrak{x}}), where ℰ_{𝔵,𝔵𝔫+1} (.) is the Mittag-Leffler function (Mittag-Leffler [37]).

Proof

The proof of Theorem 2 is parallel to the proof of Theorem 1, thus the details are omitted.

Theorem 3

If 𝔵 > 0; λ, μ, δ ∈ ℂ be such that $a, ω \in ℂ \ ℤ_{0}^{-}$ {\kern 1pt} a,{\kern 1pt} \omega {\kern 1pt} \in {\kern 1pt} \mathbb{C}\backslash \mathbb{Z}_0^ - ; σ, ρ, κ ∈ ℝ⁺, then the solution of the following fractional equation (21) $z (y) - z_{0} Φ_{λ, μ; ω}^{(σ, ρ, κ)} (y, s, a) = - δ^{x}_{0} D_{y}^{- x}$ \mathfrak{z}(\mathfrak{y}) - {\mathfrak{z}_0}{\kern 1pt} \Phi_{\lambda ,{\kern 1pt} \mu ;{\kern 1pt} \omega }^{(\sigma ,{\kern 1pt} \rho ,{\kern 1pt} \kappa )}(\mathfrak{y},s,a) = {\kern 1pt} - {\delta^\mathfrak{x}}{{\kern 1pt}_0}\mathcal{D}_\mathfrak{y}^{ - \mathfrak{x}} is given by (22) $z (y) = z_{0} \sum_{n = 0}^{\infty} \frac{{(λ)}_{σ n} {(μ)}_{ρ n} Γ (n + 1) t^{n}}{{(ω)}_{κ n} n! {(n + a)}^{s}} ℰ_{x, n + 1} (- δ^{x} t^{x}),$ \mathfrak{z}(\mathfrak{y}) = {\mathfrak{z}_0}{\kern 1pt} {\kern 1pt} \sum\limits_{n = 0}^\infty \frac{{{{(\lambda )}_{\sigma n}}{\kern 1pt} {{(\mu )}_{\rho n}}{\kern 1pt} \Gamma (n + 1){\kern 1pt} {t^n}}}{{{{(\omega )}_{\kappa n}}{\kern 1pt} n!{\kern 1pt} {{(n + a)}^s}}}{\kern 1pt} {\mathcal{E}_{\mathfrak{x},n + 1}}( - {\delta^\mathfrak{x}}{\kern 1pt} {t^\mathfrak{x}}), where ℰ_𝔵,𝔫+1 (.) is the Mittag-Leffler function (Mittag-Leffler [37]).

Proof

Theorem 3 can be easily acquired from Theorem 1, so the details are omitted.

2.2

Special cases

Choosing λ = σ = 1 in the equation (4), which is the generalized Hurwitz-Lerch Zeta function $Φ_{μ; ω}^{ρ, κ} (z, s, a)$ \Phi_{\mu ;{\kern 1pt} \omega }^{\rho ,{\kern 1pt} \kappa }(z,s,a) introduced and studied by Lin and Srivastava [4].

Applying λ = σ = 1 in Theorem 1, Theorem 2, and Theorem 3 obtained the following forms:

Corollary 4

If 𝔡 > 0, 𝔵 > 0; μ, δ ∈ ℂ, and 𝔡 ≠ δ be such that $a, ω \in ℂ \ ℤ_{0}^{-}$ {\kern 1pt} a,{\kern 1pt} \omega {\kern 1pt} \in {\kern 1pt} \mathbb{C}\backslash \mathbb{Z}_0^ - ; ρ, κ ∈ ℝ⁺, then the solution of the following fractional equation (23) $z (y) - z_{0} Φ_{μ; ω}^{(ρ, κ)} (d^{x} y^{x}, s, a) = - δ^{x}_{0} D_{t}^{- x}$ \mathfrak{z}(\mathfrak{y}) - {\mathfrak{z}_0}{\kern 1pt} \Phi_{\mu ;{\kern 1pt} \omega }^{(\rho ,{\kern 1pt} \kappa )}({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}},s,a) = {\kern 1pt} - {\delta^\mathfrak{x}}{{\kern 1pt}_0}\mathcal{D}_t^{ - \mathfrak{x}} is given by (24) $z (y) = z_{0} \sum_{n = 0}^{\infty} \frac{{(μ)}_{ρ n} Γ (x n + 1) {(d^{x} y^{x})}^{n}}{{(ω)}_{κ n} {(n + a)}^{s}} ℰ_{x, x n + 1} (- δ^{x} y^{x}) .$ \mathfrak{z}(\mathfrak{y}) = {\mathfrak{z}_0}{\kern 1pt} {\kern 1pt} \sum\limits_{n = 0}^\infty \frac{{{{(\mu )}_{\rho n}}{\kern 1pt} \Gamma (\mathfrak{x}n + 1){{({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}})}^n}}}{{{{(\omega )}_{\kappa n}}{\kern 1pt} {{(n + a)}^s}}}{\kern 1pt} {\mathcal{E}_{\mathfrak{x},\mathfrak{x}n + 1}}( - {\delta^\mathfrak{x}}{\kern 1pt} {\mathfrak{y}^\mathfrak{x}}).

Corollary 5

If 𝔡 > 0, x > 0; μ ∈ ℂ be such that $a, ω \in ℂ \ ℤ_{0}^{-}$ {\kern 1pt} a,{\kern 1pt} \omega {\kern 1pt} \in {\kern 1pt} \mathbb{C}\backslash \mathbb{Z}_0^ - ; ρ, κ ∈ ℝ⁺, then the solution of the following fractional equation (25) $z (y) - z_{0} Φ_{μ; ω}^{(ρ, κ)} (d^{x} y^{x}, s, a) = - d^{x}_{0} D_{y}^{- x}$ \mathfrak{z}(\mathfrak{y}) - {\mathfrak{z}_0}{\kern 1pt} \Phi_{\mu ;{\kern 1pt} \omega }^{(\rho ,{\kern 1pt} \kappa )}({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}},s,a) = {\kern 1pt} - {\mathfrak{d}^\mathfrak{x}}{{\kern 1pt}_0}\mathcal{D}_\mathfrak{y}^{ - \mathfrak{x}} is given by (26) $z (y) = z_{0} \sum_{n = 0}^{\infty} \frac{{(μ)}_{ρ n} Γ (x n + 1) {(d^{x} y^{x})}^{n}}{{(ω)}_{κ n} {(n + a)}^{s}} Ε_{x, x n + 1} (- d^{x} y^{x}) .$ \mathfrak{z}(\mathfrak{y}) = {\mathfrak{z}_0}{\kern 1pt} {\kern 1pt} \sum\limits_{n = 0}^\infty \frac{{{{(\mu )}_{\rho n}}{\kern 1pt} \Gamma (\mathfrak{x}n + 1){{({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}})}^n}}}{{{{(\omega )}_{\kappa n}}{\kern 1pt} {{(n + a)}^s}}}{\kern 1pt} {{\mathcal E}_{\mathfrak{x},\mathfrak{x}n + 1}}( - {\mathfrak{d}^\mathfrak{x}}{\kern 1pt} {\mathfrak{y}^\mathfrak{x}}).

Corollary 6

If μ, δ∈ ℂ be such that $a, ω \in ℂ \ ℤ_{0}^{-}$ {\kern 1pt} a,{\kern 1pt} \omega {\kern 1pt} \in {\kern 1pt} \mathbb{C}\backslash \mathbb{Z}_0^ - ; ρ, κ ∈ ℝ⁺, then the solution of the following fractional equation (27) $z (y) - z_{0} Φ_{μ; ω}^{(ρ, κ)} (y, s, a) = - δ^{x}_{0} D_{y}^{- x}$ \mathfrak{z}(\mathfrak{y}) - {\mathfrak{z}_0}{\kern 1pt} \Phi_{\mu ;{\kern 1pt} \omega }^{(\rho ,{\kern 1pt} \kappa )}(\mathfrak{y},s,a) = {\kern 1pt} - {\delta^\mathfrak{x}}{{\kern 1pt}_0}\mathcal{D}_\mathfrak{y}^{ - \mathfrak{x}} is given by (28) $z (y) = z_{0} \sum_{n = 0}^{\infty} \frac{{(μ)}_{ρ n} Γ (n + 1) y^{n}}{{(ω)}_{κ n} {(n + a)}^{s}} ℰ_{x, n + 1} (- δ^{x} y^{x}) .$ \mathfrak{z}(\mathfrak{y}) = {\mathfrak{z}_0}{\kern 1pt} {\kern 1pt} \sum\limits_{n = 0}^\infty \frac{{{{(\mu )}_{\rho n}}{\kern 1pt} \Gamma (n + 1){\mathfrak{y}^n}}}{{{{(\omega )}_{\kappa n}}{\kern 1pt} {{(n + a)}^s}}}{\kern 1pt} {\mathcal{E}_{\mathfrak{x},n + 1}}( - {\delta^\mathfrak{x}}{\kern 1pt} {\mathfrak{y}^\mathfrak{x}}).

Setting σ = ρ = κ = 1 in the equation (4), which is the generalized Hurwitz-Lerch Zeta function Φ_λ,μ;ω(z, s, a) introduced and studied by Garg et al. [8].

Applying σ = ρ = κ = 1 in Theorem 1, Theorem 2, and Theorem 3 obtained the following forms:

Corollary 7

If 𝔡 > 0, 𝔵 > 0; λ, μ, δ ∈ ℂ, and 𝔡 ≠ δ be such that $a, ω \in ℂ \ ℤ_{0}^{-}$ {\kern 1pt} a,{\kern 1pt} \omega {\kern 1pt} \in {\kern 1pt} \mathbb{C}\backslash \mathbb{Z}_0^ - , then the solution of the following fractional equation (29) $z (y) - z_{0} Φ_{λ, μ; ω} (d^{x} y^{x}, s, a) = - δ^{x}_{0} D_{y}^{- x}$ \mathfrak{z}(\mathfrak{y}) - {\mathfrak{z}_0}{\kern 1pt} {\Phi_{\lambda ,{\kern 1pt} \mu ;{\kern 1pt} \omega }}({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}},s,a) = {\kern 1pt} - {\delta^\mathfrak{x}}{{\kern 1pt}_0}\mathcal{D}_\mathfrak{y}^{ - \mathfrak{x}} is given by (30) $z (y) = z_{0} \sum_{n = 0}^{\infty} \frac{{(λ)}_{n} {(μ)}_{n} Γ (x n + 1) {(d^{x} y^{x})}^{n}}{{(ω)}_{n} n! {(n + a)}^{s}} ℰ_{x, x n + 1} (- δ^{x} y^{x}) .$ \mathfrak{z}(\mathfrak{y}) = {\mathfrak{z}_0}{\kern 1pt} {\kern 1pt} \sum\limits_{n = 0}^\infty \frac{{{{(\lambda )}_n}{\kern 1pt} {{(\mu )}_n}{\kern 1pt} \Gamma (\mathfrak{x}n + 1){{({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}})}^n}}}{{{{(\omega )}_n}{\kern 1pt} n!{\kern 1pt} {{(n + a)}^s}}}{\kern 1pt} {\mathcal{E}_{\mathfrak{x},\mathfrak{x}n + 1}}( - {\delta^\mathfrak{x}}{\kern 1pt} {\mathfrak{y}^\mathfrak{x}}).

Corollary 8

If 𝔡 > 0, 𝔵 > 0; λ, μ ∈ ℂ be such that $a, ω \in ℂ \ ℤ_{0}^{-}$ {\kern 1pt} a,{\kern 1pt} \omega {\kern 1pt} \in {\kern 1pt} \mathbb{C}\backslash \mathbb{Z}_0^ - , then the solution of the following fractional equation (31) $z (y) - z_{0} Φ_{λ, μ; ω} (d^{x} y^{x}, s, a) = - d^{x}_{0} D_{y}^{- x}$ \mathfrak{z}(\mathfrak{y}) - {\mathfrak{z}_0}{\kern 1pt} {\Phi_{\lambda ,{\kern 1pt} \mu ;{\kern 1pt} \omega }}({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}},s,a) = {\kern 1pt} - {\mathfrak{d}^\mathfrak{x}}{{\kern 1pt}_0}\mathcal{D}_\mathfrak{y}^{ - \mathfrak{x}} is given by (32) $z (y) = z_{0} \sum_{n = 0}^{\infty} \frac{{(λ)}_{n} {(μ)}_{n} Γ (x n + 1) {(d^{x} y^{x})}^{n}}{{(ω)}_{n} n! {(n + a)}^{s}} ℰ_{x, x n + 1} (- d^{x} y^{x}) .$ \mathfrak{z}(\mathfrak{y}) = {\mathfrak{z}_0}{\kern 1pt} {\kern 1pt} \sum\limits_{n = 0}^\infty \frac{{{{(\lambda )}_n}{\kern 1pt} {{(\mu )}_n}{\kern 1pt} \Gamma (\mathfrak{x}n + 1){{({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}})}^n}}}{{{{(\omega )}_n}{\kern 1pt} n!{\kern 1pt} {{(n + a)}^s}}}{\kern 1pt} {\mathcal{E}_{\mathfrak{x},\mathfrak{x}n + 1}}( - {\mathfrak{d}^\mathfrak{x}}{\kern 1pt} {\mathfrak{y}^\mathfrak{x}}).

Corollary 9

If λ, μ, δ ∈ ℂ be such that $a, ω \in ℂ \ ℤ_{0}^{-}$ {\kern 1pt} a,{\kern 1pt} \omega {\kern 1pt} \in {\kern 1pt} \mathbb{C}\backslash \mathbb{Z}_0^ - , then the solution of the following fractional equation (33) $z (y) - z_{0} Φ_{λ, μ; ω} (y, s, a) = - δ^{x}_{0} D_{y}^{- x}$ \mathfrak{z}(\mathfrak{y}) - {\mathfrak{z}_0}{\kern 1pt} {\Phi_{\lambda ,{\kern 1pt} \mu ;{\kern 1pt} \omega }}(\mathfrak{y},s,a) = {\kern 1pt} - {\delta^\mathfrak{x}}{{\kern 1pt}_0}\mathcal{D}_\mathfrak{y}^{ - \mathfrak{x}} is given by (34) $z (y) = z_{0} \sum_{n = 0}^{\infty} \frac{{(λ)}_{n} {(μ)}_{n} Γ (n + 1) t^{n}}{{(ω)}_{n} n! {(n + a)}^{s}} ℰ_{x, n + 1} (- δ^{x} t^{x}) .$ \mathfrak{z}(\mathfrak{y}) = {\mathfrak{z}_0}{\kern 1pt} {\kern 1pt} \sum\limits_{n = 0}^\infty \frac{{{{(\lambda )}_n}{\kern 1pt} {{(\mu )}_n}{\kern 1pt} \Gamma (n + 1){t^n}}}{{{{(\omega )}_n}{\kern 1pt} n!{\kern 1pt} {{(n + a)}^s}}}{\kern 1pt} {\mathcal{E}_{\mathfrak{x},n + 1}}( - {\delta^\mathfrak{x}}{\kern 1pt} {t^\mathfrak{x}}).

Upon taking σ = ρ = κ = 1 and λ = ω in the equation (4), which is the generalized Hurwitz-Lerch Zeta function $Φ_{μ}^{⋆} (z, s, a)$ \Phi_\mu^ \star (z,s,a) introduced and studied by Goyal and Laddha [3].

Applying σ = ρ = κ = 1 and λ = ω in Theorem 1, Theorem 2, and Theorem 3 obtained the following forms:

Corollary 10

If 𝔡 > 0, 𝔵 > 0; μ, δ ∈ ℂ, and 𝔡 ≠ δ be such that $a \in ℂ \ ℤ_{0}^{-}$ {\kern 1pt} a \in {\kern 1pt} \mathbb{C}\backslash \mathbb{Z}_0^ - , then the solution of the following fractional equation (35) $z (y) - z_{0} Φ_{μ}^{⋆} (d^{x} y^{x}, s, a) = - δ^{x}_{0} D_{y}^{- x}$ \mathfrak{z}(\mathfrak{y}) - {\mathfrak{z}_0}{\kern 1pt} \Phi_\mu^ \star ({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}},s,a) = {\kern 1pt} - {\delta^\mathfrak{x}}{{\kern 1pt}_0}\mathcal{D}_\mathfrak{y}^{ - \mathfrak{x}} is given by (36) $z (y) = z_{0} \sum_{n = 0}^{\infty} \frac{{(μ)}_{n} Γ (x n + 1) {(d^{x} y^{x})}^{n}}{n! {(n + a)}^{s}} ℰ_{x, x n + 1} (- δ^{x} y^{x}) .$ \mathfrak{z}(\mathfrak{y}) = {\mathfrak{z}_0}{\kern 1pt} {\kern 1pt} \sum\limits_{n = 0}^\infty \frac{{{{(\mu )}_n}{\kern 1pt} \Gamma (\mathfrak{x}n + 1){{({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}})}^n}}}{{n!{\kern 1pt} {{(n + a)}^s}}}{\kern 1pt} {\mathcal{E}_{\mathfrak{x},\mathfrak{x}n + 1}}( - {\delta^\mathfrak{x}}{\kern 1pt} {\mathfrak{y}^\mathfrak{x}}).

Corollary 11

If 𝔡 > 0, 𝔵 > 0; μ ∈ ℂ be such that $a \in ℂ \ ℤ_{0}^{-}$ {\kern 1pt} a \in {\kern 1pt} \mathbb{C}\backslash \mathbb{Z}_0^ - , then the solution of the following fractional equation (37) $z (y) - z_{0} Φ_{μ}^{⋆} (d^{x} y^{x}, s, a) = - d^{x}_{0} D_{y}^{- x}$ \mathfrak{z}(\mathfrak{y}) - {\mathfrak{z}_0}{\kern 1pt} \Phi_\mu^ \star ({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}},s,a) = {\kern 1pt} - {\mathfrak{d}^\mathfrak{x}}{{\kern 1pt}_0}\mathcal{D}_\mathfrak{y}^{ - \mathfrak{x}} is given by (38) $z (y) = z_{0} \sum_{n = 0}^{\infty} \frac{{(μ)}_{n} Γ (x n + 1) {(d^{x} y^{x})}^{n}}{n! {(n + a)}^{s}} ℰ_{x, x n + 1} (- d^{x} y^{x}) .$ \mathfrak{z}(\mathfrak{y}) = {\mathfrak{z}_0}{\kern 1pt} {\kern 1pt} \sum\limits_{n = 0}^\infty \frac{{{{(\mu )}_n}{\kern 1pt} \Gamma (\mathfrak{x}n + 1){{({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}})}^n}}}{{n!{\kern 1pt} {{(n + a)}^s}}}{\kern 1pt} {\mathcal{E}_{\mathfrak{x},\mathfrak{x}n + 1}}( - {\mathfrak{d}^\mathfrak{x}}{\kern 1pt} {\mathfrak{y}^\mathfrak{x}}).

Corollary 12

If λ, μ, δ ∈ ℂ be such that $a \in ℂ \ ℤ_{0}^{-}$ {\kern 1pt} a{\kern 1pt} \in {\kern 1pt} \mathbb{C}\backslash \mathbb{Z}_0^ - , then the solution of the following fractional equation (39) $z (y) - z_{0} Φ_{μ}^{⋆} (y, s, a) = - δ^{x}_{0} D_{y}^{- x}$ \mathfrak{z}(\mathfrak{y}) - {\mathfrak{z}_0}{\kern 1pt} \Phi_\mu^ \star (\mathfrak{y},s,a) = {\kern 1pt} - {\delta^\mathfrak{x}}{{\kern 1pt}_0}\mathcal{D}_\mathfrak{y}^{ - \mathfrak{x}} is given by (40) $z (y) = z_{0} \sum_{n = 0}^{\infty} \frac{{(μ)}_{n} Γ (n + 1) t^{n}}{n! {(n + a)}^{s}} ℰ_{x, n + 1} (- δ^{x} t^{x}) .$ \mathfrak{z}(\mathfrak{y}) = {\mathfrak{z}_0}{\kern 1pt} {\kern 1pt} \sum\limits_{n = 0}^\infty \frac{{{{(\mu )}_n}{\kern 1pt} \Gamma (n + 1){t^n}}}{{n!{\kern 1pt} {{(n + a)}^s}}}{\kern 1pt} {\mathcal{E}_{\mathfrak{x},n + 1}}( - {\delta^\mathfrak{x}}{\kern 1pt} {t^\mathfrak{x}}).

Upon taking σ = ρ = μ = 1 and $z = \frac{z}{λ}$ z = {\kern 1pt} \frac{z}{\lambda } . Then, the limit case of (4) when λ → ∞, would yield the Mittag-Leffler type function $ℰ_{κ, ω}^{(a)} (s; t)$ \mathcal{E}_{\kappa ,{\kern 1pt} \omega }^{(a)}(s;t) studied by Barnes [38], that is, (41) $\begin{matrix} ℰ_{κ, ω}^{(a)} (s; z) = \sum_{n = 0}^{\infty} \frac{z^{n}}{{(n + a)}^{s} Γ (ω + κ n)}, \\ (a, ω \in ℂ \ ℤ_{0}^{-}; ℜ (κ) > 0; s, z \in ℂ) . \end{matrix}$ \matrix{{\mathcal{E}_{\kappa ,{\kern 1pt} \omega }^{(a)}(s;z) = {\kern 1pt} \sum\limits_{n = 0}^\infty {\kern 1pt} \frac{{{z^n}}}{{{{(n + a)}^s}{\kern 1pt} \Gamma (\omega + \kappa n)}},}\\{(a,{\kern 1pt} \omega \in {\kern 1pt} \mathbb{C}\backslash \mathbb{Z}_0^ - ;{\kern 1pt} \Re (\kappa ) > 0;{\kern 1pt} s,{\kern 1pt} z{\kern 1pt} \in {\kern 1pt} \mathbb{C}).}}

Applying σ = ρ = μ = 1 and $z = \frac{z}{λ}$ z = {\kern 1pt} \frac{z}{\lambda } . Then, the limit case of (4) when λ → ∞ in Theorem 1, Theorem 2, and Theorem 3 obtained the following forms:

Corollary 13

If 𝔡 > 0, 𝔵 > 0; κ, δ ∈ ℂ, and 𝔡 ≠ δ be such that $a, ω \in ℂ \ ℤ_{0}^{-}$ {\kern 1pt} a,{\kern 1pt} \omega {\kern 1pt} \in {\kern 1pt} \mathbb{C}\backslash \mathbb{Z}_0^ - , then the solution of the following fractional equation (42) $z (y) - z_{0} ℰ_{κ, ω}^{(a)} (s; d^{x} y^{x}) = - δ^{x}_{0} D_{y}^{- x}$ \mathfrak{z}(\mathfrak{y}) - {\mathfrak{z}_0}{\kern 1pt} \mathcal{E}_{\kappa ,{\kern 1pt} \omega }^{(a)}(s;{\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}}) = {\kern 1pt} - {\delta^\mathfrak{x}}{{\kern 1pt}_0}\mathcal{D}_\mathfrak{y}^{ - \mathfrak{x}} is given by (43) $z (y) = z_{0} \sum_{n = 0}^{\infty} \frac{Γ (x n + 1) {(d^{x} y^{x})}^{n}}{{(n + a)}^{s} Γ (ω + κ n)} ℰ_{x, x n + 1} (- δ^{x} y^{x}) .$ \mathfrak{z}(\mathfrak{y}) = {\mathfrak{z}_0}{\kern 1pt} {\kern 1pt} \sum\limits_{n = 0}^\infty \frac{{\Gamma (\mathfrak{x}n + 1){{({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}})}^n}}}{{{{(n + a)}^s}{\kern 1pt} \Gamma (\omega + \kappa n)}}{\kern 1pt} {\mathcal{E}_{\mathfrak{x},\mathfrak{x}n + 1}}( - {\delta^\mathfrak{x}}{\kern 1pt} {\mathfrak{y}^\mathfrak{x}}).

Corollary 14

If 𝔡 > 0, 𝔵 > 0; κ ∈ ℂ be such that $a, ω \in ℂ \ ℤ_{0}^{-}$ {\kern 1pt} a,{\kern 1pt} \omega {\kern 1pt} \in {\kern 1pt} \mathbb{C}\backslash \mathbb{Z}_0^ - , then the solution of the following fractional equation (44) $z (y) - z_{0} ℰ_{κ, ω}^{(a)} (s; d^{x} y^{x}) = - d^{x}_{0} D_{y}^{- x}$ \mathfrak{z}(\mathfrak{y}) - {\mathfrak{z}_0}{\kern 1pt} \mathcal{E}_{\kappa ,{\kern 1pt} \omega }^{(a)}(s;{\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}}) = {\kern 1pt} - {\mathfrak{d}^\mathfrak{x}}{{\kern 1pt}_0}\mathcal{D}_\mathfrak{y}^{ - \mathfrak{x}} is given by (45) $z (y) = z_{0} \sum_{n = 0}^{\infty} \frac{Γ (x n + 1) {(d^{x} y^{x})}^{n}}{{(n + a)}^{s} Γ (ω + κ n)} ℰ_{x, x n + 1} (- d^{x} y^{x}) .$ \mathfrak{z}(\mathfrak{y}) = {\mathfrak{z}_0}{\kern 1pt} {\kern 1pt} \sum\limits_{n = 0}^\infty \frac{{\Gamma (\mathfrak{x}n + 1){{({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}})}^n}}}{{{{(n + a)}^s}{\kern 1pt} \Gamma (\omega + \kappa n)}}{\kern 1pt} {\mathcal{E}_{\mathfrak{x},\mathfrak{x}n + 1}}( - {\mathfrak{d}^\mathfrak{x}}{\kern 1pt} {\mathfrak{y}^\mathfrak{x}}).

Corollary 15

If κ, δ ∈ ℂ be such that $a, ω \in ℂ \ ℤ_{0}^{-}$ {\kern 1pt} a,{\kern 1pt} \omega {\kern 1pt} \in {\kern 1pt} \mathbb{C}\backslash \mathbb{Z}_0^ - , then the solution of the following fractional equation (46) $z (y) - z_{0} ℰ_{κ, ω}^{(a)} (s; y) = - δ^{x}_{0} D_{y}^{- x}$ \mathfrak{z}(\mathfrak{y}) - {\mathfrak{z}_0}{\kern 1pt} \mathcal{E}_{\kappa ,{\kern 1pt} \omega }^{(a)}(s;\mathfrak{y}) = {\kern 1pt} - {\delta^\mathfrak{x}}{{\kern 1pt}_0}\mathcal{D}_\mathfrak{y}^{ - \mathfrak{x}} is given by (47) $z (y) = z_{0} \sum_{n = 0}^{\infty} \frac{Γ (n + 1) t^{n}}{{(n + a)}^{s} Γ (ω + κ n)} ℰ_{x, n + 1} (- δ^{x} y^{x}) .$ \mathfrak{z}(\mathfrak{y}) = {\mathfrak{z}_0}{\kern 1pt} {\kern 1pt} \sum\limits_{n = 0}^\infty \frac{{\Gamma (n + 1){t^n}}}{{{{(n + a)}^s}{\kern 1pt} \Gamma (\omega + \kappa n)}}{\kern 1pt} {\mathcal{E}_{\mathfrak{x},n + 1}}( - {\delta^\mathfrak{x}}{\kern 1pt} {\mathfrak{y}^\mathfrak{x}}).

Finally, upon setting λ, μ, ω, σ, ρ, κ = 1 in the equation (4) gives the equation (1) [1, 2].

Choosing λ, μ, ω, σ, ρ, κ = 1 in Theorem 1, Theorem 2, and Theorem 3 obtained the following forms:

Corollary 16

If 𝔡 > 0; δ, 𝔵 ℂ, $a \in ℂ \ ℤ_{0}^{-}$ {\kern 1pt} a \in {\kern 1pt} \mathbb{C}\backslash \mathbb{Z}_0^ - , and 𝔡 = δ, then the solution of the following fractional equation (48) $z (y) - z_{0} Φ (d^{x} y^{x}, s, a) = - δ^{x}_{0} D_{y}^{- x}$ \mathfrak{z}(\mathfrak{y}) - {\mathfrak{z}_0}{\kern 1pt} \Phi ({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}},s,a) = {\kern 1pt} - {\delta^\mathfrak{x}}{{\kern 1pt}_0}\mathcal{D}_\mathfrak{y}^{ - \mathfrak{x}} is given by (49) $z (y) = z_{0} \sum_{n = 0}^{\infty} \frac{Γ (x n + 1) {(d^{x} y^{x})}^{n}}{{(n + a)}^{s}} ℰ_{x, x n + 1} (- δ^{x} y^{x}) .$ \mathfrak{z}(\mathfrak{y}) = {\mathfrak{z}_0}{\kern 1pt} {\kern 1pt} \sum\limits_{n = 0}^\infty \frac{{\Gamma (\mathfrak{x}n + 1){{({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}})}^n}}}{{{{(n + a)}^s}}}{\kern 1pt} {\mathcal{E}_{\mathfrak{x},\mathfrak{x}n + 1}}( - {\delta^\mathfrak{x}}{\kern 1pt} {\mathfrak{y}^\mathfrak{x}}).

Corollary 17

If 𝔡 > 0; 𝔵 ∈ ℂ, $a \in ℂ \ ℤ_{0}^{-}$ {\kern 1pt} a \in {\kern 1pt} \mathbb{C}\backslash \mathbb{Z}_0^ - , then the solution of the following fractional equation (50) $z (y) - z_{0} Φ (d^{x} y^{x}, s, a) = - d^{x}_{0} D_{y}^{- x}$ \mathfrak{z}(\mathfrak{y}) - {\mathfrak{z}_0}{\kern 1pt} \Phi ({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}},s,a) = {\kern 1pt} - {\mathfrak{d}^\mathfrak{x}}{{\kern 1pt}_0}\mathcal{D}_\mathfrak{y}^{ - \mathfrak{x}} is given by (51) $z (y) = z_{0} \sum_{n = 0}^{\infty} \frac{Γ (x n + 1) {(d^{x} y^{x})}^{n}}{{(n + a)}^{s}} ℰ_{x, x n + 1} (- d^{x} y^{x}) .$ \mathfrak{z}(\mathfrak{y}) = {\mathfrak{z}_0}{\kern 1pt} {\kern 1pt} \sum\limits_{n = 0}^\infty \frac{{\Gamma (\mathfrak{x}n + 1){{({\mathfrak{d}^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}})}^n}}}{{{{(n + a)}^s}}}{\kern 1pt} {\mathcal{E}_{\mathfrak{x},\mathfrak{x}n + 1}}( - {\mathfrak{d}^\mathfrak{x}}{\kern 1pt} {\mathfrak{y}^\mathfrak{x}}).

Corollary 18

If δ ∈ ℂ, $a \in ℂ \ ℤ_{0}^{-}$ {\kern 1pt} a \in {\kern 1pt} \mathbb{C}\backslash \mathbb{Z}_0^ - , then the solution of the following fractional equation (52) $z (y) - z_{0} Φ (y, s, a) = - δ^{x}_{0} D_{y}^{- x}$ \mathfrak{z}(\mathfrak{y}) - {\mathfrak{z}_0}{\kern 1pt} \Phi (\mathfrak{y},s,a) = {\kern 1pt} - {\delta^\mathfrak{x}}{{\kern 1pt}_0}\mathcal{D}_\mathfrak{y}^{ - \mathfrak{x}} is given by (53) $z (y) = z_{0} \sum_{n = 0}^{\infty} \frac{Γ (n + 1) t^{n}}{{(n + a)}^{s}} ℰ_{x, n + 1} (- δ^{x} t^{x}) .$ \mathfrak{z}(\mathfrak{y}) = {\mathfrak{z}_0}{\kern 1pt} {\kern 1pt} \sum\limits_{n = 0}^\infty \frac{{\Gamma (n + 1){t^n}}}{{{{(n + a)}^s}}}{\kern 1pt} {\mathcal{E}_{\mathfrak{x},n + 1}}( - {\delta^\mathfrak{x}}{\kern 1pt} {t^\mathfrak{x}}).

Numerical result and graphics

This section obtains the general solutions of the fractional Kinetic equation involving the generalized Hurwitz-Lerch Zeta function using Laplace transform.

Application Let λ, μ, ω, ρ, κ, σ, a, s = 1, 𝔡, δ = 4, and 𝔷₀ = 3 in the (13). Then, we have the following equation: (54) $z (y) - 3 Φ_{1, 1; 1}^{(1, 1, 1)} (4^{0, 4} y^{x}, 1, 1) = - 4^{x}_{0} D_{y}^{- x} .$ \mathfrak{z}(\mathfrak{y}) - 3{\kern 1pt} \Phi_{1,{\kern 1pt} 1;{\kern 1pt} 1}^{(1,{\kern 1pt} 1,{\kern 1pt} 1)}({4^{0,4}}{\mathfrak{y}^\mathfrak{x}},1,1) = {\kern 1pt} - {4^\mathfrak{x}}{{\kern 1pt}_0}\mathcal{D}_\mathfrak{y}^{ - \mathfrak{x}}.

Applying the Laplace transform to the both sides of (54), we get (55) $\begin{array}{l} L {z (y); p} = 3 L {Φ_{1, 1; 1}^{(1, 1, 1)} (4^{x} y^{x}, 1, 1); p} - 4^{x} L {_{0} D_{y}^{- x} z (y); p} \\ z (p) = 3 \sum_{n = 0}^{\infty} \frac{{(1)}_{n} {(1)}_{n} Γ (x n + 1) 4^{x n}}{{(1)}_{n} (n + 1)!} p^{- (x n + 1)} \sum_{r = 0}^{\infty} {[- {(\frac{p}{4})}^{- x}]}^{r} . \end{array}$ \matrix{{\mathfrak{L}{\kern 1pt} \{ \mathfrak{z}(\mathfrak{y});p\} = 3{\kern 1pt} \mathfrak{L}{\kern 1pt} \{ {\kern 1pt} \Phi_{1,{\kern 1pt} 1;{\kern 1pt} 1}^{(1,{\kern 1pt} 1,{\kern 1pt} 1)}({4^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}},1,1);p\} - {\kern 1pt} {4^\mathfrak{x}}{\kern 1pt} \mathfrak{L}{\kern 1pt} {\{_0}\mathcal{D}_\mathfrak{y}^{ - \mathfrak{x}}{\kern 1pt} \mathfrak{z}(\mathfrak{y});p\} }\\{\mathfrak{z}(p) = 3{\kern 1pt} {\kern 1pt} \sum\limits_{n = 0}^\infty \frac{{{{(1)}_n}{\kern 1pt} {{(1)}_n}{\kern 1pt} \Gamma (\mathfrak{x}n + 1){\kern 1pt} {4^{\mathfrak{x}n}}}}{{{{(1)}_n}{\kern 1pt} (n + 1)!}}{\kern 1pt} {p^{ - (\mathfrak{x}n + 1)}}\sum\limits_{r = 0}^\infty {{[ - {{(\frac{p}{4})}^{ - \mathfrak{x}}}]}^r}.}}

Then, using the inverse Laplace transform of (55), we obtain (56) $\begin{array}{l} L^{- 1} {z (p)} = 3 \sum_{n = 0}^{\infty} \frac{{(1)}_{n} {(1)}_{n} Γ (x n + 1) 4^{x n}}{{(1)}_{n} (n + 1)!} \\ \times L^{- 1} {p^{- (x n + 1)} \sum_{r = 0}^{\infty} {[- {(\frac{p}{4})}^{- x}]}^{r}} \\ z (y) = \sum_{n = 0}^{\infty} \frac{{(1)}_{n} {(1)}_{n} Γ (x n + 1) {(4^{x} y^{x})}^{n}}{{(1)}_{n} (n + 1)!} \sum_{r = 0}^{\infty} {(- 1)}^{r} 1^{x r} \frac{y^{x r}}{Γ (x n + x r + 1)} . \end{array}$ \matrix{{{L^{ - 1}}\{ \mathfrak{z}(p)\} = {\kern 1pt} 3{\kern 1pt} \sum\limits_{n = 0}^\infty \frac{{{{(1)}_n}{\kern 1pt} {{(1)}_n}{\kern 1pt} \Gamma (\mathfrak{x}n + 1){\kern 1pt} {4^{\mathfrak{x}n}}}}{{{{(1)}_n}{\kern 1pt} (n + 1)!}}}\\{\,\,\,\,\,\,\,\,\,\,\, \times {\kern 1pt} {L^{ - 1}}\{ {p^{ - (\mathfrak{x}n + 1)}}\sum\limits_{r = 0}^\infty {{[ - {{(\frac{p}{4})}^{ - \mathfrak{x}}}]}^r}\} }\\{\mathfrak{z}(\mathfrak{y}) = {\kern 1pt} \sum\limits_{n = 0}^\infty \frac{{{{(1)}_n}{\kern 1pt} {{(1)}_n}{\kern 1pt} \Gamma (\mathfrak{x}n + 1){{({4^\mathfrak{x}}{\mathfrak{y}^\mathfrak{x}})}^n}}}{{{{(1)}_n}{\kern 1pt} (n + 1)!}}\sum\limits_{r = 0}^\infty {\kern 1pt} {{( - 1)}^r}{\kern 1pt} {1^{\mathfrak{x}r}}{\kern 1pt} \frac{{{\mathfrak{y}^{\mathfrak{x}r}}}}{{\Gamma (\mathfrak{x}n + \mathfrak{x}r + 1)}}.}}

In Figure (1), we plotted the 2D graphs of solution yielded by (56), for the different values of 𝔵 using Mathematica.

Conclusions

The fractional Kinetic equation involving the generalized Hurwitz-Lerch Zeta function was studied using the Laplace transform. The results obtained in this study have remarkable significance as the solution of the equations are general and many new and known solutions of FKEs can be reproduced involving various types of special functions.

Declarations

5.1

Conflict of interest

The authors hereby declare that there is no conflict of interests regarding the publication of this paper.

5.2

Funding

Not applicable.

5.3

Author's contribution

O.Y.-Writing Original Draft; R.Ş.-Supervision. K.S.N.-Writing Review Editing, Methodology. All authors read and approved the final submitted version of this manuscript.

5.4

Acknowledgement

We thank the reviewers for their constructive comments in improving the quality of this paper.

5.5

Data availability statement

All data that support the findings of this study are included within the article.

5.6

Using of AI tools

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

New results for generalized Hurwitz-Lerch Zeta functions using Laplace transform

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