A New Generalization of Pochhammer Symbol and Its Applications

The classical Pochhammer symbol (λ)_ν is given as follows: [1, 7, 9, 10, 18, 22, 23, 26, 29, 30, 33] (1) $\begin{array}{l} {(λ)}_{ν} & = \frac{Γ (λ + ν)}{Γ (λ)} & (λ, ν \in ℂ \ ℤ_{0}^{-}) \\ = {\begin{array}{l} 1 \\ λ (λ + 1) \dots (λ + n - 1) \end{array} & \begin{array}{l} (ν = 0) \\ (ν = n \in ℕ) \end{array} . \end{array}$ \matrix{ {(\lambda )_\nu } \hfill & { = {{\Gamma (\lambda + \nu )} \over {\Gamma (\lambda )}}} \hfill & {(\lambda, \, \nu \in {\mathbb{C}} \backslash {\mathbb{Z}}_0^ - )} \hfill \cr {} \hfill & { = \left\{ {\matrix{ 1 \hfill \cr {\lambda (\lambda + 1) \cdots (\lambda + n - 1)} \hfill \cr } } \right.} \hfill & {\matrix{ {\,\,\,\,\,(\nu = 0)} \hfill \cr {\,\,\,\,\,(\nu = n \in {\mathbb{N}})}} .}} and Γ(λ ) is the familiar Gamma function whose Euler’s integral is (see, e.g., [1, 7, 9, 10, 18, 22, 23, 30, 33]) (2) $\begin{matrix} Γ (z) = \int_{0}^{\infty} e^{- t} t^{z - 1} dt & (ℜ (z) > 0) \end{matrix} .$ \matrix{ {\Gamma (z) = \int\limits_0^\infty \, e^{ - t} \, t^{z - 1} \, dt} & {(\Re (z) > 0)}} . From (1) and (2), it is easy to see the following integral formula (3) $\begin{matrix} {(λ)}_{ν} = \frac{1}{Γ (λ)} \int_{0}^{\infty} e^{- t} t^{λ + ν - 1} dt & (ℜ (λ + ν) > 0) \end{matrix} .$ \matrix{ {(\lambda )_\nu = {1 \over {\Gamma (\lambda )}} \, \int\limits_0^\infty \, e^{ - t} \, t^{\lambda + \nu - 1} \, dt} & {(\Re (\lambda + \nu ) > 0)}} .

Here and in the following, let ℂ, $ℤ_{0}^{-}$ {\mathbb{Z}}_0^ - , and ℕ be the sets of complex numbers, non-positive integers and positive integers, respectively and assume that min {ℜ(p), ℜ(q), ℜ(κ), ℜ(μ)} > 0. Recently, various generalization of beta functions have been introduced and investigated (see, e.g., [2, 3, 4, 5, 6, 9, 13, 14, 15, 16, 17, 19, 20, 21, 24, 25, 26, 28, 29] and the references cited therein). Very recently, Şahin et al. [31] introduced and studied following generalization of the extended gamma function as follows: (4) $\begin{matrix} Γ_{p, q}^{(κ, μ)} (z) = \int_{0}^{\infty} t^{z - 1} exp (- \frac{t^{κ}}{p} - \frac{q}{t^{μ}}) dt, \\ (ℜ (z) > 0, ℜ (p) > 0, ℜ (q) > 0, ℜ (κ) > 0, ℜ (μ) > 0) . \end{matrix}$ \matrix{ {\Gamma _{p,q}^{(\kappa, \mu )} (z) = \, \int_0^\infty \, t^{z - 1} \, \exp \left( { - {{t^\kappa } \over p} - {q \over {t^\mu }}} \right)dt,} \cr {(\Re (z) > 0, \, \Re (p) > 0, \, \Re (q) > 0, \, \Re (\kappa ) > 0, \, \Re (\mu ) > 0).}} It is easily seen that the special cases of (4) returns to other forms of gamma functions. For example, $Γ_{1, 0}^{(1, 1)} (x) = Γ (x)$ \Gamma _{1,0}^{(1,1)} (x)\, = \,\Gamma (x) , $Γ_{1, q}^{(1, 1)} (x) = Γ_{q} (x)$ \Gamma _{1,q}^{(1,1)} (x)\, = \,\Gamma _q (x) .

First, by selecting a known generalization of the gamma function (4), systematicaly, we goal to introduce new Pochhammer symbol by using the gamma function (4). Also, we give some properties for the extended Pochhammer symbol. Next, using this Pochhammer symbol, we define a new generalization of the extended hypergeometric functions one or several variables such as Gauss hypergeometric function, confluent hypergeometric function, Appell hypergeometric function, Humbert hypergeometric function. Finally, for this a new generalization of the extended hypergeometric functions, we give some properties such as integral representations, derivative formulas and recuurence relations.

A New Generalization of the Pochhammer Symbol

In this section, we denote a new generalization of Pochhammer symbol (5). Also, we give some useful properties.

Definition 1

Let λ, μ ∈ ℂ and ℜ(p) > 0, ℜ(q) > 0, ℜ(κ) > 0, ℜ(μ) > 0, the generalization of the extended Pochhammer symbol (λ; p,q;κ, μ)_ν is given by (5) ${(λ; p, q; κ, μ)}_{ν} : = {\begin{array}{l} \frac{Γ_{p, q}^{(κ, μ)} (λ + ν)}{Γ (λ)} & , ℜ (p) > 0, ℜ (q) > 0, ℜ (κ) > 0, ℜ (μ) > 0, \\ {(λ)}_{ν} & , p = 1, q = 0, κ = 1, μ = 0, \end{array}$ (\lambda ;p,q;\kappa, \mu )_\nu : = \left\{ {\matrix{ {{{\Gamma _{p,q}^{(\kappa, \mu )} (\lambda + \nu )} \over {\Gamma (\lambda )}}} \hfill & {, \, \Re (p) > 0, \, \Re (q) > 0, \, \Re (\kappa ) > 0, \, \Re (\mu ) > 0,} \hfill \cr \, \, \, \, \, \, \, {(\lambda )_\nu } \hfill & {, \, p = 1, \, q = 0, \, \kappa = 1, \, \mu = 0,} \hfill \cr } } \right. where $Γ_{p, q}^{(κ, μ)}$ \Gamma _{p,q}^{(\kappa, \mu )} is the generalization of the extended gamma function (4) [31

Theorem 1

For the generalization of the Pochhammer symbol(5)following integral representation holds true:(6) $\begin{matrix} {(λ; p, q; κ, μ)}_{ν} : = \frac{1}{Γ (λ)} \int_{0}^{\infty} t^{λ + ν - 1} exp (- \frac{t^{κ}}{p} - \frac{q}{t^{μ}}) dt \\ (ℜ (p) > 0, ℜ (q) > 0, ℜ (κ) > 0, ℜ (μ) > 0) . \end{matrix}$ \matrix{ {(\lambda ;p,q;\kappa, \mu )_\nu : = {1 \over {\Gamma (\lambda )}} \, \int_0^\infty \, t^{\lambda + \nu - 1} \, \exp \left( { - {{t^\kappa } \over p} - {q \over {t^\mu }}} \right)dt} \cr {(\Re (p) > 0, \, \Re (q) > 0, \, \Re (\kappa ) > 0, \, \Re (\mu ) > 0).}}

Proof

Using the equality (4) in the definition of the (5), we get the desired result (6).

Theorem 2

Let λ, m, n ∈ ℂ. Then, (7) ${(λ; p, q; κ, μ)}_{n + m} : = {(λ)}_{n} {(λ + n; p, q; κ, μ)}_{m}$ (\lambda ;p,q;\kappa, \mu )_{n + m} : = (\lambda )_n (\lambda + n;p,q;\kappa, \mu )_m

Proof

From the equations (1) and (5), we obtain that (8) $\begin{array}{l} {(λ; p, q; κ, μ)}_{m + n} & : = \frac{Γ_{p, q}^{(κ, μ)} (λ + m + n)}{Γ (λ)} = \frac{Γ (λ + n)}{Γ (λ + n)} \frac{Γ_{p, q}^{(κ, μ)} (λ + m + n)}{Γ (λ)} \\ = {(λ)}_{n} {(λ + n; p, q; κ, μ)}_{m} \end{array}$ \matrix{ {(\lambda ;p,q;\kappa, \mu )_{m + n} } \hfill & {: = {{\Gamma _{p,q}^{(\kappa, \mu )} (\lambda + m + n)} \over {\Gamma (\lambda )}} = {{\Gamma (\lambda + n)} \over {\Gamma (\lambda + n)}}{{\Gamma _{p,q}^{(\kappa, \mu )} (\lambda + m + n)} \over {\Gamma (\lambda )}}} \hfill \cr {} \hfill & { = (\lambda )_n (\lambda + n;p,q;\kappa, \mu )_m }}

By appealing the well-known properties of the classical Pochhammer symbol in (7) following features of generalization of the Pochhammer symbol can be easily obtained.

Corollary 3

Let k,l,m,n ∈ ℕ₀and N ∈ ℕ. Then, $\begin{array}{l} {(λ; p, q; κ, μ)}_{m + n + l} : = {(λ)}_{m} {(λ + m)}_{n} {(λ + m + n; p, q; κ, μ)}_{l} \\ {(λ; p, q; κ, μ)}_{m - n + l} : = \frac{{(- 1)}^{n} {(λ)}_{m}}{{(1 - λ - m)}_{n}} {(λ + m - n; p, q; κ, μ)}_{l} \\ {(λ; p, q; κ, μ)}_{2 m + l} : = 2^{2 m} {(\frac{λ}{2})}_{m} {(\frac{λ + 1}{2})}_{m} {(λ + 2 m; p, q; κ, μ)}_{l} \\ {(λ; p, q; κ, μ)}_{N m + l} : = N^{N m} {(\frac{λ}{N})}_{m} {(\frac{λ + 1}{N})}_{m} \dots {(\frac{λ + N - 1}{N})}_{m} {(λ + N m; p, q; κ, μ)}_{l} \\ {(λ + n; p, q; μ)}_{n + 1} : = {(λ + n)}_{n} {(λ + 2 n; p, q; κ, μ)}_{l} = \frac{{(λ)}_{2 n}}{{(λ)}_{n}} {(λ + 2 n; p, q, κ, μ)}_{l} \\ {(λ + m; p, q; κ, μ)}_{n + l} : = \frac{{(λ)}_{n} {(λ + n)}_{m}}{{(λ)}_{m}} {(λ + m + n; p, q; κ, μ)}_{l} \\ {(λ + km; p, q; κ, μ)}_{kn + l} : = \frac{{(λ)}_{km + kn}}{{(λ)}_{km}} {(λ + km + kn; p, q; κ, μ)}_{l} \\ {(λ - n; p, q; κ, μ)}_{n + 1} : = {(- 1)}^{n} {(1 - λ)}_{n} {(λ; p, q; κ, μ)}_{l} \\ {(λ - m; p, q; κ, μ)}_{n + 1} : = \frac{{(1 - λ)}_{m} {(λ)}_{n}}{{(1 - λ - n)}_{m}} {(λ + n -; p, q, κ, μ)}_{l} \\ {(λ - km; p, q; κ, μ)}_{kn + 1} : = {(- 1)}^{km} {(λ)}_{kn - km} {(1 - λ)}_{km} {(λ + kn - km; p, q, κ, μ)}_{l} \\ {(λ + m; p, q; κ, μ)}_{n - m + l} : = \frac{{(λ)}_{n}}{{(λ)}_{m}} {(λ + n +; p, q; κ, μ)}_{l} \\ {(λ - m; p, q; κ, μ)}_{n - m + 1} : = {(- 1)}^{m} \frac{{(λ)}_{n} {(1 - λ)}_{m}}{{(1 - λ - n)}_{2 m}} {(λ + n - 2 m; p, q, κ, μ)}_{l} \\ {(- λ; p, q; κ, μ)}_{n + 1} : = {(- 1)}^{n} (λ - n + 1) {(- λ + n; p, q, κ, μ)}_{l} \end{array}$ \eqalign{ & \left( {\lambda ;p,q;\kappa ,\mu } \right)_{m + n + l} : = \left( \lambda \right)_m \left( {\lambda + m} \right)_n \left( {\lambda + m + n;p,q;\kappa ,\mu } \right)_l \cr & \left( {\lambda ;p,q;\kappa ,\mu } \right)_{m - n + l} : = {{\left( { - 1} \right)^n \left( \lambda \right)_m } \over {\left( {1 - \lambda - m} \right)_n }}\left( {\lambda + m - n;p,q;\kappa ,\mu } \right)_l \cr & \left( {\lambda ;p,q;\kappa ,\mu } \right)_{2m + l} : = 2^{2m} \left( {{\lambda \over 2}} \right)_m \left( {{{\lambda + 1} \over 2}} \right)_m \left( {\lambda + 2m;p,q;\kappa ,\mu } \right)_l \cr & \left( {\lambda ;p,q;\kappa ,\mu } \right)_{{\rm {N}}m + l} : = {\rm {N}}^{{\rm {N}}m} \left( {{\lambda \over {\rm {N}}}} \right)_m \left( {{{\lambda + 1} \over {\rm {N}}}} \right)_m \ldots \left( {{{\lambda + {\rm {N}} - 1} \over {\rm {N}}}} \right)_m \left( {\lambda + {\rm {N}}m;p,q;\kappa ,\mu } \right)_l \cr & \left( {\lambda + n;p,q;\mu } \right)_{n + 1} : = \left( {\lambda + n} \right)_n \left( {\lambda + 2n;p,q;\kappa ,\mu } \right)_l = {{\left( \lambda \right)_{2n} } \over {\left( \lambda \right)_n }}\left( {\lambda + 2n;p,q,\kappa ,\mu } \right)_l \cr & \left( {\lambda + m;p,q;\kappa ,\mu } \right)_{n + l} : = {{\left( \lambda \right)_n \left( {\lambda + n} \right)_m } \over {\left( \lambda \right)_m }}\left( {\lambda + m + n;p,q;\kappa ,\mu } \right)_l \cr & \left( {\lambda + km;p,q;\kappa ,\mu } \right)_{kn + l} : = {{\left( \lambda \right)_{km + kn} } \over {\left( \lambda \right)_{km} }}\left( {\lambda + km + kn;p,q;\kappa ,\mu } \right)_l \cr & \left( {\lambda - n;p,q;\kappa ,\mu } \right)_{n + 1} : = \left( { - 1} \right)^n \left( {1 - \lambda } \right)_n \left( {\lambda ;p,q;\kappa ,\mu } \right)_l \cr & \left( {\lambda - m;p,q;\kappa ,\mu } \right)_{n + 1} : = {{\left( {1 - \lambda } \right)_m \left( \lambda \right)_n } \over {\left( {1 - \lambda - n} \right)_m }}\left( {\lambda + n - ;p,q,\kappa ,\mu } \right)_l \cr & \left( {\lambda - km;p,q;\kappa ,\mu } \right)_{kn + 1} : = \left( { - 1} \right)^{km} \left( \lambda \right)_{kn - km} \left( {1 - \lambda } \right)_{km} \left( {\lambda + kn - km;p,q,\kappa ,\mu } \right)_l \cr & \left( {\lambda + m;p,q;\kappa ,\mu } \right)_{n - m + l} : = {{\left( \lambda \right)_n } \over {\left( \lambda \right)_m }}\left( {\lambda + n + ;p,q;\kappa ,\mu } \right)_l \cr & \left( {\lambda - m;p,q;\kappa ,\mu } \right)_{n - m + 1} : = \left( { - 1} \right)^m {{\left( \lambda \right)_n \left( {1 - \lambda } \right)_m } \over {\left( {1 - \lambda - n} \right)_{2m} }}\left( {\lambda + n - 2m;p,q,\kappa ,\mu } \right)_l \cr & \left( { - \lambda ;p,q;\kappa ,\mu } \right)_{n + 1} : = \left( { - 1} \right)^n \left( {\lambda - n + 1} \right)\left( { - \lambda + n;p,q,\kappa ,\mu } \right)_l .}

Remark 1

Taking p = κ = μ = 1 in the Corollary 3, it is easily seen that the special case of extended Pochhammer symbol [29].

A New Generalization of the extended hypergeometric function

According to the generalization of the extended Pochhammer symbol (λ; p,q; κ, μ)_n (n ∈ ℕ₀), a generalization of the extended hypergeometric function _rF_s of r numerator parameters a₁,⋯,a_r and s denominator parameters b₁,⋯,b_s can be given as follows: (9) $_{r} F_{s} [(a_{1}; p, q; κ, μ), a_{2}, \dots, a_{r} b_{1}, b_{2}, \dots, b_{s}; z] : = \sum_{n = 0}^{\infty} \frac{{(a_{1}; p, q; κ, μ)}_{n} {(a_{2})}_{n} \dots {(a_{r})}_{n}}{{(b_{1})}_{n} \dots {(b_{s})}_{n}} \cdot \frac{z^{n}}{n!},$ _r F_s \left[ {(a_1 ;p,q;\kappa, \mu ),a_2, \, \cdots \,, a_r \, b_1, b_2, \cdots, b_s \, ;z} \right]: = \sum\limits_{n = 0}^\infty {{(a_1 ;p,q;\kappa, \mu )_n (a_2 )_n \cdots (a_r )_n } \over {(b_1 )_n \cdots (b_s )_n }} \cdot {{z^n } \over {n!}}, on condition that the series on the right-hand side converges, it making sense that a_j ∈ ℂ ( j = 1,..., r) and $b_{j} \in ℂ \ ℤ_{0}^{-} (j = 1, ..., s; ℤ_{0}^{-} = {0, - 1, - 2, ...})$ b_j \in {\mathbb{C}} \backslash {\mathbb{Z}}_0^ - (j = 1,...,s;\, \, {\mathbb{Z}}_0^ - = \{ 0, - 1, - 2,...\} ) .

Particularly, the corresponding generalization of the extended confluent hypergeometric function $Φ_{p, q}^{κ, μ}$ \Phi _{p,q}^{\kappa, \mu } and the Gauss hypergeometric function $F_{p, q}^{κ, μ}$ F_{p,q}^{\kappa, \mu } are given by (10) $Φ_{p, q}^{κ, μ} (a; b; z) : = \sum_{n = 0}^{\infty} \frac{{(a; p, q; κ, μ)}_{n}}{{(b)}_{n}} \cdot \frac{z^{n}}{n!}$ \Phi _{p,q}^{\kappa, \mu } (a;b;z): = \, \sum\limits_{n = 0}^\infty \, {{(a;p,q;\kappa, \mu )_n } \over {(b)_n }} \cdot {{z^n } \over {n!}} and (11) $F_{p, q}^{κ, μ} (a, b, c; z) : = \sum_{n = 0}^{\infty} \frac{{(a; p, q; κ, μ)}_{n} {(b)}_{n}}{{(c)}_{n}} \cdot \frac{z^{n}}{n!},$ F_{p,q}^{\kappa, \mu } (a,b,c;z): = \sum\limits_{n = 0}^\infty \, {{(a;p,q;\kappa, \mu )_n (b)_n } \over {(c)_n }} \cdot {{z^n } \over {n!}}, respectively.

Theorem 4

The following integral representation holds true:(12) $\begin{array}{l} _{r} F_{s} [(a_{1}; p, q; κ, μ), a_{2}, \dots, a_{r} b_{1}, b_{2}, \dots, b_{s}; z] \\ : = \frac{1}{Γ (a_{1})} \int_{0}^{\infty} t^{a_{1} - 1} exp (- \frac{t^{κ}}{p} - \frac{q}{t^{μ}}) \times_{r - 1} F_{s} [a_{2}, \dots, a_{r} b_{1}, b_{2}, \dots, b_{s}; zt] dt, \\ (ℜ (p) > 0, ℜ (q) > 0, ℜ (κ) > 0, ℜ (μ) > 0; ℜ (b_{s}) > ℜ (a_{r}) > 0,) . \end{array}$ \eqalign{ & _r F_s \left[ {(a_1 ;\,p,q;\,\kappa, \,\mu ),a_2, \, \cdots \,, a_r \, b_1, b_2, \cdots, b_s \, ;z} \right] \cr & : = {1 \over {\Gamma (a_1 )}}\int\limits_0^\infty t^{a_1 - 1} \exp \left( { - {{t^\kappa } \over p} - {q \over {t^\mu }}} \right) \times _{r - 1} F_s \left[ {a_2, \, \cdots \,, a_r \, b_1, b_2, \cdots, b_s \, ;zt} \right]dt, \cr & \, \, \, \, \, \, \, \, (\Re (p) > 0, \, \, \,\Re (q) > 0, \, \,\Re (\kappa ) > 0, \, \,\Re (\mu ) > 0; \, \,\Re (b_s ) > \Re (a_r ) > 0,).}

Proof

Using the integral representation given by (6) in the definition (9), we led to desired result (12).

Theorem 5

The following integral representation holds true:(13) $\begin{array}{l} \begin{array}{l} _{r} F_{s} [(a_{1}; p, q; κ, μ), a_{2}, \dots, a_{r} b_{1}, b_{2}, \dots, b_{s}; z] & : = \frac{1}{B (a_{r}, b_{s} - a_{r})} \int_{0}^{1} t^{a_{r} - 1} {(1 - t)}^{b_{s} - a_{r} - 1} \\ \times_{r - 1} F_{s - 1} [(a_{1}; p, q; κ, μ), a_{2}, \dots, a_{r - 1} b_{1}, b_{2}, \dots, b_{s - 1}; zt] dt, \end{array} \\ (ℜ (p) > 0, ℜ (q) > 0, ℜ (κ) > 0, ℜ (μ) > 0; ℜ (b_{s}) > ℜ (a_{r}) > 0) . \end{array}$ \eqalign{ & \matrix{ {_r F_s \left[ {(a_1 ;p,q;\kappa, \mu ),a_2, \, \cdots \,, a_r \, b_1, b_2, \cdots, b_s \, ;z} \right]} \hfill & {: = {1 \over {B(a_r, b_s - a_r )}}\int\limits_0^1 \, t^{a_r - 1} (1 - t)^{b_s - a_r - 1} } \hfill \cr {} \hfill & { \times \, _{r - 1} F_{s - 1} \left[ {(a_1 ;p,q;\kappa, \mu ),a_2, \, \cdots \,, a_{r - 1} \, b_1, b_2, \cdots, b_{s - 1} \, ;zt} \right]dt,} \hfill \cr } \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\Re (p) > 0, \, \,\Re (q) > 0, \, \,\Re (\kappa ) > 0,\, \, \Re (\mu ) > 0; \, \,\Re (b_s ) > \Re (a_r ) > 0).}

Proof

The classical Beta function B(α,β ) defined by [1, 7, 9, 10, 18, 22, 23, 26, 29, 30, 33], (14) $B (α, β) = {\begin{array}{l} \int_{0}^{1} t^{α - 1} {(1 - t)}^{β - 1} dt & (min {ℝ (α), ℝ (β)} > 0) \\ \frac{Γ (α) Γ (β)}{Γ (α + β)} & (α, β \in ℂ \ ℤ_{0}^{-}) . \end{array}$ B(\alpha, \beta ) = \left\{ {\matrix{ {\int\limits_0^1 t^{\alpha - 1} (1 - t)^{\beta - 1} dt} \hfill & {(\min \{ {\mathbb{R}}(\alpha ),{\mathbb{R}}(\beta )\} > 0)} \hfill \cr {{{\Gamma (\alpha )\Gamma (\beta )} \over {\Gamma (\alpha + \beta )}}} \hfill & {(\alpha, \beta \in {\mathbb{C}} \backslash {\mathbb{Z}}_0^ - ).}} } \right. Also, we have the following equation (15) $\begin{matrix} \frac{{(a_{r})}_{n}}{{(b_{s})}_{n}} = \frac{1}{B (a_{r}, b_{s} - a_{r})} \int_{0}^{1} t^{a_{r} + n - 1} {(1 - t)}^{b_{s} - a_{r} - 1} dt, \\ (ℝ (b_{s}) > ℝ (a_{r}) > 0; n \in ℕ_{0}) \end{matrix}$ \matrix{ {{{(a_r )_n } \over {(b_s )_n }} = {1 \over {B(a_r,\ b_s - a_r )}}\int\limits_0^1 t^{a_r + n - 1} (1 - t)^{b_s - a_r - 1} dt,} \cr {({\mathbb{R}}(b_s ) > {\mathbb{R}}(a_r ) > 0;n \in {\mathbb{N}}_0 )}}

Using the equalities (14), (15) in the generization of the extended hypergeometric function (9), we get the desired result (13).

Corollary 6

Each of the integral representations hold true:(16) $Φ_{p, q}^{κ, μ} (a; b; z) = \frac{1}{Γ (a)} \int_{0}^{\infty} t^{a - 1} exp (- \frac{t^{κ}}{p} - \frac{q}{t^{μ}})_{0} F_{1} (-; b; zt) dt,$ \Phi _{p,q}^{\kappa, \mu } (a;b;z) = {1 \over {\Gamma (a)}}\int\limits_0^\infty t^{a - 1} \exp \left( { - {{t^\kappa } \over p} - {q \over {t^\mu }}} \right) \, _0 F_1 ( - ;b;zt)dt,(17) $F_{p, q}^{κ, μ} (a, b, c; z) = \frac{1}{Γ (a)} \int_{0}^{\infty} t^{a - 1} exp (- \frac{t^{κ}}{p} - \frac{q}{t^{μ}})_{1} F_{1} (b; c; zt) dt,$ F_{p,q}^{\kappa, \mu } (a,b,c;z) = {1 \over {\Gamma (a)}}\int\limits_0^\infty t^{a - 1} \exp \left( { - {{t^\kappa } \over p} - {q \over {t^\mu }}} \right) \, _1 F_1 (b;c;zt)dt,and(18) $F_{p, q}^{κ, μ} (a, b, c; z) = \frac{1}{B (b, c - b)} \int_{0}^{1} t^{b - 1} {(1 - t)}^{c - b - 1}_{1} F_{0} ((a; p, q; κ, μ); -; zt) dt,$ F_{p,q}^{\kappa, \mu } (a,b,c;z) = {1 \over {B(b,c - b)}} \, \int\limits_0^1 t^{b - 1} (1 - t)^{c - b - 1} \, _1 F_0 ((a;p,q;\kappa, \mu ); - ;zt)dt,on condition that the integrals involved are convergent.

Theorem 7

The following derivative formula holds true:(19) $\begin{matrix} \frac{d^{n}}{{dz}^{n}} {_{r} F_{s} [(a_{1}; p, q; κ, μ), a_{2}, \dots, a_{r} b_{1}, b_{2}, \dots, b_{s}; z]} : = \frac{{(a_{1})}_{n} {(a_{2})}_{n} \dots {(a_{r})}_{n}}{{(b_{1})}_{n} \dots {(b_{s})}_{n}} \\ \times_{r} F_{s} [(a_{1} + n; p, q; κ, μ), a_{2} + n, \dots, a_{r} + n b_{1} + n, b_{2} + n, \dots, b_{s} + n; z] \\ (ℜ (p) > 0, ℜ (q) > 0, ℜ (κ) > 0, ℜ (μ) > 0; n \in ℕ_{0}) . \end{matrix}$ \eqalign{ & {{{d^n } \over {dz^n }}\left\{ {_r F_s \left[ {(a_1 ;p,q;\kappa, \mu ),a_2, \, \cdots \,, a_r \, b_1, b_2, \cdots, b_s \, ;z} \right]} \right\}\,:\, = \,{{(a_1 )_n (a_2 )_n \cdots \, (a_r )_n } \over {(b_1 )_n \cdots \, (b_s )_n }}} \cr & { \times _r F_s \left[ {(a_1 \, + \,n;p,q;\kappa, \mu ),a_2 \, + \,n, \, \cdots \,, a_r \, + \,n \, b_1 \, + \,n,b_2 \, + \,n, \cdots, b_s \, + \,n \, ;z} \right]} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, {(\Re (p) > 0, \, \,\Re (q) > 0,\, \, \Re (\kappa ) > 0, \, \,\Re (\mu ) > 0; \, \,n \in {\mathbb{N}}_0 ).}}

Proof

Differentiating (9) with respect to z and then replacing n → n + 1 in the right-hand side term, we obtain (20) $\begin{array}{l} \frac{d}{dz} {_{r} F_{s} [(a_{1}; p, q; κ, μ), a_{2}, \dots, a_{r} b_{1}, b_{2}, \dots, b_{s}; z]} : = \sum_{n = 0}^{\infty} \frac{{(a_{1}; p, q; κ, μ)}_{n + 1} {(a_{2})}_{n + 1} \dots {(a_{r})}_{n + 1}}{{(b_{1})}_{n + 1} {(b_{2})}_{n + 1} \dots {(b_{s})}_{n + 1}} \frac{z^{n + 1}}{(n + 1)} \\ = \frac{a_{1} \dots a_{r}}{b_{1} \dots b_{s}}_{r} F_{s} [(a_{1} + 1; p, q; κ, μ), a_{2} + 1, \dots, a_{r} + 1 b_{1} + 1, b_{2} + 1, \dots, b_{s} + 1; z], \end{array}$ \eqalign{ & {d \over {dz}}\left\{ {{}_rF_s \left[ {(a_1 ;p,q;\kappa, \mu ),a_2, \, \cdots \,, a_r \, b_1, b_2, \cdots, b_s \, ;z} \right]} \right\}\,: = \,\sum\limits_{n = 0}^\infty {{(a_1 ;p,q;\kappa, \mu )_{n + 1} (a_2 )_{n + 1} \cdots (a_r )_{n + 1} } \over {(b_1 )_{n + 1} (b_2 )_{n + 1} \cdots (b_s )_{n + 1} }}{{z^{n + 1} } \over {(n + 1)}} \cr & = {{a_1 \cdots a_r } \over {b_1 \cdots b_s }} \, {}_rF_s \left[ {(a_1 + 1;p,q;\kappa, \mu ),a_2 + 1, \, \cdots \,, a_r + 1 \, b_1 + 1,b_2 + 1, \cdots, b_s + 1 \, ;z} \right],} repeating the same procedure n-times gives the formula (19).

Choosing r = s = 1 and r = 2, s = 1 in (19), we have the derivative formulas for the (10) and (11), respectively.

Corollary 8

The following derivative formulas hold true:(21) $\frac{d^{n}}{{dz}^{n}} {Φ_{p, q}^{κ, μ} (a; b; z)} = \frac{{(a)}_{n}}{{(b)}_{n}} Φ_{p, q}^{κ, μ} (a + n; b + n; z)$ {{d^n } \over {dz^n }} \, \{ \Phi _{p,q}^{\kappa, \mu } (a;b;z)\} = {{(a)_n } \over {(b)_n }} \, \Phi _{p,q}^{\kappa, \mu } (a + n;b + n;z)and(22) $\frac{d^{n}}{{dz}^{n}} {F_{p, q}^{κ, μ} (a, b, c; z)} = \frac{{(a)}_{n} {(b)}_{n}}{{(c)}_{n}} F_{p, q}^{κ, μ} (a + n, b + n, c + n; z) .$ {{d^n } \over {dz^n }} \, \{ F_{p,q}^{\kappa, \mu } (a,b,c;z)\} = {{(a)_n (b)_n } \over {(c)_n }} \, F_{p,q}^{\kappa, \mu } (a + n,b + n,c + n;z).

The Bessel function J_ν(z) and the modified Bessel function I_ν(z) are expressible as hypergeometric functions as follows [11, 12, 33]: (23) $\begin{array}{l} J_{ν} (z) = \frac{{(\frac{z}{2})}^{ν}}{Γ (ν + 1)}_{0} F_{1} (-; ν + 1; - \frac{1}{4} z^{2}) \\ (ν \in ℂ \ ℤ^{-} (ℤ^{-} = {- 1, - 2, - 3, ...})) \end{array}$ \matrix{ {J_\nu (z) = {{({z \over 2})^\nu } \over {\Gamma (\nu + 1)}} \, _0 F_1 ( - ;\nu + 1; - {1 \over 4}z^2 )} \hfill \cr {(\nu \in {\mathbb{C}} \, \backslash \, {\mathbb{Z}}^ - ({\mathbb{Z}}^ - = \{ - 1, - 2, - 3,...\} ))}} and (24) $\begin{array}{l} I_{ν} (z) = \frac{{(\frac{z}{2})}^{ν}}{Γ (ν + 1)}_{0} F_{1} (-; ν + 1; \frac{1}{4} z^{2}) \\ (ν \in ℂ \ ℤ^{-}) . \end{array}$ \eqalign{ & {I_\nu (z) = {{({z \over 2})^\nu } \over {\Gamma (\nu + 1)}} \, _0 F_1 ( - ;\nu + 1;{1 \over 4}z^2 )} \hfill \cr & {(\nu \in {\mathbb{C}} \, \backslash \, {\mathbb{Z}}^ - ).}}

Additionally, for the incomplete gamma function γ(s,x) defined by [10], (25) $\begin{array}{l} γ (s, x) = \int_{0}^{x} t^{s - 1} exp (- t) dt \\ (ℝ (s) > 0; x \geq 0) . \end{array}$ \eqalign{ & {\gamma (s,x) = \int\limits_0^x t^{s - 1} \exp ( - t)dt} \hfill \cr & {({\mathbb{R}}(s) > 0;x \ge 0).}}

Also, we know that [7, 9, 10] (26) $_{1} F_{1} [s; s + 1; - x] = {sx}^{- s} γ (s, x) .$ _1 F_1 [s;s + 1; - x] = sx^{ - s} \gamma (s,x).

So, we can deduce Corollary 9 and Corollary 10 by performing the relationships (23)–(26) in the equations (16) and (17).

Corollary 9

Each of the following integral representations hold true:(27) $Φ_{p, q}^{κ, μ} (a; b + 1; - z) = \frac{Γ (b + 1)}{Γ (a)} z^{- \frac{b}{2}} \int_{0}^{\infty} t^{a - \frac{b}{2} - 1} exp (- \frac{t^{κ}}{p} - \frac{q}{t^{μ}}) J_{b} (2 \sqrt{zt}) dt$ \Phi _{p,q}^{\kappa, \mu } (a;b + 1; - z) = {{\Gamma (b + 1)} \over {\Gamma (a)}}z^{ - {b \over 2}} \int\limits_0^\infty t^{a - {b \over 2} - 1} \exp ( - {{t^\kappa } \over p} - {q \over {t^\mu }})J_b (2\sqrt {zt} )dtand(28) $Φ_{p, q}^{κ, μ} (a; b + 1; z) = \frac{Γ (b + 1)}{Γ (a)} z^{- \frac{b}{2}} \int_{0}^{\infty} t^{a - \frac{b}{2} - 1} exp (- \frac{t^{κ}}{p} - \frac{q}{t^{μ}}) I_{b} (2 \sqrt{zt}) dt$ \Phi _{p,q}^{\kappa, \mu } (a;b + 1;z) = {{\Gamma (b + 1)} \over {\Gamma (a)}}z^{ - {b \over 2}} \int\limits_0^\infty t^{a - {b \over 2} - 1} \exp ( - {{t^\kappa } \over p} - {q \over {t^\mu }})I_b (2\sqrt {zt} )dton condition that the integrals involves are convergent.

Corollary 10

The following integral representation holds true:(29) $F_{p, q}^{κ, μ} (a, b, b + 1; - z) = \frac{{bz}^{- b}}{Γ (a)} \int_{0}^{\infty} t^{a - b - 1} exp (- \frac{t^{κ}}{p} - \frac{q}{t^{μ}}) γ (b, zt) dt,$ F_{p,q}^{\kappa, \mu } (a,b,b + 1; - z) = {{bz^{ - b} } \over {\Gamma (a)}}\int\limits_0^\infty t^{a - b - 1} \exp ( - {{t^\kappa } \over p} - {q \over {t^\mu }})\gamma (b,zt)dt,on condition that the integrals involves are convergent.

A New Generalization of the extended Appell hypergeometric functions

In this section, we introduce extended Appell hypergeometric series and some extended multivariable hypergeometric functions.

Let us introduce the extensions of the Appell’s functions and extended Lauricella’s hypergeometric function and other functions defined by (30) $\begin{matrix} _{p, q} F_{1}^{(κ, μ)} [a, b, c; d; x, y] : = \sum_{m, n = 0}^{\infty} \frac{{(a; p, q; κ, μ)}_{m + n} {(b)}_{m} {(c)}_{n}}{{(d)}_{m + n}} \frac{x^{m}}{m!} \frac{y^{n}}{n!} \\ max (| x |, | y |) < 1, \end{matrix}$ \matrix{ {{}_{p,q}F_1^{(\kappa, \mu )} [a,b,c;d;x,y]: = \, \sum\limits_{m,n = 0}^\infty {{(a;p,q;\kappa, \mu )_{m + n} \, (b)_m \, (c)_n } \over {(d)_{m + n} }} \, {{x^m } \over {m!}} \, {{y^n } \over {n!}}} \cr {\max (|x|, \, |y|) < \, 1,}}(31) $\begin{matrix} _{p, q} F_{2}^{(κ, μ)} [a, b, c; d, e; x, y] : = \sum_{m, n = 0}^{\infty} \frac{{(a; p, q; κ, μ)}_{m + n} {(b)}_{m} {(c)}_{n}}{{(d)}_{m} {(e)}_{n}} \frac{x^{m}}{m!} \frac{y^{n}}{n!} \\ | x | + | y | < 1, \end{matrix}$ \matrix{ {_{p,q} F_2^{(\kappa, \mu )} [a,b,c;d,e;x,y]: = \, \sum\limits_{m,n = 0}^\infty {{(a;p,q;\kappa, \mu )_{m + n} \, (b)_m \, (c)_n } \over {(d)_m \, (e)_n }} \, {{x^m } \over {m!}} \, {{y^n } \over {n!}}} \cr {|x| + |y| < \, 1,}}(32) $\begin{matrix} _{p, q} F_{3}^{(κ, μ)} [a, b, c, d; e; x, y] : = \sum_{m, n = 0}^{\infty} \frac{{(a; p, q; κ, μ)}_{m} {(b)}_{n} {(c)}_{m} {(d)}_{n}}{{(e)}_{m + n}} \frac{x^{m}}{m!} \frac{y^{n}}{n!} \\ max (| x |, | y |) < 1, \end{matrix}$ \matrix{ {_{p,q} F_3^{(\kappa, \mu )} [a,b,c,d;e;x,y]: = \, \sum\limits_{m,n = 0}^\infty {{(a;p,q;\kappa, \mu )_m \, (b)_n \, (c)_m \, (d)_n } \over {(e)_{m + n} }} \, {{x^m } \over {m!}} \, {{y^n } \over {n!}}} \cr {\max (|x|, \, |y|) < \, 1,}}(33) $\begin{matrix} _{p, q} F_{4}^{(κ, μ)} [a, b; c, d; x, y] : = \sum_{m, n = 0}^{\infty} \frac{{(a; p, q; κ, μ)}_{m + n} {(b)}_{m + n}}{{(c)}_{m} {(d)}_{n}} \frac{x^{m}}{m!} \frac{y^{n}}{n!} \\ \sqrt{| x |} + \sqrt{| y |} < 1, \end{matrix}$ \matrix{ {{}_{p,q}F_4^{(\kappa, \mu )} [a,b;c,d;x,y]: = \, \sum\limits_{m,n = 0}^\infty {{(a;p,q;\kappa, \mu )_{m + n} \, (b)_{m + n} } \over {(c)_m \, (d)_n }} \, {{x^m } \over {m!}} \, {{y^n } \over {n!}}} \cr {\sqrt {|x|} + \sqrt {|y|} < \, 1,}}(34) $\begin{matrix} _{p, q} F_{D}^{(κ, μ; 3)} [a, b, c, d; e; x, y, z] : = \sum_{m, n, r = 0}^{\infty} \frac{{(a; p, q; κ, μ)}_{m + n + r} {(b)}_{m} {(c)}_{n} {(d)}_{r}}{{(e)}_{m + n + r}} \frac{x^{m}}{m!} \frac{y^{n}}{n!} \frac{z^{r}}{r!} \\ max (| x |, | y |, | z |) < 1, \end{matrix}$ \matrix{ {_{p,q} F_D^{(\kappa, \mu ;3)} [a,b,c,d;e;x,y,z]: = \, \sum\limits_{m,n,r = 0}^\infty {{(a;p,q;\kappa, \mu )_{m + n + r} \, (b)_m \, (c)_n \, (d)_r } \over { \, (e)_{m + n + r} }} \, {{x^m } \over {m!}} \, {{y^n } \over {n!}} \, {{z^r } \over {r!}}} \cr {\max (|x|, \, |y|, \, |z|) < \, 1,}}(35) $\begin{matrix} _{p, q} Φ_{1}^{(κ, μ)} [a, b; d; x, y] : = \sum_{m, n = 0}^{\infty} \frac{{(a; p, q; κ, μ)}_{m + n} {(b)}_{m}}{{(d)}_{m + n}} \frac{x^{m}}{m!} \frac{y^{n}}{n!} \\ max (| x |, | y |) < 1, \end{matrix}$ \matrix{ {_{p,q} \Phi _1^{(\kappa, \mu )} [a,b;d;x,y]: = \, \sum\limits_{m,n = 0}^\infty \, {{(a;p,q;\kappa, \mu )_{m + n} \, (b)_m } \over {(d)_{m + n} }} \, {{x^m } \over {m!}} \, {{y^n } \over {n!}}} \cr {\max (|x|, \, |y|) < \, 1,}}(36) $\begin{matrix} _{p, q} Ψ_{1}^{(κ, μ)} [a, b; d, e; x, y] : = \sum_{m, n = 0}^{\infty} \frac{{(a; p, q; κ, μ)}_{m + n} {(b)}_{m}}{{(d)}_{m} {(e)}_{n}} \frac{x^{m}}{m!} \frac{y^{n}}{n!} \\ | x | + | y | < 1, \end{matrix}$ \matrix{ {_{p,q} \Psi _1^{(\kappa, \mu )} [a,b;d,e;x,y]: = \, \sum\limits_{m,n = 0}^\infty {{(a;p,q;\kappa, \mu )_{m + n} \, (b)_m } \over {(d)_m \, (e)_n }} \, {{x^m } \over {m!}} \, {{y^n } \over {n!}}} \cr {|x| + |y| < \, 1,}}(37) $\begin{matrix} _{p, q} Ξ_{1}^{(κ, μ)} [a, c, d; e; x, y] : = \sum_{m, n = 0}^{\infty} \frac{{(a; p, q; κ, μ)}_{m} {(c)}_{m} {(d)}_{n}}{{(e)}_{m + n}} \frac{x^{m}}{m!} \frac{y^{n}}{n!} \\ max (| x |, | y |) < 1 \end{matrix}$ \matrix{ {_{p,q} \Xi _1^{(\kappa, \mu )} [a,c,d;e;x,y]: = \, \sum\limits_{m,n = 0}^\infty {{(a;p,q;\kappa, \mu )_m \, (c)_m \, (d)_n } \over {(e)_{m + n} }} \, {{x^m } \over {m!}} \, {{y^n } \over {n!}}} \cr {\max (|x|, \, |y|) < \, 1}} and (38) $\begin{matrix} _{p, q} Φ_{D}^{(κ, μ; 3)} [a, b, c; e; x, y, z] : = \sum_{m, n, r = 0}^{\infty} \frac{{(a; p, q; κ, μ)}_{m + n + r} {(b)}_{m} {(c)}_{n}}{{(e)}_{m + n + r}} \frac{x^{m}}{m!} \frac{y^{n}}{n!} \frac{z^{r}}{r!} \\ max (| x |, | y |, | z |) < 1, \end{matrix}$ \matrix{ {_{p,q} \Phi _D^{(\kappa, \mu ;3)} [a,b,c;e;x,y,z]: = \, \sum\limits_{m,n,r = 0}^\infty {{(a;p,q;\kappa, \mu )_{m + n + r} \, (b)_m \, (c)_n } \over { \, (e)_{m + n + r} }} \, {{x^m } \over {m!}} \, {{y^n } \over {n!}} \, {{z^r } \over {r!}}} \cr {\max (|x|, \, |y|, \, |z|) < \, 1,}} respectively. Note that taking p = 1, q = 0, κ = 0 and μ = 0 gives the original ones [1, 7, 8, 9, 10, 18, 22, 23, 26, 29, 30, 32, 33]. Now, we obtain the integral representations of the functions (30)–(34).

Theorem 11

The following integral representations for(30)hold true:(39) $_{p, q} F_{1}^{(κ, μ)} [a, b, c; d; x, y] = \frac{1}{Γ (a)} \int_{0}^{\infty} t^{a - 1} exp (- \frac{t^{κ}}{p} - \frac{q}{t^{μ}}) Φ_{2} [b, c; d; xt, yt] dt$ _{p,q} F_1^{(\kappa, \mu )} [a,b,c;d;x,y] = \, {1 \over {\Gamma (a)}}\int_0^\infty \, t^{a - 1} \, \exp ( - {{t^\kappa } \over p} - {q \over {t^\mu }}) \, \Phi _2 [b,c;d;xt,yt]dtand(40) $_{p, q} F_{1}^{(κ, μ)} [a, b, c; d; x, y] = \frac{1}{Γ (c)} \int_{0}^{\infty} t^{c - 1} exp (- t)_{p, q} Φ_{1}^{(κ, μ)} [a, b; d; x, yt] dt .$ _{p,q} F_1^{(\kappa, \mu )} [a,b,c;d;x,y] = \, {1 \over {\Gamma (c)}}\int_0^\infty \, t^{c - 1} \, \exp ( - t) \, _{p,q} \Phi _1^{(\kappa, \mu )} [a,b;d;x,yt]dt.

Proof

Using the generalization of the extended Pochhammer symbol (a₁; p,q;κ,μ) in the definition (30) by its integral representation given by (6), we led to desired result (39). Similar way, we can prove the (40).

Theorem 12

The following integral representations for(31)hold true:(41) $_{p, q} F_{2}^{(κ, μ)} [a, b, c; d, e; x, y] = \frac{1}{Γ (a)} \int_{0}^{\infty} t^{a - 1} exp (- \frac{t^{κ}}{p} - \frac{q}{t^{μ}})_{1} F_{1} [b; d; xt]_{1} F_{1} [c; e; yt] dt$ _{p,q} F_2^{(\kappa, \mu )} [a,b,c;d,e;x,y] = \, {1 \over {\Gamma (a)}}\int_0^\infty \, t^{a - 1} \, \exp ( - {{t^\kappa } \over p} - {q \over {t^\mu }}) \, _1 F_1 [b;d;xt] \, _1 F_1 [c;e;yt]dtand(42) $_{p, q} F_{2}^{(κ, μ)} [a, b, c; d, e; x, y] = \frac{1}{Γ (c)} \int_{0}^{\infty} t^{c - 1} exp (- t)_{p, q} Ψ_{1}^{(κ, μ)} [a, b; d, e; x, yt] dt .$ _{p,q} F_2^{(\kappa, \mu )} [a,b,c;d,e;x,y] = \, {1 \over {\Gamma (c)}}\int_0^\infty \, t^{c - 1} \, \exp ( - t) \, _{p,q} \Psi _1^{(\kappa, \mu )} [a,b;d,e;x,yt]dt.

Proof

Using the generalization of the extended Pochhammer symbol (a₁; p,q;κ, μ) in the definition (31) by its integral representation given by (6), we led to desired result (41). Similar way, we can prove the (42).

Theorem 13

The following integral representation for(32)holds true:(43) $_{p, q} F_{3}^{(κ, μ)} [a, b, c, d; e; x, y] = \frac{1}{Γ (b)} \int_{0}^{\infty} t^{b - 1} exp (- t)_{p, q} Ξ_{1}^{(κ, μ)} [a, c, d; e; x, yt] dt .$ _{p,q} F_3^{(\kappa, \mu )} [a,b,c,d;e;x,y] = \, {1 \over {\Gamma (b)}}\int_0^\infty \, t^{b - 1} \, \exp ( - t) \, _{p,q} \Xi _1^{(\kappa, \mu )} [a,c,d;e;x,yt]dt.

Proof

Using the generalization of the extended Pochhammer symbol (a₁; p,q;κ, μ) in the definition (32) by its integral representation given by (3), we led to desired result (43).

Theorem 14

The following integral representation for(33)holds true:(44) $_{p, q} F_{4}^{(κ, μ)} [a, b; c, d; x, y] = \frac{1}{Γ (a)} \int_{0}^{\infty} t^{a - 1} exp (- \frac{t^{κ}}{p} - \frac{q}{t^{μ}}) Ψ_{2} [b; c, d; xt, yt] dt .$ _{p,q} F_4^{(\kappa, \mu )} [a,b;c,d;x,y] = \, {1 \over {\Gamma (a)}}\int_0^\infty \, t^{a - 1} \, \exp ( - {{t^\kappa } \over p} - {q \over {t^\mu }}) \, \Psi _2 [b;c,d;xt,yt]dt.

Proof

Using the generalization of the extended Pochhammer symbol (a₁; p,q;κ, μ) in the definition (33) by its integral representation given by (6), we led to desired result (44).

Theorem 15

The following integral representations for(34)hold true:(45) $_{p, q} F_{D}^{(κ, μ; 3)} [a, b, c, d; e; x, y, z] = \frac{1}{Γ (a)} \int_{0}^{\infty} t^{a - 1} exp (- \frac{t^{κ}}{p} - \frac{q}{t^{μ}}) Φ_{2}^{(3)} [b, c, d; e; xt, yt, zt] dt$ _{p,q} F_D^{(\kappa, \mu ;3)} [a,b,c,d;e;x,y,z] = \, {1 \over {\Gamma (a)}}\int_0^\infty \, t^{a - 1} \, \exp ( - {{t^\kappa } \over p} - {q \over {t^\mu }}) \, \Phi _2^{(3)} [b,c,d;e;xt,yt,zt]dtand(46) $_{p, q} F_{D}^{(κ, μ; 3)} [a, b, c, d; e; x, y, z] = \frac{1}{Γ (d)} \int_{0}^{\infty} t^{d - 1} exp (- t)_{p, q} Φ_{D}^{(κ, μ; 3)} [a, b, c; e; x, y, zt] dt .$ _{p,q} F_D^{(\kappa, \mu ;3)} [a,b,c,d;e;x,y,z] = \, {1 \over {\Gamma (d)}}\int_0^\infty \, t^{d - 1} \, \exp ( - t) \, _{p,q} \Phi _D^{(\kappa, \mu ;3)} [a,b,c;e;x,y,zt]dt.

Proof

Using the generalization of the extended Pochhammer symbol (a₁; p,q;κ, μ) in the definition (34) by its integral representation given by (6), we led to desired result (45). Similar way, we can prove the (46).

Theorem 16

The following derivative formulas for(30)–(34)hold true:(47) $D_{x, y}^{m, n} {_{p, q} F_{1}^{(κ, μ)} [a, b, c; d; x, y]} : = {\frac{{(a)}_{m + n} {(b)}_{m} {(c)}_{n}}{{(d)}_{m + n}}}_{p, q} F_{1}^{(κ, μ)} [a + m + n, b + m, c + n; d + m + n; x, y],$ D_{x,y}^{m,n} \left\{ { \, _{p,q} F_1^{(\kappa, \mu )} [a,b,c;d;x,y]} \right\}: = {{(a)_{m + n} (b)_m \, (c)_n } \over {(d)_{m + n} }}_{p,q} F_1^{(\kappa, \mu )} [a + m + n,b + m,c + n;d + m + n;x,y],(48) $D_{x, y}^{m, n} {_{p, q} F_{2}^{(κ, μ)} [a, b, c; d, e; x, y]} : = {\frac{{(a)}_{m + n} {(b)}_{m} {(c)}_{n}}{{(d)}_{m} {(e)}_{n}}}_{p, q} F_{2}^{(κ, μ)} [a + m + n, b + m, c + n; d + m, e + n; x, y],$ D_{x,y}^{m,n} \left\{ { \, _{p,q} F_2^{(\kappa, \mu )} [a,b,c;d,e;x,y]} \right\}: = {{(a)_{m + n} (b)_m \, (c)_n } \over {(d)_m \, (e)_n }}_{p,q} F_2^{(\kappa, \mu )} [a + m + n,b + m,c + n;d + m,e + n;x,y],(49) $D_{x, y}^{m, n} {_{p, q} F_{3}^{(κ, μ)} [a, b, c, d; e; x, y]} : = {\frac{{(a)}_{m} {(b)}_{n} {(c)}_{m} {(d)}_{n}}{{(e)}_{m + n}}}_{p, q} F_{3}^{(κ, μ)} [a + m, b + n, c + m, d + n; e + m + n; x, y],$ D_{x,y}^{m,n} \left\{ { \, _{p,q} F_3^{(\kappa, \mu )} [a,b,c,d;e;x,y]} \right\}: = {{(a)_m (b)_n \, (c)_m \, (d)_n } \over {(e)_{m + n} }}_{p,q} F_3^{(\kappa, \mu )} [a + m,b + n,c + m,d + n;e + m + n;x,y],(50) $D_{x, y}^{m, n} {_{p, q} F_{4}^{(κ, μ)} [a, b; c, d; x, y]} : = {\frac{{(a)}_{m + n} {(b)}_{m + n}}{{(c)}_{m} {(c)}_{n}}}_{p, q} F_{4}^{(κ, μ)} [a + m + n, b + m + n; c + m, d + n; x, y]$ D_{x,y}^{m,n} \left\{ { \, _{p,q} F_4^{(\kappa, \mu )} [a,b;c,d;x,y]} \right\}: = {{(a)_{m + n} (b)_{m + n} } \over {(c)_m \, (c)_n }}_{p,q} F_4^{(\kappa, \mu )} [a + m + n,b + m + n;c + m,d + n;x,y]and(51) $\begin{array}{l} D_{x, y, z}^{m, n, r} {_{p, q} F_{D}^{(κ, μ; 3)} [a, b, c, d; e; x, y]} : = \frac{{(a)}_{m + n + r} {(b)}_{m} {(c)}_{n} {(d)}_{r}}{{(e)}_{m + n + r}} \\ \times_{p, q} F_{D}^{(κ, μ; 3)} [a + m + n + r, b + m, c + n, d + r; e + m + n + r; x, y, z] . \end{array}$ \eqalign{ & D_{x,y,z}^{m,n,r} \{ \, _{p,q} F_D^{(\kappa, \mu ;3)} [a,b,c,d;e;x,y]\} : = {{(a)_{m + n + r} (b)_m \, (c)_n \, (d)_r } \over {(e)_{m + n + r} }} \cr & \times \, _{p,q} F_D^{(\kappa, \mu ;3)} [a + m + n + r,b + m,c + n,d + r;e + m + n + r;x,y,z].}

Proof

Differentiating (30)–(33) with respect to x and y, then repeating same procedure n-times and making some simple calculation, we can obtain the (47)–(50) results. Similiarly, taking differentiation (34) with respect to x, y and z, we can get the derivative formula (51)

Theorem 17

The following derivative formulas for(30)hold true:(52) $D_{y}^{n} {y^{c + n - 1}_{p, q} F_{1}^{(κ, μ)} [a, b, c; d; x, y]} : = {(c)}_{n} y_{p, q}^{c - 1} F_{1}^{(κ, μ)} [a, b, c + n; d; x, y],$ D_y^n \left\{ {y^{c + n - 1} \, _{p,q} F_1^{(\kappa, \mu )} [a,b,c;d;x,y]} \right\}: = (c)_n \, y_{p,q}^{c - 1} F_1^{(\kappa, \mu )} [a,b,c + n;d;x,y],(53) $D_{y}^{n} {y^{d - 1}_{p, q} F_{1}^{(κ, μ)} [a, b, c; d; x, y]} : = {(- 1)}^{n} {(1 - d)}_{n} y^{d - n - 1}_{p, q} F_{1}^{(κ, μ)} [a, b, c; d - n; x, y]$ D_y^n \left\{ {y^{d - 1} \, _{p,q} F_1^{(\kappa, \mu )} [a,b,c;d;x,y]} \right\}: = ( - 1)^n (1 - d)_n \, y^{d - n - 1} \, _{p,q} F_1^{(\kappa, \mu )} [a,b,c;d - n;x,y]and(54) $\begin{matrix} D_{y}^{n} {y^{d - b - 1}_{p, q} F_{1}^{(κ, μ)} [a, b, c; d; x, y]} : = {(- 1)}^{n} {(b - d - 1)}_{n} y^{d - b - n - 1} \\ \times \sum_{n = 0}^{m} (\begin{matrix} m \\ n \end{matrix}) \frac{{(a)}_{n} {(c)}_{n} y^{n}}{{(d)}_{n} {(d - b - m)}_{n}}_{p, q} F_{1}^{(κ, μ)} [a + n, b, c + n; d + n; x, y] . \end{matrix}$ \matrix{ {D_y^n \left\{ {y^{d - b - 1} \, _{p,q} F_1^{(\kappa, \mu )} [a,b,c;d;x,y]} \right\}: = ( - 1)^n (b - d - 1)_n \, y^{d - b - n - 1} } \cr {\, \times \, \, \sum\limits_{n = 0}^m \left( {\matrix{ m \cr n \cr } } \right) \, {{(a)_n \, (c)_n \, y^n } \over {(d)_n \, (d - b - m)_n }} \, _{p,q} F_1^{(\kappa, \mu )} [a + n,b,c + n;d + n;x,y].}}

Proof

Multiplying the (30) with y^c+n−1 and taking the derivative n-times with respect to y, we have (55) $\begin{array}{l} D_{y}^{n} {y^{c + n - 1}_{p, q} F_{1}^{(κ, μ)} [a, b, c; d; x, y]} : & = D_{y}^{n} {y^{c + n - 1} \sum_{m, n = 0}^{\infty} \frac{{(a; p, q; κ, μ)}_{m + n} {(b)}_{m} {(c)}_{n}}{{(d)}_{m + n}} \frac{x^{m}}{m!} \frac{y^{n}}{n!}} \\ = \sum_{m = 0}^{\infty} \frac{{(a)}_{m} {(b)}_{m} x^{m}}{{(d)}_{m} m!} D_{y}^{n} {y^{c + n - 1} \sum_{n = 0}^{\infty} \frac{{(a + m; p, q; κ, μ)}_{n} {(c)}_{n}}{{(d + m)}_{n} n!} y^{n}} \\ = \sum_{m = 0}^{\infty} \frac{{(a)}_{m} {(b)}_{m} x^{m}}{{(d)}_{m} m!} {(c)}_{n} y^{c - 1} \sum_{n = 0}^{\infty} \frac{{(a + m; p, q; κ, μ)}_{n} {(c + n)}_{n}}{{(d + m)}_{n} n!} y^{n} \\ = {(c)}_{n} y^{c - 1} \sum_{m, n = 0}^{\infty} \frac{{(a; p, q; κ, μ)}_{m + n} {(b)}_{m} {(c + n)}_{n}}{{(d)}_{m + n}} \frac{x^{m}}{m!} \frac{y^{n}}{n!} . \end{array}$ \matrix{ {D_y^n \left\{ {y^{c + n - 1} \, _{p,q} F_1^{(\kappa, \mu )} [a,b,c;d;x,y]} \right\}:} \hfill & { = D_y^n \left\{ {y^{c + n - 1} \, \sum\limits_{m,n = 0}^\infty {{(a;p,q;\kappa, \mu )_{m + n} \, (b)_m \, (c)_n } \over {(d)_{m + n} }} \, {{x^m } \over {m!}} \, {{y^n } \over {n!}}} \right\}} \hfill \cr {} \hfill & { = \sum\limits_{m = 0}^\infty {{(a)_m \, (b)_m \, x^m } \over {(d)_m \, m!}} \, D_y^n \left\{ {y^{c + n - 1} \, \sum\limits_{n = 0}^\infty {{(a + m;p,q;\kappa, \mu )_n \, (c)_n } \over {(d + m)_n \, n!}}y^n } \right\}} \hfill \cr {} \hfill & { = \sum\limits_{m = 0}^\infty {{(a)_m \, (b)_m \, x^m } \over {(d)_m \, m!}} \, (c)_n \, y^{c - 1} \, \sum\limits_{n = 0}^\infty {{(a + m;p,q;\kappa, \mu )_n \, (c + n)_n } \over {(d + m)_n \, n!}}y^n } \hfill \cr {} \hfill & { = \, (c)_n \, y^{c - 1} \, \sum\limits_{m,n = 0}^\infty {{(a;p,q;\kappa, \mu )_{m + n} \, (b)_m \, (c + n)_n } \over {(d)_{m + n} }} \, {{x^m } \over {m!}} \, {{y^n } \over {n!}}.}} Thus, we obtain the (52) result. Similar way, we can prove the equations (53) and (54).

Theorem 18

The following derivative formulas for(31)hold true:(56) $D_{y}^{n} {y^{c + n - 1}_{p, q} F_{2}^{(κ, μ)} [a, b, c; d, e; x, y]} : = {(c)}_{n} y_{p, q}^{c - 1} F_{2}^{(κ, μ)} [a, b, c + n; d, e; x, y]$ D_y^n \left\{ {y^{c + n - 1} \, _{p,q} F_2^{(\kappa, \mu )} [a,b,c;d,e;x,y]} \right\}: = (c)_n \, y_{p,q}^{c - 1} F_2^{(\kappa, \mu )} [a,b,c + n;d,e;x,y]and(57) $D_{y}^{n} {y^{e - 1}_{p, q} F_{2}^{(κ, μ)} [a, b, c; d, e; x, y]} : = {(- 1)}^{n} {(1 - e)}_{n} y^{e - n - 1}_{p, q} F_{2}^{(κ, μ)} [a, b, c; d, e - n; x, y] .$ D_y^n \left\{ {y^{e - 1} \, _{p,q} F_2^{(\kappa, \mu )} [a,b,c;d,e;x,y]} \right\}: = ( - 1)^n (1 - e)_n \, y^{e - n - 1} \, _{p,q} F_2^{(\kappa, \mu )} [a,b,c;d,e - n;x,y].

Proof

The proof of theorem would be parallel to those of the Theorem 17.

Theorem 19

The following derivative formulas for(32)hold true:(58) $D_{y}^{n} {y^{d + n - 1}_{p, q} F_{3}^{(κ, μ)} [a, b, c, d; e; x, y]} : = {(d)}_{n} y_{p, q}^{d - 1} F_{3}^{(κ, μ)} [a, b, c, d + n; e; x, y],$ D_y^n \left\{ {y^{d + n - 1} \, _{p,q} F_3^{(\kappa, \mu )} [a,b,c,d;e;x,y]} \right\}: = (d)_n \, y_{p,q}^{d - 1} F_3^{(\kappa, \mu )} [a,b,c,d + n;e;x,y],(59) $\begin{matrix} D_{y}^{n} {{(1 - y)}^{b + n - 1}_{p, q} F_{3}^{(κ, μ)} [a, b, e - c, c; e; x, y]} : = {(- 1)}^{n} \frac{{(b)}_{n} {(e - c)}_{n}}{{(e)}_{n}} {(1 - y)}^{b - 1} \\ \times_{p, q} F_{3}^{(κ, μ)} [a, b + n, e - c + n, d; e + n; x, y] \end{matrix}$ \matrix{ {D_y^n \left\{ {(1 - y)^{b + n - 1} \, _{p,q} F_3^{(\kappa, \mu )} [a,b,e - c,c;e;x,y]} \right\}: = ( - 1)^n {{(b)_n \, (e - c)_n } \over {(e)_n }} \, (1 - y)^{b - 1} } \cr { \times \, _{p,q} F_3^{(\kappa, \mu )} [a,b + n,e - c + n,d;e + n;x,y]}}and(60) $\begin{matrix} D_{y}^{n} {y^{e - c - 1}_{p, q} F_{3}^{(κ, μ)} [a, b, c, d; e; x, y]} : = {(- 1)}^{n} {(c - e - 1)}_{n} y^{e - c - n - 1} \\ \times \sum_{n = 0}^{m} (\begin{matrix} m \\ n \end{matrix}) \frac{{(b)}_{n} {(d)}_{n} y^{n}}{{(e)}_{n} {(e - c - m)}_{n}}_{p, q} F_{3}^{(κ, μ)} [a, b + n, c, d + n; e + n; x, y] . \end{matrix}$ \matrix{ {D_y^n \left\{ {y^{e - c - 1} \, _{p,q} F_3^{(\kappa, \mu )} [a,b,c,d;e;x,y]} \right\}: = ( - 1)^n (c - e - 1)_n \, y^{e - c - n - 1} } \cr { \times \, \sum\limits_{n = 0}^m \left( {\matrix{ m \cr n \cr } } \right) \, {{(b)_n \, (d)_n \, y^n } \over {(e)_n \, (e - c - m)_n }} \, _{p,q} F_3^{(\kappa, \mu )} [a,b + n,c,d + n;e + n;x,y].}}

Proof

The proof of theorem would be parallel to those of the Theorem 17.

Theorem 20

The following derivative formulas for(33)hold true:(61) $D_{x}^{n} {x^{c - 1}_{p, q} F_{4}^{(κ, μ)} [a, b; c, d; x, y]} : = {(- 1)}^{n} {(1 - c)}_{n} x_{p, q}^{c - n - 1} F_{4}^{(κ, μ)} [a, b; c - n, d; x, y]$ D_x^n \left\{ {x^{c - 1} \, _{p,q} F_4^{(\kappa, \mu )} [a,b;c,d;x,y]} \right\}: = ( - 1)^n (1 - c)_n \, x_{p,q}^{c - n - 1} F_4^{(\kappa, \mu )} [a,b;c - n,d;x,y]and(62) $D_{y}^{n} {y^{d - 1}_{p, q} F_{4}^{(κ, μ)} [a, b; c, d; x, y]} : = {(- 1)}^{n} {(1 - d)}_{n} y_{p, q}^{d - n - 1} F_{4}^{(κ, μ)} [a, b; c, d - n; x, y] .$ D_y^n \left\{ {y^{d - 1} \, _{p,q} F_4^{(\kappa, \mu )} [a,b;c,d;x,y]} \right\}: = ( - 1)^n (1 - d)_n \, y_{p,q}^{d - n - 1} F_4^{(\kappa, \mu )} [a,b;c,d - n;x,y].

Proof

The proof of theorem would be parallel to those of the Theorem 17.

Recursion Formulas for Extended Appell Hypergeometric Functions

In this section, we present some recursion formulas for Appell hypergeometric functions. Let’s we start following theorem.

Theorem 21

The following recursion formulas for(30)hold true:(63) $_{p, q} F_{1}^{(κ, μ)} [a, b + n, c; d; x, y] =_{p, q} F_{1}^{(κ, μ)} [a, b, c; d; x, y] + \frac{ax}{d} \sum_{k = 1}^{n}_{p, q} F_{1}^{(κ, μ)} [a + 1, b + k, c; d + 1; x, y],$ _{p,q} F_1^{(\kappa, \mu )} [a,b + n,c;d;x,y] = _{p,q} F_1^{(\kappa, \mu )} [a,b,c;d;x,y] + \, {{ax} \over d} \, \sum\limits_{k = 1}^n \, _{p,q} F_1^{(\kappa, \mu )} [a + 1,b + k,c;d + 1;x,y],(64) $_{p, q} F_{1}^{(κ, μ)} [a, b - n, c; d; x, y] =_{p, q} F_{1}^{(κ, μ)} [a, b, c; d; x, y] - \frac{ax}{d} {\sum_{k = 0}^{n - 1}}_{p, q} F_{1}^{(κ, μ)} [a + 1, b - k, c; d + 1; x, y],$ _{p,q} F_1^{(\kappa, \mu )} [a,b - n,c;d;x,y] = _{p,q} F_1^{(\kappa, \mu )} [a,b,c;d;x,y] - \, {{ax} \over d} \, \sum\limits_{k = 0}^{n - 1} \, _{p,q} F_1^{(\kappa, \mu )} [a + 1,b - k,c;d + 1;x,y],and(65) $\begin{array}{l} _{p, q} F_{1}^{(κ, μ)} [a, b, c; d - n; x, y] =_{p, q} F_{1}^{(κ, μ)} [a, b, c; d; x, y] \\ + abx \sum_{k = 1}^{n} \frac{_{p, q} F_{1}^{(κ, μ)} [a + 1, b + 1, c; d + 2 - k; x, y]}{(d - k) (d - k - 1)} \\ + acy \sum_{k = 1}^{n} \frac{_{p, q} F_{1}^{(κ, μ)} [a + 1, b, c + 1; d + 2 - k; x, y]}{(d - k) (d - k - 1)} . \end{array}$ \eqalign{ & _{p,q} F_1^{(\kappa, \mu )} [a,b,c;d - n;x,y] = \,_{p,q} F_1^{(\kappa, \mu )} [a,b,c;d;x,y] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \, abx \, \sum\limits_{k = 1}^n \, {{\,_{p,q} F_1^{(\kappa, \mu )} [a + 1,b + 1,c;d + 2 - k;x,y]} \over {(d - k)(d - k - 1)}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \, acy \, \sum\limits_{k = 1}^n \, {{\,_{p,q} F_1^{(\kappa, \mu )} [a + 1,b,c + 1;d + 2 - k;x,y]} \over {(d - k)(d - k - 1)}}.}

Proof

Applying the transformation formula ${(b + 1)}_{m} = {(b)}_{m} \times (1 + \frac{m}{b})$ (b + 1)_m = (b)_m \times (1 + {m \over b}) in the definition of the extension of the Appell hypergeometric function $_{p, q} F_{1}^{(κ, μ)} (.)$ \,_{p,q} F_1^{(\kappa, \mu )} (.) in (30) and we have following contiguous formula: (66) $p, q F_{1}^{(κ, μ)} [a, b + 1, c; d; x, y] =_{p, q} F_{1}^{(κ, μ)} [a, b, c; d; x, y] + \frac{ax}{d}_{p, q} F_{1}^{(κ, μ)} [a + 1, b + 1, c; d + 1; x, y] .$ \,_{p,q}F_1^{(\kappa, \mu )} [a,b + 1,c;d;x,y] = \,_{p,q} F_1^{(\kappa, \mu )} [a,b,c;d;x,y] + \, {{ax} \over d} \, _{p,q} F_1^{(\kappa, \mu )} [a + 1,b + 1,c;d + 1;x,y]. Calculating the function $_{p, q} F_{1}^{(κ, μ)} (.)$ \,_{p,q} F_1^{(\kappa, \mu )} (.) with the parameter b+n by equation (66) for n times, we obtain the required result (63). Setting the b = b − n in the equation (66) and making same calculation as above equation, we can be yield the desired result (64). The proof of (65) is omitted to readers because it is similar to the proof of (63).

Theorem 22

The following recursion formulas for(31)hold true:(67) $_{p, q} F_{2}^{(κ, μ)} [a, b + n, c; d, e; x, y] =_{p, q} F_{2}^{(κ, μ)} [a, b, c; d, e; x, y] + \frac{ax}{d} {\sum_{k = 1}^{n}}_{p, q} F_{2}^{(κ, μ)} [a + 1, b + k, c; d + 1, e; x, y],$ \,_{p,q} F_2^{(\kappa, \mu )} [a,b + n,c;d,e;x,y] = \,_{p,q} F_2^{(\kappa, \mu )} [a,b,c;d,e;x,y] + \, {{ax} \over d} \, \sum\limits_{k = 1}^n \,_{p,q} F_2^{(\kappa, \mu )} [a + 1,b + k,c;d + 1,e;x,y],(68) $_{p, q} F_{2}^{(κ, μ)} [a, b - n, c; d, e; x, y] =_{p, q} F_{2}^{(κ, μ)} [a, b, c; d, e; x, y] - \frac{ax}{d} {\sum_{k = 0}^{n - 1}}_{p, q} F_{2}^{(κ, μ)} [a + 1, b - k, c; d + 1, e; x, y],$ \,_{p,q} F_2^{(\kappa, \mu )} [a,b - n,c;d,e;x,y] = \,_{p,q} F_2^{(\kappa, \mu )} [a,b,c;d,e;x,y] - \, {{ax} \over d} \, \sum\limits_{k = 0}^{n - 1} \,_{p,q} F_2^{(\kappa, \mu )} [a + 1,b - k,c;d + 1,e;x,y],and(69) $_{p, q} F_{2}^{(κ, μ)} [a, b, c; d - n, e; x, y] =_{p, q} F_{2}^{(κ, μ)} [a, b, c; d, e; x, y] + abx \sum_{k = 1}^{n} \frac{_{p, q} F_{2}^{(κ, μ)} [a + 1, b + 1, c; d + 2 - k, e; x, y]}{(d - k) (d - k - 1)} .$ \,_{p,q} F_2^{(\kappa, \mu )} [a,b,c;d - n,e;x,y] = \,_{p,q} F_2^{(\kappa, \mu )} [a,b,c;d,e;x,y] + \, abx \, \sum\limits_{k = 1}^n \, {{_{p,q} F_2^{(\kappa, \mu )} [a + 1,b + 1,c;d + 2 - k,e;x,y]} \over {(d - k)(d - k - 1)}}.

Proof

The proof of the Theorem 22 is similar to the proof of Theorem 21.

Theorem 23

The following recursion formula for(32)holds true:(70) $\begin{array}{l} _{p, q} F_{3}^{(κ, μ)} [a, b, c, d; e - n; x, y] =_{p, q} F_{3}^{(κ, μ)} [a, b, c, d; e; x, y] \\ + acx \sum_{k = 1}^{n} \frac{_{p, q} F_{3}^{(κ, μ)} [a + 1, b, c + 1, d; e + 2 - k; x, y]}{(e - k) (e - k - 1)} \\ + bdy \sum_{k = 1}^{n} \frac{_{p, q} F_{3}^{(κ, μ)} [a, b + 1, c, d + 1; e + 2 - k; x, y]}{(e - k) (e - k - 1)} . \end{array}$ \eqalign{ & _{p,q} F_3^{(\kappa, \mu )} [a,b,c,d;e - n;x,y] = _{p,q} F_3^{(\kappa, \mu )} [a,b,c,d;e;x,y] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \, acx \, \sum\limits_{k = 1}^n \, {{_{p,q} F_3^{(\kappa, \mu )} [a + 1,b,c + 1,d;e + 2 - k;x,y]} \over {(e - k)(e - k - 1)}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \, bdy \, \sum\limits_{k = 1}^n \, {{_{p,q} F_3^{(\kappa, \mu )} [a,b + 1,c,d + 1;e + 2 - k;x,y]} \over {(e - k)(e - k - 1)}}.}

Proof

The proof of the Theorem 23 is parallel to the proof of Theorem 21.

Theorem 24

The following recursion formula for(33)holds true:(71) $_{p, q} F_{4}^{(κ, μ)} [a, b; c - n, d; x, y] =_{p, q} F_{4}^{(κ, μ)} [a, b; c, d; x, y] + abx \sum_{k = 0}^{n - 1} \frac{_{p, q} F_{4}^{(κ, μ)} [a + 1, b + 1; c + 1 - k, d; x, y]}{(c - k) (c - k - 1)} .$ \,_{p,q} F_4^{(\kappa, \mu )} [a,b;c - n,d;x,y] = \, _{p,q} F_4^{(\kappa, \mu )} [a,b;c,d;x,y] + \, abx \, \sum\limits_{k = 0}^{n - 1} \, {{\,_{p,q} F_4^{(\kappa, \mu )} [a + 1,b + 1;c + 1 - k,d;x,y]} \over {(c - k)(c - k - 1)}}.

Proof

The proof of the Theorem 24 is same as the proof of Theorem 21.

Remark 2

Taking p = 1 and q = κ = μ = 0 in the relation Theorem 21–Theorem 24, it is easily seen that the special case of recursion formulas of Appell hypergeometric functions [32].

Conclusions

We may also give point to that results obtained in this work are of general character and can appropriate to give a new generalization of the Pochhammer symbol by means of the generalization of extended gamma function (4) [31]. Using the generalization of Pochhammer symbol, we give a generalization of the extended hypergeometric functions one or several variables. Also, we obtain various integral representations, derivative formulas and certain properties of these functions.

eISSN:: 2444-8656
Language:: English

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Journal Subjects:: Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics

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A New Generalization of Pochhammer Symbol and Its Applications

Published Online: Mar 31, 2020

Page range: 255 - 266

Received: Jun 14, 2019

Accepted: Jul 10, 2019

DOI: https://doi.org/10.2478/amns.2020.1.00023

Keywordsgamma function, beta function, Pochhammer symbol, Gauss hypergeometric function, confluent hypergeometric function, Appell hypergeometric functions, Humbert hypergeometric functions of two variables, integral representations, derivative formulas, recurrence relation

© 2020 Recep Şahin et al., published by Sciendo

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

Keywords
gamma function, beta function, Pochhammer symbol, Gauss hypergeometric function, confluent hypergeometric function, Appell hypergeometric functions, Humbert hypergeometric functions of two variables, integral representations, derivative formulas, recurrence relation