A study on certain properties of generalized special functions defined by Fox-Wright function

On the last quarter century, some generalizations of special functions, which frequently used in applied mathematics, have been studied by many scientists [1, 7,8, 9, 10, 11, 12, 13,14, 17, 19,20,21,22,23,24,25,26,27,28,29, 31, 32, 33]. Chaudhry and Zubair [10] defined the extended gamma function in 1994 as $Γ_{p} (x) = \int_{0}^{\infty} t^{x - 1} \exp [- t - \frac{p}{t}] dt,$ {\Gamma _p}(x) = \int_0^\infty {t^{x - 1}}\exp \left[ { - t - {p \over t}} \right]dt, where Re(p) > 0. Three years later, Chaudhry et al. [7] defined the extended beta function as $B_{p} (x, y) = \int_{0}^{1} t^{x - 1} {(1 - t)}^{y - 1} \exp [- \frac{p}{t (1 - t)}] dt,$ {B_p}(x,y) = \int_0^1 {t^{x - 1}}{(1 - t)^{y - 1}}\exp \left[ { - {p \over {t(1 - t)}}} \right]dt, where Re(p) > 0, Re(x) > 0, Re(y) > 0. It clearly seems that, for p = 0, Γ₀(x) = Γ(x) and B₀(x,y) = B(x,y), where Γ(x) and B(x,y) are the classical gamma and beta functions [6].

In 2004, Chaudhry et al. [8] used B_p(x,y) to extend the Gauss and confluent hypergeometric functions as follows: (1) $_{} F_{p} (a, b; c; z) = \sum_{n = 0}^{\infty} {(a)}_{n} \frac{B_{p} (b + n, c - b)}{B (b, c - b)} \frac{z^{n}}{n!},$ _{}{F_p}(a,b;c;z) = \sum\limits_{n = 0}^\infty {(a)_n}{{{B_p}(b + n,c - b)} \over {B(b,c - b)}}{{{z^n}} \over {n!}},(2) $_{} Φ_{p} (b; c; z) = \sum_{n = 0}^{\infty} \frac{B_{p} (b + n, c - b)}{B (b, c - b)} \frac{z^{n}}{n!},$ _{}{\Phi _p}(b;c;z) = \sum\limits_{n = 0}^\infty {{{B_p}(b + n,c - b)} \over {B(b,c - b)}}{{{z^n}} \over {n!}}, where p ≥ 0, Re(c) > Re(b) > 0. In the same paper, the authors also gave the integral representations of (1) and (2) as $_{} F_{p} (a, b; c; z) = \frac{1}{B (b, c - b)} \int_{0}^{1} t^{b - 1} {(1 - t)}^{c - b - 1} {(1 - zt)}^{- a} \exp [- \frac{p}{t (1 - t)}] dt,$ _{}{F_p}(a,b;c;z) = {1 \over {B(b,c - b)}}\int_0^1 {t^{b - 1}}{(1 - t)^{c - b - 1}}{(1 - zt)^{ - a}}\exp \left[ { - {p \over {t(1 - t)}}} \right]dt, where p > 0, p = 0 and |arg(1 − z)| < π < p, Re(c) > Re(b) > 0, and $_{} Φ_{p} (b; c; z) = \frac{1}{B (b, c - b)} \int_{0}^{1} t^{b - 1} {(1 - t)}^{c - b - 1} \exp [zt - \frac{p}{t (1 - t)}] dt,$ _{}{\Phi _p}(b;c;z) = {1 \over {B(b,c - b)}}\int_0^1 {t^{b - 1}}{(1 - t)^{c - b - 1}}\exp \left[ {zt - {p \over {t(1 - t)}}} \right]dt, where p > 0, p = 0 and Re(c) > Re(b) > 0. Here (a)_n is the Pochhammer symbol which defined as ${(a)}_{ν} = \frac{Γ (a + ν)}{Γ (a)}, a, ν \in ℂ$ {(a)_\nu } = {{\Gamma (a + \nu )} \over {\Gamma (a)}},\;a,\nu \in \mathbb{C} with the assume (a)₀ ≣ 1.

The Fox-Wright function is given in [18] as (3) $_{ξ} Ψ_{η} (z {) =}_{ξ} Ψ_{η} [ \begin{matrix} {(β_{i}, α_{i})}_{1, ξ} \\ {(μ_{j}, κ_{j})}_{1, η} \end{matrix} | z] = \sum_{n = 0}^{\infty} \frac{\prod_{i = 1}^{ξ} Γ (α_{i} n + β_{i})}{\prod_{j = 1}^{η} Γ (κ_{j} n + μ_{j})} \frac{z^{n}}{n!},$ _\xi {\Psi _\eta }(z{) = _\xi }{\Psi _\eta }\left[ {\matrix{ {{{({\beta _i},{\alpha _i})}_{1,\xi }}} \cr {{{({\mu _j},{\kappa _j})}_{1,\eta }}} \cr } |z} \right] = \sum\limits_{n = 0}^\infty {{\prod\limits_{i = 1}^\xi \Gamma ({\alpha _i}n + {\beta _i})} \over {\prod\limits_{j = 1}^\eta \Gamma ({\kappa _j}n + {\mu _j})}}{{{z^n}} \over {n!}}, where z,β_i, µ_j∈ ℂ,α_i,κ_j∈ ℝ,i = 1...ξ and j = 1...η. The asymptotic behaviour of the above function was studied by Fox [15, 16] and Wright [34, 35, 36] for the large values of z, considering the condition $\sum_{j = 1}^{η} κ_{j} - \sum_{i = 1}^{ξ} α_{i} > - 1 .$ \sum\limits_{j = 1}^\eta {\kappa _j} - \sum\limits_{i = 1}^\xi {\alpha _i} > - 1. If these conditions are met, for any z ∈ ℂ the series (3) is convergent. For κ, µ,z ∈ ℂ,Re(κ) > −1, the classic Wright function [18] $_{0} Ψ_{1} (z {) =}_{0} Ψ_{1} [\begin{matrix} \underline{} \\ (μ, κ) \end{matrix} | z] = \sum_{n = 0}^{\infty} \frac{1}{Γ (κ n + μ)} \frac{z^{n}}{n!}$ _0{\Psi _1}(z{) = _0}{\Psi _1}\left[ {\matrix{ {\underline \quad } \cr {(\mu ,\kappa )} \cr } |z} \right] = \sum\limits_{n = 0}^\infty {1 \over {\Gamma (\kappa n + \mu )}}{{{z^n}} \over {n!}} can obtained by choosing ξ = 0 and η = 1 in equation (3).

Inspired by the aforementioned studies and motivated by the frequent use of Fox-Wright function in the theory of special functions, we defined two new functions as generalizations of gamma and beta functions.

Generalized functions and their properties

Throughout the study, we assume that x,y,z ∈ ℂ, k,m,n ∈ ℕ, α_i,κ_j ∈ ℝ, β_i, µ_j,a,b,c, p ∈ ℂ, Re(p) > 0, Re(x) > 0, Re(y) > 0, and Re(c) > Re(b) > 0. For the sake of shortness, we did not wrote these conditions for the rest of the article, unless otherwise stated.

Let us defined the new generalizations as (4) $^{Ψ} {\hat{Γ}}_{p} (x {) : =}^{Ψ} Γ_{p} [ \begin{matrix} {(β_{i}, α_{i})}_{1, ξ} \\ {(μ_{j}, κ_{j})}_{1, η} \end{matrix} | x] = \int_{0}^{\infty} t^{x - 1}_{ξ} Ψ_{η} (- t - \frac{p}{t}) dt$ ^\Psi {\Gamma _p}(x{): = ^\Psi }{\Gamma _p}\left[ {\matrix{{{{({\beta _i},{\alpha _i})}_{1,\xi }}} \cr {{{({\mu _j},{\kappa _j})}_{1,\eta }}} \cr } |x} \right] = \int_0^\infty {t^{x - 1}}_\xi {\Psi _\eta }\left( { - t - {p \over t}} \right)dt and (5) $^{Ψ} {\hat{B}}_{p} (x, y {) : =}^{Ψ} B_{p} [ \begin{matrix} {(β_{i}, α_{i})}_{1, ξ} \\ {(μ_{j}, κ_{j})}_{1, η} \end{matrix} | x, y] = \int_{0}^{1} t^{x - 1} {(1 - t)}^{y - 1}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) dt .$ ^\Psi {\hat B_p}(x,y{): = ^\Psi }{B_p}\left[ {\matrix{{{{({\beta _i},{\alpha _i})}_{1,\xi }}} \cr {{{({\mu _j},{\kappa _j})}_{1,\eta }}} \cr } |x,y} \right] = \int_0^1 {t^{x - 1}}{(1 - t)^{y - 1}}_\xi {\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right)dt. We called them as _ξΨ_η-gamma and _ξΨ_η-beta functions.

Our first theorem is about the current relationship of the two _ξ Ψ_η-gamma functions.

Theorem 1

The following equality holds true: $\begin{array}{l} ^{Ψ} {\hat{Γ}}_{p} (^{x) Ψ} {\hat{Γ}}_{p} (y) = 4 \int_{0}^{\frac{π}{2}} \int_{0}^{\infty} r^{2 (x + y) - 1} {(\cos θ)}^{2 x - 1} {(\sin θ)}^{2 y - 1} \\ \times_{ξ} Ψ_{η} (- r^{2} {(\cos θ)}^{2} - \frac{p}{r^{2} {(\cos θ)}^{2}}) \\ \times_{ξ} Ψ_{η} (- r^{2} {(\sin θ)}^{2} - \frac{p}{r^{2} {(\sin θ)}^{2}}) drd θ . \end{array}$ \eqalign{ & ^\Psi {\Gamma _p}{(x)^\Psi }{\Gamma _p}(y) = 4\int_0^{{\pi \over 2}} \int_0^\infty {r^{2(x + y) - 1}}{(\cos \theta )^{2x - 1}}{(\sin \theta )^{2y - 1}} \cr & { \times _\xi }{\Psi _\eta }\left( { - {r^2}{{(\cos \theta )}^2} - {p \over {{r^2}{{(\cos \theta )}^2}}}} \right) \cr & { \times _\xi }{\Psi _\eta }\left( { - {r^2}{{(\sin \theta )}^2} - {p \over {{r^2}{{(\sin \theta )}^2}}}} \right)drd\theta . \cr}

Proof

Substituting t = u² in (4), we get $^{Ψ} {\hat{Γ}}_{p} (x) = 2 \int_{0}^{\infty} u^{2 x - 1}_{ξ} Ψ_{η} (- u^{2} - \frac{p}{u^{2}}) du .$ ^\Psi {\Gamma _p}(x) = 2\int_0^\infty {u^{2x - 1}}_\xi {\Psi _\eta }\left( { - {u^2} - {p \over {{u^2}}}} \right)du. Therefore, $^{Ψ} {\hat{Γ}}_{p} {(x)}^{Ψ} {\hat{Γ}}_{p} (y) = 4 \int_{0}^{\infty} \int_{0}^{\infty} u^{2 x - 1} v^{2 y - 1}_{ξ} Ψ_{η} {(- u^{2} - \frac{p}{u^{2}})}_{ξ} Ψ_{η} (- v^{2} - \frac{p}{v^{2}}) dudv .$ ^\Psi {\Gamma _p}{(x)^\Psi }{\Gamma _p}(y) = 4\int_0^\infty \int_0^\infty {u^{2x - 1}}{v^{2y - 1}}_\xi {\Psi _\eta }{\left( { - {u^2} - {p \over {{u^2}}}} \right)_\xi }{\Psi _\eta }\left( { - {v^2} - {p \over {{v^2}}}} \right)dudv. In the above equality, taking u = r(cosθ) and v = r(sinθ) yields $\begin{array}{l} ^{Ψ} {\hat{Γ}}_{p} {(x)}^{Ψ} {\hat{Γ}}_{p} (y) = 4 \int_{0}^{\frac{π}{2}} \int_{0}^{\infty} r^{2 (x + y) - 1} {(\cos θ)}^{2 x - 1} {(\sin θ)}^{2 y - 1} \\ \times_{ξ} Ψ_{η} (- r^{2} {(\cos θ)}^{2} - \frac{p}{r^{2} {(\cos θ)}^{2}}) \\ \times_{ξ} Ψ_{η} (- r^{2} {(\sin θ)}^{2} - \frac{p}{r^{2} {(\sin θ)}^{2}}) drd θ, \end{array}$ \eqalign{ & ^\Psi {\Gamma _p}{(x)^\Psi }{\Gamma _p}(y) = 4\int_0^{{\pi \over 2}} \int_0^\infty {r^{2(x + y) - 1}}{(\cos \theta )^{2x - 1}}{(\sin \theta )^{2y - 1}} \cr & { \times _\xi }{\Psi _\eta }\left( { - {r^2}{{(\cos \theta )}^2} - {p \over {{r^2}{{(\cos \theta )}^2}}}} \right) \cr & { \times _\xi }{\Psi _\eta }\left( { - {r^2}{{(\sin \theta )}^2} - {p \over {{r^2}{{(\sin \theta )}^2}}}} \right)drd\theta , \cr} which completes the proof.

Theorem 2

The_ξΨ_η-beta function has the following integral representations: $\begin{array}{l} ^{Ψ} {\hat{B}}_{p} (x, y) = 2 \int_{0}^{\frac{π}{2}} {(\sin θ)}^{2 x - 1} {(\cos θ)}^{2 y - 1}_{ξ} Ψ_{η} (- p {(\sec θ)}^{2} {(\csc θ)}^{2}) d θ, \\ ^{Ψ} {\hat{B}}_{p} (x, y) = \int_{0}^{\infty} \frac{u^{x - 1}}{{(1 + u)}^{x + y}} ξ Ψ_{η} (- 2 p - p (u + \frac{1}{u})) du, \\ ^{Ψ} {\hat{B}}_{p} (x, y) = (c - a)^{1 - x - y} \int_{a}^{c} {(u - a)}^{x - 1} {(c - u)}^{y - 1}_{ξ} Ψ_{η} (\frac{- p {(c - a)}^{2}}{(u - a) (c - u)}) du . \end{array}$ \eqalign{ & ^\Psi {{\hat B}_p}(x,y) = 2\int_0^{{\pi \over 2}} {(\sin \theta )^{2x - 1}}{(\cos \theta )^{2y - 1}}_\xi {\Psi _\eta }\left( { - p{{(\sec \theta )}^2}{{(\csc \theta )}^2}} \right)d\theta , \cr & ^\Psi {{\hat B}_p}(x,y) = \int_0^\infty {{{u^{x - 1}}} \over {{{(1 + u)}^{x + y}}}}{ _\xi }{\Psi _\eta }\left( { - 2p - p\left( {u + {1 \over u}} \right)} \right)du, \cr & ^\Psi {{\hat B}_p}(x,y) = (c - a{)^{1 - x - y}}\int_a^c {(u - a)^{x - 1}}{(c - u)^{y - 1}}_\xi {\Psi _\eta }\left( {{{ - p{{(c - a)}^2}} \over {(u - a)(c - u)}}} \right)du. \cr}

Proof

Taking t = (sinθ)² in (5), we get $\begin{array}{l} ^{Ψ} {\hat{B}}_{p} (x, y) = \int_{0}^{1} t^{x - 1} {(1 - t)}^{y - 1}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) dt \\ = 2 \int_{0}^{\frac{π}{2}} {(\sin θ)}^{2 x - 1} {(\cos θ)}^{2 y - 1}_{ξ} Ψ_{η} (- p {(\sec θ)}^{2} {(\csc θ)}^{2}) d θ . \end{array}$ \eqalign{ & ^\Psi {{\hat B}_p}(x,y) = \int_0^1 {t^{x - 1}}{(1 - t)^{y - 1}}_\xi {\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right)dt \cr & = 2\int_0^{{\pi \over 2}} {(\sin \theta )^{2x - 1}}{(\cos \theta )^{2y - 1}}_\xi {\Psi _\eta }\left( { - p{{(\sec \theta )}^2}{{(\csc \theta )}^2}} \right)d\theta . \cr}

Taking $t = \frac{u}{1 + u}$ t = {u \over {1 + u}} in (5), we get $\begin{array}{l} ^{Ψ} {\hat{B}}_{p} (x, y) = \int_{0}^{\infty} {( \frac{u}{1 + u} )}^{x - 1} {( \frac{1}{1 + u} )}^{y - 1} {( \frac{1}{1 + u} )}^{2} _{ξ} Ψ_{η} (- \frac{p}{(\frac{u}{1 + u}) (\frac{1}{1 + u})}) du \\ = \int_{0}^{\infty} \frac{u^{x - 1}}{{(1 + u)}^{x + y}} ξ Ψ_{η} (- 2 p - p (u + \frac{1}{u})) du . \end{array}$ \eqalign{ & ^\Psi {{\hat B}_p}(x,y) = \int_0^\infty {\left( {{u \over {1 + u}}} \right)^{x - 1}}{\left( {{1 \over {1 + u}}} \right)^{y - 1}}{\left( {{1 \over {1 + u}}} \right)^2}{_\xi }{\Psi _\eta }\left( { - {p \over {\left( {{u \over {1 + u}}} \right)\left( {{1 \over {1 + u}}} \right)}}} \right)du \cr & = \int_0^\infty {{{u^{x - 1}}} \over {{{(1 + u)}^{x + y}}}}{ _\xi }{\Psi _\eta }\left( { - 2p - p\left( {u + {1 \over u}} \right)} \right)du. \cr} Taking $t = \frac{u - a}{c - a}$ t = {{u - a} \over {c - a}} in (5), we get $\begin{array}{l} ^{Ψ} {\hat{B}}_{p} (x, y) = \int_{a}^{c} {(\frac{u - a}{c - a})}^{x - 1} {(1 - \frac{u - a}{c - a})}^{y - 1} \frac{1}{c - a} ξ Ψ_{η} (\frac{- p {(c - a)}^{2}}{(u - a) (c - u)}) du \\ = {(c - a)}^{1 - x - y} \int_{a}^{c} {(u - a)}^{x - 1} {(c - u)}^{y - 1} ξ Ψ_{η} (- \frac{p {(c - a)}^{2}}{(u - a) (c - u)}) du, \end{array}$ \eqalign{ & ^\Psi {{\hat B}_p}(x,y) = \int_a^c {\left( {{{u - a} \over {c - a}}} \right)^{x - 1}}{\left( {1 - {{u - a} \over {c - a}}} \right)^{y - 1}}{1 \over {c - a}}{ _\xi }{\Psi _\eta }\left( {{{ - p{{(c - a)}^2}} \over {(u - a)(c - u)}}} \right)du \cr & = (c - a{)^{1 - x - y}}\int_a^c {(u - a)^{x - 1}}{(c - u)^{y - 1}}{ _\xi }{\Psi _\eta }\left( { - {{p{{(c - a)}^2}} \over {(u - a)(c - u)}}} \right)du, \cr} which gives the result.

Theorem 3

The following derivative formula is provided for Re(x) > m, Re(y) > m: $\frac{d^{m}}{{dp}^{m}} {^{Ψ} {\hat{B}}_{p} (x, y)} = {(- 1)}^{m} Ψ B_{p} [ \begin{matrix} {(α_{i} m + β_{i}, α_{i})}_{1, ξ} \\ {(κ_{j} m + μ_{j}, κ_{j})}_{1, η} \end{matrix} | x - m, y - m] .$ {{{d^m}} \over {d{p^m}}}\left\{ {^\Psi {{\hat B}_p}(x,y)} \right\} = {( - 1)^m}{ ^\Psi }{B_p}\left[ {\matrix{ {{{({\alpha _i}m + {\beta _i},{\alpha _i})}_{1,\xi }}} \cr {{{({\kappa _j}m + {\mu _j},{\kappa _j})}_{1,\eta }}} \cr } |x - m,y - m} \right].

Proof

It is done by induction. The first order derivative of (5) is as follows: $\begin{array}{l} \frac{d}{dp} {^{Ψ} {\hat{B}}_{p} (x, y)} = \frac{d}{dp} {\int_{0}^{1} t^{x - 1} {(1 - t)}^{y - 1}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) dt} \\ = (- 1^{) Ψ} B_{p} [ \begin{matrix} {(α_{i} + β_{i}, α_{i})}_{1, ξ} \\ {(κ_{j} + μ_{j}, κ_{j})}_{1, η} \end{matrix} | x - 1, y - 1] . \end{array}$ \eqalign{& {d \over {dp}}\left\{ {^\Psi {{\hat B}_p}(x,y)} \right\} = {d \over {dp}}\left\{ {\int_0^1 {t^{x - 1}}{{(1 - t)}^{y - 1}}_\xi {\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right)dt} \right\} \cr & = ( - {1) ^\Psi }{B_p}\left[ {\matrix{{{{({\alpha _i} + {\beta _i},{\alpha _i})}_{1,\xi }}} \cr {{{({\kappa _j} + {\mu _j},{\kappa _j})}_{1,\eta }}} \cr } |x - 1,y - 1} \right]. \cr} Let us assume that the k-order derivative of (5) is (6) $\frac{d^{k}}{{dp}^{k}} {^{Ψ} {\hat{B}}_{p} (x, y)} = {(- 1)}^{k}^{Ψ} B_{p} [ \begin{matrix} {(α_{i} k + β_{i}, α_{i})}_{1, ξ} \\ {(κ_{j} k + μ_{j}, κ_{j})}_{1, η} \end{matrix} | x - k, y - k] .$ {{{d^k}} \over {d{p^k}}}\left\{ {^\Psi {{\hat B}_p}(x,y)} \right\} = {( - 1)^k}{ ^\Psi }{B_p}\left[ {\matrix{{{{({\alpha _i}k + {\beta _i},{\alpha _i})}_{1,\xi }}} \cr {{{({\kappa _j}k + {\mu _j},{\kappa _j})}_{1,\eta }}} \cr } |x - k,y - k} \right]. From the first order derivative of (6), the k + 1-order derivative is found as follows: $\begin{array}{l} \frac{d^{k + 1}}{{dp}^{k + 1}} {^{Ψ} {\hat{B}}_{p} (x, y)} = \frac{d}{dp} {\frac{d^{k}}{{dp}^{k}} {^{Ψ} {\hat{B}}_{p} (x, y)}} \\ = {(- 1)}^{k + 1}^{Ψ} B_{p} [ \begin{matrix} {(α_{i} (k + 1) + β_{i}, α_{i})}_{1, ξ} \\ {(κ_{j} (k + 1) + μ_{j}, κ_{j})}_{1, η} \end{matrix} | x - (k + 1), y - (k + 1)] . \end{array}$ \eqalign{& {{{d^{k + 1}}} \over {d{p^{k + 1}}}}\left\{ {^\Psi {{\hat B}_p}(x,y)} \right\} = {d \over {dp}}\left\{ {{{{d^k}} \over {d{p^k}}}\left\{ {^\Psi {{\hat B}_p}(x,y)} \right\}} \right\} \cr & = ( - {1)^{k + 1}}{ ^\Psi }{B_p}\left[ {\matrix{{{{({\alpha _i}(k + 1) + {\beta _i},{\alpha _i})}_{1,\xi }}} \cr {{{({\kappa _j}(k + 1) + {\mu _j},{\kappa _j})}_{1,\eta }}} \cr } |x - (k + 1),y - (k + 1)} \right]. \cr} This gives the result.

Theorem 4

The following equality is provided for Re(s) > 0: $ℳ [^{Ψ} {\hat{B}}_{p} (x, y)] = B {(x + s, y + s)}^{Ψ} {\hat{Γ}}_{p} (s) .$ {\cal M}\left[ {^\Psi {{\hat B}_p}(x,y)} \right] = B{(x + s,y + s)^\Psi }{\Gamma _p}(s).

Proof

If we apply Mellin transformation according to argument p in equation (5), we have (7) $\begin{array}{l} ℳ [^{Ψ} {\hat{B}}_{p} (x, y)] = \int_{0}^{\infty} p^{s - 1} \int_{0}^{1} t^{x - 1} {(1 - t)}^{y - 1}_{ξ} Ψ_{η} (\frac{- p}{t (1 - t)}) dtdp \\ = \int_{0}^{1} t^{x - 1} {(1 - t)}^{y - 1} \int_{0}^{\infty} p^{s - 1}_{ξ} Ψ_{η} (\frac{- p}{t (1 - t)}) dpdt . \end{array}$ \eqalign{ & {\cal M}\left[ {^\Psi {{\hat B}_p}(x,y)} \right] = \int_0^\infty {p^{s - 1}}\int_0^1 {t^{x - 1}}{(1 - t)^{y - 1}}_\xi {\Psi _\eta }\left( {{{ - p} \over {t(1 - t)}}} \right)dtdp \cr & = \int_0^1 {t^{x - 1}}{(1 - t)^{y - 1}}\int_0^\infty {p^{s - 1}}_\xi {\Psi _\eta }\left( {{{ - p} \over {t(1 - t)}}} \right)dpdt. \cr} Letting $v = \frac{p}{t (1 - t)}$ v = {p \over {t(1 - t)}} in (7), we get $ℳ [^{Ψ} {\hat{B}}_{p} (x, y)] = \int_{0}^{1} t^{x + s - 1} {(1 - t)}^{y + s - 1} dt \int_{0}^{\infty} v^{s - 1}_{ξ} Ψ_{η} (- v) dv .$ {\cal M}\left[ {^\Psi {{\hat B}_p}(x,y)} \right] = \int_0^1 {t^{x + s - 1}}{(1 - t)^{y + s - 1}}dt\int_0^\infty {v^{s - 1}}_\xi {\Psi _\eta }\left( { - v} \right)dv.

Thus, we have $ℳ [^{Ψ} {\hat{B}}_{p} (x, y)] = B {(x + s, y + s)}^{Ψ} Γ_{p} (s),$ {\cal M}\left[ {^\Psi {{\hat B}_p}(x,y)} \right] = B{(x + s,y + s)^\Psi }{\Gamma _p}(s), which completes the proof.

Remark 1

By using the inverse Mellin transform, it is easy to see $^{Ψ} {\hat{B}}_{p} (x, y) = \frac{1}{2 π i} \int_{- i \infty}^{+ i \infty} B {(x + s, y + s)}^{Ψ} Γ_{p} (s) p^{- s} ds$ ^\Psi {\hat B_p}(x,y) = {1 \over {2\pi i}}\int_{ - i\infty }^{ + i\infty } B{(x + s,y + s)^\Psi }{\Gamma _p}(s){p^{ - s}}ds for Re(s) > 0.

Theorem 5

The following equality holds true: $^{Ψ} {\hat{B}}_{p} (x, y) = ^{Ψ} {\hat{B}}_{p} (x + 1, y) + ^{Ψ} {\hat{B}}_{p} (x, y + 1) .$ ^\Psi {\hat B_p}(x,y) = {^\Psi }{\hat B_p}(x + 1,y) + {^\Psi }{\hat B_p}(x,y + 1).

Proof

Direct calculation yields $\begin{array}{l} ^{Ψ} {\hat{B}}_{p} (x, y) = \int_{0}^{1} t^{x - 1} {(1 - t)}^{y - 1}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) dt \\ = \int_{0}^{1} t^{x} {(1 - t)}^{y} \frac{1}{t (1 - t)} ξ Ψ_{η} (- \frac{p}{t (1 - t)}) dt \\ = \int_{0}^{1} t^{x} {(1 - t)}^{y} {[{(1 - t)}^{- 1} + t^{- 1}]}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) dt \\ = \int_{0}^{1} {[t^{x} {(1 - t)}^{y - 1} + t^{x - 1} {(1 - t)}^{y}]}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) dt \\ = \int_{0}^{1} t^{x} {(1 - t)}^{y - 1}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) dt + \int_{0}^{1} t^{x - 1} {(1 - t)}^{y}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) dt \\ =^{Ψ} {\hat{B}}_{p} (x + 1, y) + ^{Ψ} {\hat{B}}_{p} (x, y + 1), \end{array}$ \eqalign{ & ^\Psi {{\hat B}_p}(x,y) = \int_0^1 {t^{x - 1}}{(1 - t)^{y - 1}}_\xi {\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right)dt \cr & = \int_0^1 {t^x}{(1 - t)^y}{1 \over {t(1 - t)}}{ _\xi }{\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right)dt \cr & = \int_0^1 {t^x}{(1 - t)^y}{\left[ {{{(1 - t)}^{ - 1}} + {t^{ - 1}}} \right]_\xi }{\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right)dt \cr & = \int_0^1 {\left[ {{t^x}{{(1 - t)}^{y - 1}} + {t^{x - 1}}{{(1 - t)}^y}} \right]_\xi }{\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right)dt \cr & = \int_0^1 {t^x}{(1 - t)^{y - 1}}_\xi {\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right)dt + \int_0^1 {t^{x - 1}}{(1 - t)^y}_\xi {\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right)dt \cr & { = ^\Psi }{{\hat B}_p}(x + 1,y) + {^\Psi }{{\hat B}_p}(x,y + 1), \cr} which is the result.

Theorem 6

The following summation formula is provided for Re(y) < 1: $^{Ψ} {\hat{B}}_{p} (x, 1 - y) = \sum_{n = 0}^{\infty} {\frac{{(y)}_{n}}{n!}}^{Ψ} {\hat{B}}_{p} (x + n, 1) .$ ^\Psi {\hat B_p}(x,1 - y) = \sum\limits_{n = 0}^\infty {{{{{(y)}_n}} \over {n!}}^\Psi }{\hat B_p}(x + n,1).

Proof

From the definition of the _ξ Ψ_η-beta function, we obtain $^{Ψ} {\hat{B}}_{p} (x, 1 - y) = \int_{0}^{1} t^{x - 1} {(1 - t)}^{- y}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) dt .$ ^\Psi {\hat B_p}(x,1 - y) = \int_0^1 {t^{x - 1}}{(1 - t)^{ - y}}_\xi {\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right)dt. With the help of the following series expression ${(1 - t)}^{- y} = \sum_{n = 0}^{\infty} {(y)}_{n} \frac{t^{n}}{n!}, | t | < 1,$ {(1 - t)^{ - y}} = \sum\limits_{n = 0}^\infty {(y)_n}{{{t^n}} \over {n!}},\quad |t| < 1, we obtain $\begin{array}{l} ^{Ψ} {\hat{B}}_{p} (x, 1 - y) = \int_{0}^{1} \sum_{n = 0}^{\infty} \frac{{(y)}_{n}}{n!} t^{x + n - 1}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) dt \\ = \sum_{n = 0}^{\infty} \frac{{(y)}_{n}}{n!} \int_{0}^{1} t^{x + n - 1}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) dt \\ = \sum_{n = 0}^{\infty} {\frac{{(y)}_{n}}{n!}}^{Ψ} {\hat{B}}_{p} (x + n, 1) . \end{array}$ \eqalign{ & ^\Psi {{\hat B}_p}(x,1 - y) = \int_0^1 \sum\limits_{n = 0}^\infty {{{{(y)}_n}} \over {n!}}{t^{x + n - 1}}_\xi {\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right)dt \cr & = \sum\limits_{n = 0}^\infty {{{{(y)}_n}} \over {n!}}\int_0^1 {t^{x + n - 1}}_\xi {\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right)dt \cr & = \sum\limits_{n = 0}^\infty {{{{{(y)}_n}} \over {n!}}^\Psi }{{\hat B}_p}(x + n,1). \cr} This completes the proof.

Theorem 7

The following equality holds true: $^{Ψ} {\hat{B}}_{p} (x, y) = {\sum_{n = 0}^{\infty}}^{Ψ} {\hat{B}}_{p} (x + n, y + 1) .$ ^\Psi {\hat B_p}(x,y) = {\sum\limits_{n = 0}^\infty ^\Psi }{\hat B_p}(x + n,y + 1).

Proof

From the definition of the _ξΨ_η-beta function, we get $^{Ψ} {\hat{B}}_{p} (x, y) = \int_{0}^{1} t^{x - 1} {(1 - t)}^{y - 1}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) dt .$ ^\Psi {\hat B_p}(x,y) = \int_0^1 {t^{x - 1}}{(1 - t)^{y - 1}}_\xi {\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right)dt. With the help of the following series expression ${(1 - t)}^{y - 1} = {(1 - t)}^{y} \sum_{n = 0}^{\infty} t^{n}, | t | < 1,$ {(1 - t)^{y - 1}} = {(1 - t)^y}\sum\limits_{n = 0}^\infty {t^n},\quad |t| < 1, we obtain $\begin{array}{l} ^{Ψ} {\hat{B}}_{p} (x, y) = \int_{0}^{1} t^{x - 1} {(1 - t)}^{y} \sum_{n = 0}^{\infty} t^{n}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) dt \\ = \sum_{n = 0}^{\infty} \int_{0}^{1} t^{x + n - 1} {(1 - t)}^{y}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) dt \\ = {\sum_{n = 0}^{\infty}}^{Ψ} {\hat{B}}_{p} (x + n, y + 1), \end{array}$ \eqalign{& ^\Psi {{\hat B}_p}(x,y) = \int_0^1 {t^{x - 1}}{(1 - t)^y}\sum\limits_{n = 0}^\infty {t^n}_\xi {\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right)dt \cr & = \sum\limits_{n = 0}^\infty \int_0^1 {t^{x + n - 1}}{(1 - t)^y}_\xi {\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right)dt \cr & = {\sum\limits_{n = 0}^\infty ^\Psi }{{\hat B}_p}(x + n,y + 1), \cr} which gives the result.

Theorem 8

The following relation is provided for Re(x) > 1,Re(y) > 1: (8) $\begin{array}{l} x^{Ψ} {\hat{B}}_{p} (x, y + 1) = y^{Ψ} {\hat{B}}_{p} (x + 1, y) \\ + 2 p^{Ψ} B_{p} [ \begin{matrix} {(α_{i} + β_{i}, α_{i})}_{1, ξ} \\ {(κ_{j} + μ_{j}, κ_{j})}_{1, η} \end{matrix} | x, y - 1] \\ - p^{Ψ} B_{p} [ \begin{matrix} {(α_{i} + β_{i}, α_{i})}_{1, ξ} \\ {(κ_{j} + μ_{j}, κ_{j})}_{1, η} \end{matrix} | x - 1, y - 1] . \end{array}$ \eqalign{& {x^\Psi }{{\hat B}_p}(x,y + 1) = \;{y^\Psi }{{\hat B}_p}(x + 1,y) \cr & + 2{p^\Psi }{B_p}\left[ {\matrix{{{{({\alpha _i} + {\beta _i},{\alpha _i})}_{1,\xi }}} \cr {{{({\kappa _j} + {\mu _j},{\kappa _j})}_{1,\eta }}} \cr } |x,y - 1} \right] \cr & - {p^\Psi }{B_p}\left[ {\matrix{{{{({\alpha _i} + {\beta _i},{\alpha _i})}_{1,\xi }}} \cr {{{({\kappa _j} + {\mu _j},{\kappa _j})}_{1,\eta }}} \cr } |x - 1,y - 1} \right]. \cr}

Proof

(5) equality provides the following equation $^{Ψ} {\hat{B}}_{p} (x, y) = ℳ [\hat{f} (t : y; p) : x]$ ^\Psi {\hat B_p}(x,y) = {\cal M}[\hat f(t:y;p):x] where $\hat{f} (t : y; p) = {(1 - t)}^{y - 1} H (1 - t)_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)})$ \hat f(t:y;p) = {(1 - t)^{y - 1}}H(1 - t{) _\xi }{\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right) and $H (1 - t) = {\begin{matrix} 0, t > 1, \\ 1, t < 1 . \end{matrix}$ H(1 - t) = \left\{ {\matrix{{0,t > 1,} \cr {1,t < 1.} \cr }} \right. The derivative of $\hat{f} (t : y; p)$ \hat f(t:y;p) according to the parameter t provides the following equation: $\begin{array}{l} \frac{d}{dt} {\hat{f} (t : y; p)} = - δ (1 - t) (1 - t)^{y - 1}_{ξ} Ψ_{η} (\frac{- p}{t (1 - t)}) \\ - (y - 1) (1 - t)^{y - 2} H (1 - t)_{ξ} Ψ_{η} (\frac{- p}{t (1 - t)}) \\ + \frac{p (1 - 2 t)}{t^{2} {(1 - t)}^{2}} {(1 - t)}^{y - 1} H (1 - t)_{ξ} Ψ_{η} (\frac{- p}{t (1 - t)}), \end{array}$ \eqalign{& {d \over {dt}}\left\{ {\hat f(t:y;p)} \right\} = - \delta (1 - t)(1 - t{)^{y - 1}}_\xi {\Psi _\eta }\left( {{{ - p} \over {t(1 - t)}}} \right) \cr & - (y - 1)(1 - t{)^{y - 2}}H(1 - t{) _\xi }{\Psi _\eta }\left( {{{ - p} \over {t(1 - t)}}} \right) \cr & + {{p(1 - 2t)} \over {{t^2}{{(1 - t)}^2}}}{(1 - t)^{y - 1}}H(1 - t{) _\xi }{\Psi _\eta }\left( {{{ - p} \over {t(1 - t)}}} \right), \cr} where $\frac{d}{dt} H (1 - t) = - δ (1 - t)$ {d \over {dt}}H(1 - t) = - \delta (1 - t) and δ represents the Dirac delta δ (1−t) = δ (t −1) = 0 for t ≠ 1. The relationship between the derivative of a function and the Mellin transformation is as follows: $ℳ [f (x) : s] = F (s) \Rightarrow ℳ [f^{'} (x) : s] = - (s - 1) F (s - 1) .$ {\cal M}[f(x):s] = F(s) \Rightarrow {\cal M}[{f^'}(x):s] = - (s - 1)F(s - 1). From here, by arranging, we find that $\begin{array}{l} - {(x - 1)}^{Ψ} {\hat{B}}_{p} (x - 1, y) = - (^{y - 1) Ψ} {\hat{B}}_{p} (x, y - 1) \\ + p^{Ψ} B_{p} [ \begin{matrix} {(α_{i} + β_{i}, α_{i})}_{1, ξ} \\ {(κ_{j} + μ_{j}, κ_{j})}_{1, η} \end{matrix} | x - 2, y - 2] \\ - 2 p^{Ψ} B_{p} [ \begin{matrix} {(α_{i} + β_{i}, α_{i})}_{1, ξ} \\ {(κ_{j} + μ_{j}, κ_{j})}_{1, η} \end{matrix} | x - 1, y - 2] . \end{array}$ \eqalign{& - {(x - 1)^\Psi }{{\hat B}_p}(x - 1,y) = - {(y - 1)^\Psi }{{\hat B}_p}(x,y - 1) \cr & + {p^\Psi }{B_p}\left[ {\matrix{{{{({\alpha _i} + {\beta _i},{\alpha _i})}_{1,\xi }}} \cr {{{({\kappa _j} + {\mu _j},{\kappa _j})}_{1,\eta }}} \cr } |x - 2,y - 2} \right] \cr & - 2{p^\Psi }{B_p}\left[ {\matrix{{{{({\alpha _i} + {\beta _i},{\alpha _i})}_{1,\xi }}} \cr {{{({\kappa _j} + {\mu _j},{\kappa _j})}_{1,\eta }}} \cr } |x - 1,y - 2} \right]. \cr} Finally if x replaced by x + 1 and y replaced by y + 1 we get (8).

_xiΨ_η-generalization of Gauss and confluent hypergeometric functions

We used the _ξ Ψ_η-beta function (5) to define the generalizations of Gauss and confluent hypergeometric functions as $^{Ψ} {\hat{F}}_{p} (a, b; c; z) : = ^{Ψ} F_{p} [ \begin{matrix} {(β_{i}, α_{i})}_{1, ξ} \\ {(μ_{j}, κ_{j})}_{1, η} \end{matrix} | a, b; c; z] = \sum_{n = 0}^{\infty} {(a)}_{n} \frac{^{Ψ} {\hat{B}}_{p} (b + n, c - b)}{B (b, c - b)} \frac{z^{n}}{n!}$ ^\Psi {\hat F_p}(a,b;c;z): = {^\Psi }{F_p}\left[ {\matrix{{{{({\beta _i},{\alpha _i})}_{1,\xi }}} \cr {{{({\mu _j},{\kappa _j})}_{1,\eta }}} \cr } | a,b;c;z} \right] = \sum\limits_{n = 0}^\infty {(a)_n}{{^\Psi {{\hat B}_p}(b + n,c - b)} \over {B(b,c - b)}}{{{z^n}} \over {n!}} and $^{Ψ} {\hat{Φ}}_{p} (b; c; z {) : =}^{Ψ} Φ_{p} [ \begin{matrix} {(β_{i}, α_{i})}_{1, ξ} \\ {(μ_{j}, κ_{j})}_{1, η} \end{matrix} | b; c; z] = \sum_{n = 0}^{\infty} \frac{^{Ψ} {\hat{B}}_{p} (b + n, c - b)}{B (b, c - b)} \frac{z^{n}}{n!},$ ^\Psi {\hat \Phi _p}(b;c;z{): = ^\Psi }{\Phi _p}\left[ {\matrix{{{{({\beta _i},{\alpha _i})}_{1,\xi }}} \cr {{{({\mu _j},{\kappa _j})}_{1,\eta }}} \cr } |b;c;z} \right] = \sum\limits_{n = 0}^\infty {{^\Psi {{\hat B}_p}(b + n,c - b)} \over {B(b,c - b)}}{{{z^n}} \over {n!}}, respectively. We call $^{Ψ} {\hat{F}}_{p} (a, b; c; z)$ ^\Psi {\hat F_p}(a,b;c;z) as _ξΨ_η-Gauss hypergeometric function and $^{Ψ} {\hat{Φ}}_{p} (b; c; z)$ ^\Psi {\hat \Phi _p}(b;c;z) as _ξΨ_η-confluent hypergeometric function.

The following two theorems are about the integral representations of _ξ Ψ_η-Gauss and _ξ Ψ_η-confluent hypergeometric functions.

Theorem 9

The_ξ Ψ_η-Gauss hypergeometric function has the following integral representations:(9) $\begin{array}{l} ^{Ψ} {\hat{F}}_{p} (a, b; c; z) = \frac{1}{B (b, c - b)} \int_{0}^{1} t^{b - 1} {(1 - t)}^{c - b - 1} {(1 - zt)}^{- a}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) dt, \\ ^{Ψ} {\hat{F}}_{p} (a, b; c; z) = \frac{1}{B (b, c - b)} \int_{0}^{\infty} u^{b - 1} {(1 + u)}^{a - c} {[1 + u (1 - z)]}^{- a}_{ξ} Ψ_{η} (- 2 p - p (u + \frac{1}{u})) du, \\ ^{Ψ} {\hat{F}}_{p} (a, b; c; z) = \frac{2}{B (b, c - b)} \int_{0}^{\frac{π}{2}} {(\sin θ)}^{2 b - 1} {(\cos θ)}^{2 c - 2 b - 1} {(1 - z {(\sin θ)}^{2})}^{- a}_{ξ} Ψ_{η} (- p {(\sec θ)}^{2} {(\csc θ)}^{2}) d θ . \end{array}$ \eqalign{& ^\Psi {{\hat F}_p}(a,b;c;z) = {1 \over {B(b,c - b)}}\int_0^1 {t^{b - 1}}{(1 - t)^{c - b - 1}}{(1 - zt)^{ - a}}_\xi {\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right)dt, \cr & ^\Psi {{\hat F}_p}(a,b;c;z) = {1 \over {B(b,c - b)}}\int_0^\infty {u^{b - 1}}{(1 + u)^{a - c}}{\left[ {1 + u(1 - z)} \right]^{ - a}}_\xi {\Psi _\eta }\left( { - 2p - p\left( {u + {1 \over u}} \right)} \right)du, \cr & ^\Psi {{\hat F}_p}(a,b;c;z) = {2 \over {B(b,c - b)}}\int_0^{{\pi \over 2}} {(\sin \theta )^{2b - 1}}{(\cos \theta )^{2c - 2b - 1}}{\left( {1 - z{{(\sin \theta )}^2}} \right)^{ - a}}_\xi {\Psi _\eta }\left( { - p{{(\sec \theta )}^2}{{(\csc \theta )}^2}} \right)d\theta . \cr}

Proof

Direct calculation yields $\begin{array}{l} ^{Ψ} {\hat{F}}_{p} (a, b; c; z) = \sum_{n = 0}^{\infty} {(a)}_{n} \frac{^{Ψ} {\hat{B}}_{p} (b + n, c - b)}{B (b, c - b)} \frac{z^{n}}{n!} \\ = \frac{1}{B (b, c - b)} \sum_{n = 0}^{\infty} {(a)}_{n} \int_{0}^{1} t^{b + n - 1} {(1 - t)}^{c - b - 1}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) \frac{z^{n}}{n!} dt \\ = \frac{1}{B (b, c - b)} \int_{0}^{1} t^{b - 1} {(1 - t)}^{c - b - 1}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) \sum_{n = 0}^{\infty} {(a)}_{n} \frac{{(zt)}^{n}}{n!} dt \\ = \frac{1}{B (b, c - b)} \int_{0}^{1} t^{b - 1} {(1 - t)}^{c - b - 1}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) {(1 - zt)}^{- a} dt . \end{array}$ \eqalign{& ^\Psi {{\hat F}_p}(a,b;c;z) = \sum\limits_{n = 0}^\infty {(a)_n}{{^\Psi {{\hat B}_p}(b + n,c - b)} \over {B(b,c - b)}}{{{z^n}} \over {n!}} \cr & = {1 \over {B(b,c - b)}}\sum\limits_{n = 0}^\infty {(a)_n}\int_0^1 {t^{b + n - 1}}{(1 - t)^{c - b - 1}}_\xi {\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right){{{z^n}} \over {n!}}dt \cr & = {1 \over {B(b,c - b)}}\int_0^1 {t^{b - 1}}{(1 - t)^{c - b - 1}}_\xi {\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right)\sum\limits_{n = 0}^\infty {(a)_n}{{{{(zt)}^n}} \over {n!}}dt \cr & = {1 \over {B(b,c - b)}}\int_0^1 {t^{b - 1}}{(1 - t)^{c - b - 1}}_\xi {\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right){(1 - zt)^{ - a}}dt. \cr} Setting $u = \frac{t}{1 - t}$ u = {t \over {1 - t}} in (9), we get $^{Ψ} {\hat{F}}_{p} (a, b; c; z) = \frac{1}{B (b, c - b)} \int_{0}^{\infty} u^{b - 1} {(1 + u)}^{a - c} {[1 + u (1 - z)]}^{- a}_{ξ} Ψ_{η} (- 2 p - p (u + \frac{1}{u})) du .$ ^\Psi {\hat F_p}(a,b;c;z) = {1 \over {B(b,c - b)}}\int_0^\infty {u^{b - 1}}{(1 + u)^{a - c}}{\left[ {1 + u(1 - z)} \right]^{ - a}}_\xi {\Psi _\eta }\left( { - 2p - p\left( {u + {1 \over u}} \right)} \right)du. Besides, substituting t = (sinθ)² in (9), we have $^{Ψ} {\hat{F}}_{p} (a, b; c; z) = \frac{2}{B (b, c - b)} \int_{0}^{\frac{π}{2}} {(\sin θ)}^{2 b - 1} {(\cos θ)}^{2 c - 2 b - 1} {(1 - z {(\sin θ)}^{2})}^{- a}_{ξ} Ψ_{η} (- p {(\sec θ)}^{2} {(\csc θ)}^{2}) d θ .$ ^\Psi {\hat F_p}(a,b;c;z) = {2 \over {B(b,c - b)}}\int_0^{{\pi \over 2}} {(\sin \theta )^{2b - 1}}{(\cos \theta )^{2c - 2b - 1}}{\left( {1 - z{{(\sin \theta )}^2}} \right)^{ - a}}_\xi {\Psi _\eta }\left( { - p{{(\sec \theta )}^2}{{(\csc \theta )}^2}} \right)d\theta .

Similarly, the _ξΨ_η-confluent hypergeometric function is also performed.

Theorem 10

The_ξ Ψ_η-confluent hypergeometric function has the following integral representations:(10) $\begin{array}{l} ^{Ψ} {\hat{Φ}}_{p} (b; c; z) = \frac{1}{B (b, c - b)} \int_{0}^{1} t^{b - 1} {(1 - t)}^{c - b - 1} e^{zt}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) dt, \\ ^{Ψ} {\hat{Φ}}_{p} (b; c; z) = \frac{1}{B (b, c - b)} \int_{0}^{1} u^{c - b - 1} {(1 - u)}^{b - 1} e^{z (1 - u)}_{ξ} Ψ_{η} (- \frac{p}{u (1 - u)}) du . \end{array}$ \eqalign{& ^\Psi {{\hat \Phi }_p}(b;c;z) = {1 \over {B(b,c - b)}}\int_0^1 {t^{b - 1}}{(1 - t)^{c - b - 1}}{e^{zt}}_\xi {\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right)dt, \cr & ^\Psi {{\hat \Phi }_p}(b;c;z) = {1 \over {B(b,c - b)}}\int_0^1 {u^{c - b - 1}}{(1 - u)^{b - 1}}{e^{z(1 - u)}}_\xi {\Psi _\eta }\left( { - {p \over {u(1 - u)}}} \right)du. \cr}

In the following theorems, we obtained the derivative formulas of _ξ Ψ_η-Gauss and _ξ Ψ_η-confluent hypergeometric functions with the help of the following equations: (11) $\begin{array}{l} B (b, c - b) = \frac{c}{b} B (b + 1, c - b), \\ {(a)}_{n + 1} = a {(a + 1)}_{n} . \end{array}$ \eqalign{& B(b,c - b) = {c \over b}B(b + 1,c - b), \cr & {(a)_{n + 1}} = a{(a + 1)_n}. \cr}

Theorem 11

The following equality holds true: $\frac{d^{n}}{{dz}^{n}} {^{Ψ} {\hat{F}}_{p} (a, b; c; z)} = \frac{{(a)}_{n} {(b)}_{n}}{{(c)}_{n}} [^{Ψ} {\hat{F}}_{p} (a + n, b + n; c + n; z)] .$ {{{d^n}} \over {d{z^n}}}\left\{ {^\Psi {{\hat F}_p}(a,b;c;z)} \right\} = {{{{(a)}_n}{{(b)}_n}} \over {{{(c)}_n}}}\left[ {^\Psi {{\hat F}_p}(a + n,b + n;c + n;z)} \right].

Proof

The derivative of the $^{Ψ} {\hat{F}}_{p} (a, b; c; z)$ ^\Psi {\hat F_p}(a,b;c;z) according to the argument z is as follows: $\begin{array}{l} \frac{d}{dz} {^{Ψ} {\hat{F}}_{p} (a, b; c; z)} = \frac{d}{dz} {\sum_{n = 0}^{\infty} {(a)}_{n} \frac{^{Ψ} {\hat{B}}_{p} (b + n, c - b)}{B (b, c - b)} \frac{z^{n}}{n!}} \\ = \sum_{n = 1}^{\infty} {(a)}_{n} \frac{^{Ψ} {\hat{B}}_{p} (b + n, c - b)}{B (b, c - b)} \frac{z^{n - 1}}{(n - 1)!} . \end{array}$ \eqalign{& {d \over {dz}}\left\{ {^\Psi {{\hat F}_p}(a,b;c;z)} \right\} = {d \over {dz}}\left\{ {\sum\limits_{n = 0}^\infty {{(a)}_n}{{^\Psi {{\hat B}_p}(b + n,c - b)} \over {B(b,c - b)}}{{{z^n}} \over {n!}}} \right\} \cr & = \sum\limits_{n = 1}^\infty {(a)_n}{{^\Psi {{\hat B}_p}(b + n,c - b)} \over {B(b,c - b)}}{{{z^{n - 1}}} \over {(n - 1)!}}. \cr} Replacing n → n + 1, we get (12) $\begin{array}{l} \frac{d}{dz} {^{Ψ} {\hat{F}}_{p} (a, b; c; z)} = \frac{(a) (b)}{(c)} \sum_{n = 0}^{\infty} {(a + 1)}_{n} \frac{^{Ψ} {\hat{B}}_{p} (b + n + 1, c - b)}{B (b + 1, c - b)} \frac{z^{n}}{n!} \\ = \frac{(a) (b)}{(c)} [^{Ψ} {\hat{F}}_{p} (a + 1, b + 1; c + 1; z)] . \end{array}$ \eqalign{& {d \over {dz}}\left\{ {^\Psi {{\hat F}_p}(a,b;c;z)} \right\} = {{(a)(b)} \over {(c)}}\sum\limits_{n = 0}^\infty {(a + 1)_n}{{^\Psi {{\hat B}_p}(b + n + 1,c - b)} \over {B(b + 1,c - b)}}{{{z^n}} \over {n!}} \cr & = {{(a)(b)} \over {(c)}}\left[ {^\Psi {{\hat F}_p}(a + 1,b + 1;c + 1;z)} \right]. \cr} Thus, the general form of the above equation is $\frac{d^{n}}{{dz}^{n}} {^{Ψ} {\hat{F}}_{p} (a, b; c; z)} = \frac{{(a)}_{n} {(b)}_{n}}{{(c)}_{n}} [^{Ψ} {\hat{F}}_{p} (a + n, b + n; c + n; z)] .$ {{{d^n}} \over {d{z^n}}}\left\{ {^\Psi {{\hat F}_p}(a,b;c;z)} \right\} = {{{{(a)}_n}{{(b)}_n}} \over {{{(c)}_n}}}\left[ {^\Psi {{\hat F}_p}(a + n,b + n;c + n;z)} \right]. This completes the proof.

Theorem 12

The following equality is provided for Re(b) > 2,Re(c) > Re(b + 2): $\begin{array}{l} (b - 1) B {(b - 1, c - b + 1)}^{Ψ} {\hat{F}}_{p} (a, b - 1; c; z) \\ = (c - b - 1) B {(b, c - b - 1)}^{Ψ} {\hat{F}}_{p} (a, b; c - 1; z) \\ - azB {(b, c - b)}^{Ψ} {\hat{F}}_{p} (a + 1, b; c; z) \\ - pB {(b - 2, c - b - 2)}^{Ψ} F_{p} [ \begin{matrix} {(α_{i} + β_{i}, α_{i})}_{1, ξ} \\ {(κ_{j} + μ_{j}, κ_{j})}_{1, η} \end{matrix} | a, b - 2; c - 4; z] \\ + 2 pB {(b - 1, c - b - 2)}^{Ψ} F_{p} [ \begin{matrix} {(α_{i} + β_{i}, α_{i})}_{1, ξ} \\ {(κ_{j} + μ_{j}, κ_{j})}_{1, η} \end{matrix} | a, b - 1; c - 3; z] . \end{array}$ \eqalign{& (b - 1)B{(b - 1,c - b + 1)^\Psi }{{\hat F}_p}(a,b - 1;c;z) \cr & = (c - b - 1)B{(b,c - b - 1)^\Psi }{{\hat F}_p}(a,b;c - 1;z) \cr & - azB{(b,c - b)^\Psi }{{\hat F}_p}(a + 1,b;c;z) \cr & - pB{(b - 2,c - b - 2)^\Psi }{F_p}\left[ {\matrix{{{{({\alpha _i} + {\beta _i},{\alpha _i})}_{1,\xi }}} & {} & {} \cr {{{({\kappa _j} + {\mu _j},{\kappa _j})}_{1,\eta }}} & {} & {} \cr } | a,b - 2;c - 4;z} \right] \cr & + 2pB{(b - 1,c - b - 2)^\Psi }{F_p}\left[ {\matrix{{{{({\alpha _i} + {\beta _i},{\alpha _i})}_{1,\xi }}} & {} & {} \cr {{{({\kappa _j} + {\mu _j},{\kappa _j})}_{1,\eta }}} & {} & {} \cr } | a,b - 1;c - 3;z} \right]. \cr}

Proof

Since $B {(b, c - b)}^{Ψ} {\hat{F}}_{p} (a, b; c; z)$ B{(b,c - b)^\Psi }{\hat F_p}(a,b;c;z) is the Mellin transform of ${\hat{f}}_{a, b, c} (t : z; p) = {(1 - t)}^{c - b - 1} {(1 - zt)}^{- a} H (1 - t)_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}),$ {\hat f_{a,b,c}}(t:z;p) = {(1 - t)^{c - b - 1}}{(1 - zt)^{ - a}}H(1 - t{) _\xi }{\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right), $B {(b, c - b)}^{Ψ} {\hat{F}}_{p} (a, b; c; z)$ B{(b,c - b)^\Psi }{\hat F_p}(a,b;c;z) has the Mellin transform formula $B {(b, c - b)}^{Ψ} {\hat{F}}_{p} (a, b; c; z) = ℳ [{\hat{f}}_{a, b, c} (t : z; p) : b] .$ B{(b,c - b)^\Psi }{\hat F_p}(a,b;c;z) = {\cal M}[{\hat f_{a,b,c}}(t:z;p):b]. Differentiating ${\hat{f}}_{a, b, c} (t : z; p)$ {\hat f_{a,b,c}}(t:z;p) with respect to t we obtain $\begin{array}{l} \frac{d}{dt} {{\hat{f}}_{a, b, c} (t : z; p)} = - (c - b - 1) (1 - t)^{c - b - 2} {(1 - zt)}^{- a} H {(1 - t)}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) \\ + az {(1 - t)}^{c - b - 1} {(1 - zt)}^{- (a + 1)} H (1 - t)_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) \\ + p \frac{1}{t^{2}} {(1 - t)}^{c - b - 3} {(1 - zt)}^{- a} H (1 - t)_{ξ} Ψ_{η} [\begin{matrix} {(α_{i} + β_{i}, α_{i})}_{1, ξ} \\ {(κ_{j} + μ_{j}, κ_{j})}_{1, η} \end{matrix} | \frac{- p}{t (1 - t)}] \\ - 2 p \frac{1}{t} {(1 - t)}^{c - b - 3} {(1 - zt)}^{- a} H (1 - t) ξ Ψ_{η} [\begin{matrix} {(α_{i} + β_{i}, α_{i})}_{1, ξ} \\ {(κ_{j} + μ_{j}, κ_{j})}_{1, η} \end{matrix} | \frac{- p}{t (1 - t)}] . \end{array}$ \eqalign{& {d \over {dt}}\left\{ {{{\hat f}_{a,b,c}}(t:z;p)} \right\} = - (c - b - 1)(1 - t{)^{c - b - 2}}{(1 - zt)^{ - a}}H{(1 - t)_\xi }{\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right) \cr & + az{(1 - t)^{c - b - 1}}{(1 - zt)^{ - (a + 1)}}H(1 - t{) _\xi }{\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right) \cr & + p{1 \over {{t^2}}}{(1 - t)^{c - b - 3}}{(1 - zt)^{ - a}}H(1 - t{) _\xi }{\Psi _\eta }\left[ {\matrix{{{{({\alpha _i} + {\beta _i},{\alpha _i})}_{1,\xi }}} & {} & {} \cr {{{({\kappa _j} + {\mu _j},{\kappa _j})}_{1,\eta }}} & {} & {} \cr } |{{ - p} \over {t(1 - t)}}} \right] \cr & - 2p{1 \over t}{(1 - t)^{c - b - 3}}{(1 - zt)^{ - a}}H(1 - t{) _\xi }{\Psi _\eta }\left[ {\matrix{{{{({\alpha _i} + {\beta _i},{\alpha _i})}_{1,\xi }}} & {} & {} \cr {{{({\kappa _j} + {\mu _j},{\kappa _j})}_{1,\eta }}} & {} & {} \cr } |{{ - p} \over {t(1 - t)}}} \right]. \cr} Since $ℳ {f^{'} (t) : b} = - (b - 1) ℳ {f (t) : b - 1},$ {\cal M}\left\{ {{f^'}(t):b} \right\} = - (b - 1){\cal M}\left\{ {f(t):b - 1} \right\}, we get $\begin{array}{l} (b - 1) B {(b - 1, c - b + 1)}^{Ψ} {\hat{F}}_{p} (a, b - 1; c; z) \\ = (c - b - 1) B {(b, c - b - 1)}^{Ψ} {\hat{F}}_{p} (a, b; c - 1; z) \\ - azB {(b, c - b)}^{Ψ} {\hat{F}}_{p} (a + 1, b; c; z) \\ - pB {(b - 2, c - b - 2)}^{Ψ} F_{p} [ \begin{matrix} {(α_{i} + β_{i}, α_{i})}_{1, ξ} \\ {(κ_{j} + μ_{j}, κ_{j})}_{1, η} \end{matrix} | a, b - 2; c - 4; z] \\ + 2 pB {(b - 1, c - b - 2)}^{Ψ} F_{p} [ \begin{matrix} {(α_{i} + β_{i}, α_{i})}_{1, ξ} \\ {(κ_{j} + μ_{j}, κ_{j})}_{1, η} \end{matrix} | a, b - 1; c - 3; z], \end{array}$ \eqalign{& (b - 1)B{(b - 1,c - b + 1)^\Psi }{{\hat F}_p}(a,b - 1;c;z) \cr & = (c - b - 1)B{(b,c - b - 1)^\Psi }{{\hat F}_p}(a,b;c - 1;z) \cr & - azB{(b,c - b)^\Psi }{{\hat F}_p}(a + 1,b;c;z) \cr & - pB{(b - 2,c - b - 2)^\Psi }{F_p}\left[ {\matrix{{{{({\alpha _i} + {\beta _i},{\alpha _i})}_{1,\xi }}} & {} & {} \cr {{{({\kappa _j} + {\mu _j},{\kappa _j})}_{1,\eta }}} & {} & {} \cr } | a,b - 2;c - 4;z} \right] \cr & + 2pB{(b - 1,c - b - 2)^\Psi }{F_p}\left[ {\matrix{{{{({\alpha _i} + {\beta _i},{\alpha _i})}_{1,\xi }}} & {} & {} \cr {{{({\kappa _j} + {\mu _j},{\kappa _j})}_{1,\eta }}} & {} & {} \cr } | a,b - 1;c - 3;z} \right], \cr} which gives the result.

Theorem 13

The following equality holds true:(13) $\frac{d^{n}}{{dz}^{n}} {^{Ψ} {\hat{Φ}}_{p} (b; c; z)} = \frac{{(b)}_{n}}{{(c)}_{n}} [^{Ψ} {\hat{Φ}}_{p} (b + n; c + n; z)] .$ {{{d^n}} \over {d{z^n}}}\left\{ {^\Psi {{\hat \Phi }_p}(b;c;z)} \right\} = {{{{(b)}_n}} \over {{{(c)}_n}}}\left[ {^\Psi {{\hat \Phi }_p}(b + n;c + n;z)} \right].

Proof

The derivative of the $^{Ψ} {\hat{Φ}}_{p} (b; c; z)$ ^\Psi {\hat \Phi _p}(b;c;z) according to argument z is $\begin{array}{l} \frac{d}{dz} {^{Ψ} {\hat{Φ}}_{p} (b; c; z)} = \frac{d}{dz} {\sum_{n = 0}^{\infty} \frac{^{Ψ} {\hat{B}}_{p} (b + n, c - b)}{B (b, c - b)} \frac{z^{n}}{n!}} \\ = \sum_{n = 1}^{\infty} \frac{^{Ψ} {\hat{B}}_{p} (b + n, c - b)}{B (b, c - b)} \frac{z^{n - 1}}{(n - 1)!} . \end{array}$ \eqalign{& {d \over {dz}}\left\{ {^\Psi {{\hat \Phi }_p}(b;c;z)} \right\} = {d \over {dz}}\left\{ {\sum\limits_{n = 0}^\infty {{^\Psi {{\hat B}_p}(b + n,c - b)} \over {B(b,c - b)}}{{{z^n}} \over {n!}}} \right\} \cr & = \sum\limits_{n = 1}^\infty {{^\Psi {{\hat B}_p}(b + n,c - b)} \over {B(b,c - b)}}{{{z^{n - 1}}} \over {(n - 1)!}}. \cr} Replacing n → n + 1, we get $\begin{array}{l} \frac{d}{dz} {^{Ψ} {\hat{Φ}}_{p} (b; c; z)} = \frac{(b)}{(c)} \sum_{n = 0}^{\infty} \frac{^{Ψ} {\hat{B}}_{p} (b + n + 1, c - b)}{B (b + 1, c - b)} \frac{z^{n}}{n!} \\ = \frac{(b)}{(c)} [^{Ψ} {\hat{Φ}}_{p} (b + 1; c + 1; z)] . \end{array}$ \eqalign{& {d \over {dz}}\left\{ {^\Psi {{\hat \Phi }_p}(b;c;z)} \right\} = {{(b)} \over {(c)}}\sum\limits_{n = 0}^\infty {{^\Psi {{\hat B}_p}(b + n + 1,c - b)} \over {B(b + 1,c - b)}}{{{z^n}} \over {n!}} \cr & = {{(b)} \over {(c)}}\left[ {^\Psi {{\hat \Phi }_p}(b + 1;c + 1;z)} \right]. \cr} Thus, the general form of the above equation gives $\frac{d^{n}}{{dz}^{n}} {^{Ψ} {\hat{Φ}}_{p} (b; c; z)} = \frac{{(b)}_{n}}{{(c)}_{n}} [^{Ψ} {\hat{Φ}}_{p} (b + n; c + n; z)],$ {{{d^n}} \over {d{z^n}}}\left\{ {^\Psi {{\hat \Phi }_p}(b;c;z)} \right\} = {{{{(b)}_n}} \over {{{(c)}_n}}}\left[ {^\Psi {{\hat \Phi }_p}(b + n;c + n;z)} \right], which is the result.

Theorem 14

The following equality is provided for Re(b) > 2,Re(c) > Re(b + 2): $\begin{array}{l} (b - 1) B {(b - 1, c - b + 1)}^{Ψ} {\hat{Φ}}_{p} (b - 1; c; z) \\ = (c - b - 1) B {(b, c - b - 1)}^{Ψ} {\hat{Φ}}_{p} (b; c - 1; z) \\ - zB {(b, c - b)}^{Ψ} {\hat{Φ}}_{p} (b; c; z) \\ - pB {(b - 2, c - b - 2)}^{Ψ} Φ_{p} [ \begin{matrix} {(α_{i} + β_{i}, α_{i})}_{1, ξ} \\ {(κ_{j} + μ_{j}, κ_{j})}_{1, η} \end{matrix} | b - 2; c - 4; z] \\ + 2 pB {(b - 1, c - b - 2)}^{Ψ} Φ_{p} [ \begin{matrix} {(α_{i} + β_{i}, α_{i})}_{1, ξ} \\ {(κ_{j} + μ_{j}, κ_{j})}_{1, η} \end{matrix} | b - 1; c - 3; z] . \end{array}$ \eqalign{& (b - 1)B{(b - 1,c - b + 1)^\Psi }{{\hat \Phi }_p}(b - 1;c;z) \cr & = (c - b - 1)B{(b,c - b - 1)^\Psi }{{\hat \Phi }_p}(b;c - 1;z) \cr & - zB{(b,c - b)^\Psi }{{\hat \Phi }_p}(b;c;z) \cr & - pB{(b - 2,c - b - 2)^\Psi }{\Phi _p}\left[ {\matrix{{{{({\alpha _i} + {\beta _i},{\alpha _i})}_{1,\xi }}} & {} & {} \cr {{{({\kappa _j} + {\mu _j},{\kappa _j})}_{1,\eta }}} & {} & {} \cr } | b - 2;c - 4;z} \right] \cr & + 2pB{(b - 1,c - b - 2)^\Psi }{\Phi _p}\left[ {\matrix{{{{({\alpha _i} + {\beta _i},{\alpha _i})}_{1,\xi }}} & {} & {} \cr {{{({\kappa _j} + {\mu _j},{\kappa _j})}_{1,\eta }}} & {} & {} \cr } | b - 1;c - 3;z} \right]. \cr}

Proof

Since $B {(b, c - b)}^{Ψ} {\hat{Φ}}_{p} (b; c; z)$ B{(b,c - b)^\Psi }{\hat \Phi _p}(b;c;z) is the Mellin transform of ${\hat{f}}_{b, c} (t : z; p) = {(1 - t)}^{c - b - 1} e^{zt} H (1 - t)_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}),$ {\hat f_{b,c}}(t:z;p) = {(1 - t)^{c - b - 1}}{e^{zt}}H(1 - t{) _\xi }{\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right), $B {(b, c - b)}^{Ψ} {\hat{Φ}}_{p} (b; c; z)$ B{(b,c - b)^\Psi }{\hat \Phi _p}(b;c;z) has the Mellin transform formula $B {(b, c - b)}^{Ψ} {\hat{Φ}}_{p} (b; c; z) = ℳ [{\hat{f}}_{b, c} (t : z; p) : b] .$ B{(b,c - b)^\Psi }{\hat \Phi _p}(b;c;z) = {\cal M}[{\hat f_{b,c}}(t:z;p):b]. Differentiating ${\hat{f}}_{b, c} (t : z; p)$ {\hat f_{b,c}}(t:z;p) with regard to t obtain $\begin{array}{l} \frac{d}{dt} {{\hat{f}}_{b, c} (t : z; p)} = - (c - b - 1) (1 - t)^{c - b - 2} e^{zt} H (1 - t)_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) \\ + z (^{1 - t) c - b - 1} e^{zt} H (1 - t)_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) \\ + p \frac{1}{t^{2}} (^{1 - t) c - b - 3} e^{zt} H (1 - t)_{ξ} Ψ_{η} [\begin{matrix} {(α_{i} + β_{i}, α_{i})}_{1, ξ} \\ {(κ_{j} + μ_{j}, κ_{j})}_{1, η} \end{matrix} | \frac{- p}{t (1 - t)}] \\ - 2 p \frac{1}{t} (^{1 - t) c - b - 3} e^{zt} H (1 - t)_{ξ} Ψ_{η} [\begin{matrix} {(α_{i} + β_{i}, α_{i})}_{1, ξ} \\ {(κ_{j} + μ_{j}, κ_{j})}_{1, η} \end{matrix} | \frac{- p}{t (1 - t)}] . \end{array}$ \eqalign{& {d \over {dt}}\left\{ {{{\hat f}_{b,c}}(t:z;p)} \right\} = - (c - b - 1)(1 - t{)^{c - b - 2}}{e^{zt}}H(1 - t{) _\xi }{\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right) \cr & + z{(1 - t)^{c - b - 1}}{e^{zt}}H(1 - t{) _\xi }{\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right) \cr & + p{1 \over {{t^2}}}{(1 - t)^{c - b - 3}}{e^{zt}}H(1 - t{) _\xi }{\Psi _\eta }\left[ {\matrix{{{{({\alpha _i} + {\beta _i},{\alpha _i})}_{1,\xi }}} & {} & {} \cr {{{({\kappa _j} + {\mu _j},{\kappa _j})}_{1,\eta }}} & {} & {} \cr } | {{ - p} \over {t(1 - t)}}} \right] \cr & - 2p{1 \over t}{(1 - t)^{c - b - 3}}{e^{zt}}H(1 - t{) _\xi }{\Psi _\eta }\left[ {\matrix{{{{({\alpha _i} + {\beta _i},{\alpha _i})}_{1,\xi }}} & {} & {} \cr {{{({\kappa _j} + {\mu _j},{\kappa _j})}_{1,\eta }}} & {} & {} \cr } | {{ - p} \over {t(1 - t)}}} \right]. \cr} Since $ℳ {f^{'} (t) : b} = - (b - 1) ℳ {f (t) : b - 1},$ {\cal M}\left\{ {{f^'}(t):b} \right\} = - (b - 1){\cal M}\left\{ {f(t):b - 1} \right\}, we get $\begin{array}{l} (b - 1) B {(b - 1, c - b + 1)}^{Ψ} {\hat{Φ}}_{p} (b - 1; c; z) \\ = (c - b - 1) B {(b, c - b - 1)}^{Ψ} {\hat{Φ}}_{p} (b; c - 1; z) \\ - zB {(b, c - b)}^{Ψ} {\hat{Φ}}_{p} (b; c; z) \\ - pB {(b - 2, c - b - 2)}^{Ψ} Φ_{p} [ \begin{matrix} {(α_{i} + β_{i}, α_{i})}_{1, ξ} \\ {(κ_{j} + μ_{j}, κ_{j})}_{1, η} \end{matrix} | b - 2; c - 4; z] \\ + 2 pB {(b - 1, c - b - 2)}^{Ψ} Φ_{p} [ \begin{matrix} {(α_{i} + β_{i}, α_{i})}_{1, ξ} \\ {(κ_{j} + μ_{j}, κ_{j})}_{1, η} \end{matrix} | b - 1; c - 3; z], \end{array}$ \eqalign{& (b - 1)B{(b - 1,c - b + 1)^\Psi }{{\hat \Phi }_p}(b - 1;c;z) \cr & = (c - b - 1)B{(b,c - b - 1)^\Psi }{{\hat \Phi }_p}(b;c - 1;z) \cr & - zB{(b,c - b)^\Psi }{{\hat \Phi }_p}(b;c;z) \cr & - pB{(b - 2,c - b - 2)^\Psi }{\Phi _p}\left[ {\matrix{{{{({\alpha _i} + {\beta _i},{\alpha _i})}_{1,\xi }}} & {} & {} \cr {{{({\kappa _j} + {\mu _j},{\kappa _j})}_{1,\eta }}} & {} & {} \cr } | b - 2;c - 4;z} \right] \cr & + 2pB{(b - 1,c - b - 2)^\Psi }{\Phi _p}\left[ {\matrix{{{{({\alpha _i} + {\beta _i},{\alpha _i})}_{1,\xi }}} & {} & {} \cr {{{({\kappa _j} + {\mu _j},{\kappa _j})}_{1,\eta }}} & {} & {} \cr } | b - 1;c - 3;z} \right], \cr} which completes the proof.

In the following theorems we obtain the Mellin transform formulas of the _ξΨ_η-Gauss and _ξ Ψ_η-confluent hypergeometric functions.

Theorem 15

The following equality is provided for Re(s) > 0: $ℳ [^{Ψ} {\hat{F}}_{p} (a, b; c; z) : s] = {\frac{^{Ψ} \hat{Γ} (s) B (b + s, c + s - b)}{B (b, c - b)}}_{2} F_{1} (a, b + s; c + 2 s; z) .$ {\cal M}\left[ {^\Psi {{\hat F}_p}(a,b;c;z):s} \right] = {{{^\Psi \hat \Gamma (s)B(b + s,c + s - b)} \over {B(b,c - b)}}_2}{F_1}(a,b + s;c + 2s;z).

Proof

By applying Mellin transformation to equality (9), we get $\begin{array}{l} ℳ [^{Ψ} {\hat{F}}_{p} (a, b; c; z) : s] = \int_{0}^{\infty} p^{s - 1} [^{Ψ} {\hat{F}}_{p} (a, b; c; z)] dp \\ = \int_{0}^{\infty} p^{s - 1} \sum_{n = 0}^{\infty} \frac{^{Ψ} {\hat{B}}_{p} (b + n, c - b)}{B (b, c - b)} {(a)}_{n} \frac{z^{n}}{n!} dp \\ = \frac{1}{B (b, c - b)} \int_{0}^{1} t^{b - 1} {(1 - t)}^{c - b - 1} {(1 - zt)}^{- a} \int_{0}^{\infty} p^{s - 1}_{ξ} Ψ_{η} (\frac{- p}{t (1 - t)}) dpdt . \end{array}$ \eqalign{& {\cal M}{[^\Psi }{{\hat F}_p}(a,b;c;z):s] = \int_0^\infty {p^{s - 1}}\left[ {^\Psi {{\hat F}_p}(a,b;c;z)} \right]dp \cr & = \int_0^\infty {p^{s - 1}}\sum\limits_{n = 0}^\infty {{^\Psi {{\hat B}_p}(b + n,c - b)} \over {B(b,c - b)}}{(a)_n}{{{z^n}} \over {n!}}dp \cr & = {1 \over {B(b,c - b)}}\int_0^1 {t^{b - 1}}{(1 - t)^{c - b - 1}}{(1 - zt)^{ - a}}\int_0^\infty {p^{s - 1}}_\xi {\Psi _\eta }\left( {{{ - p} \over {t(1 - t)}}} \right)dpdt. \cr} Substituting $u = \frac{p}{t (1 - t)}$ u = {p \over {t(1 - t)}} in the above equation gives us $\int_{0}^{\infty} p^{s - 1}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) dp = t^{s} {(1 - t)}^{s}^{Ψ} \hat{Γ} (s) .$ \int_0^\infty {p^{s - 1}}_\xi {\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right)dp = {t^s}{(1 - t)^s}{ ^\Psi }\hat \Gamma (s). Thus, we get $ℳ [^{Ψ} {\hat{F}}_{p} (a, b; c; z) : s] = {\frac{^{Ψ} \hat{Γ} (s) B (b + s, c + s - b)}{B (b, c - b)}}_{2} F_{1} (a, b + s; c + 2 s; z) .$ {\cal M}\left[ {^\Psi {{\hat F}_p}(a,b;c;z):s} \right] = {{{^\Psi \hat \Gamma (s)B(b + s,c + s - b)} \over {B(b,c - b)}}_2}{F_1}(a,b + s;c + 2s;z).

Corollary 16

The following equality is provided for Re(s) > 0: $^{Ψ} {\hat{F}}_{p} (a, b; c; z) = \frac{1}{2 π i} \int_{- i \infty}^{+ i \infty} {\frac{^{Ψ} \hat{Γ} (s) B (b + s, c + s - b)}{B (b, c - b)}}_{2} F_{1} (a, b + s; c + 2 s; z) p^{- s} ds .$ ^\Psi {\hat F_p}(a,b;c;z) = {1 \over {2\pi i}}\int_{ - i\infty }^{ + i\infty } {{{^\Psi \hat \Gamma (s)B(b + s,c + s - b)} \over {B(b,c - b)}}_2}{F_1}(a,b + s;c + 2s;z){p^{ - s}}ds.

Theorem 17

The following equality is provided for Re(s) > 0: $ℳ [^{Ψ} {\hat{Φ}}_{p} (b; c; z) : s] = \frac{^{Ψ} \hat{Γ} (s) B (b + s, c + s - b)}{B (b, c - b)} Φ (b + s; c + 2 s; z) .$ {\cal M}\left[ {^\Psi {{\hat \Phi }_p}(b;c;z):s} \right] = {{^\Psi \hat \Gamma (s)B(b + s,c + s - b)} \over {B(b,c - b)}}\Phi (b + s;c + 2s;z).

Proof

By applying Mellin transformation to equality (10), we get $\begin{array}{l} ℳ [^{Ψ} {\hat{Φ}}_{p} (b; c; z) : s] = \int_{0}^{\infty} p^{s - 1} [^{Ψ} {\hat{Φ}}_{p} (b; c; z)] dp \\ = \int_{0}^{\infty} p^{s - 1} \sum_{n = 0}^{\infty} \frac{^{Ψ} {\hat{B}}_{p} (b + n, c - b)}{B (b, c - b)} \frac{z^{n}}{n!} d p \\ = \frac{1}{B (b, c - b)} \int_{0}^{1} t^{b - 1} {(1 - t)}^{c - b - 1} e^{zt} [\int_{0}^{\infty} p^{s - 1}_{ξ} Ψ_{η} (\frac{- p}{t (1 - t)}) dp] dt . \end{array}$ \eqalign{& {\cal M}{[^\Psi }{{\hat \Phi }_p}(b;c;z):s] = \int_0^\infty {p^{s - 1}}\left[ {^\Psi {{\hat \Phi }_p}(b;c;z)} \right]dp \cr & = \int_0^\infty {p^{s - 1}}\sum\limits_{n = 0}^\infty {{^\Psi {{\hat B}_p}(b + n,c - b)} \over {B(b,c - b)}}{{{z^n}} \over {n!}}dp \cr & = {1 \over {B(b,c - b)}}\int_0^1 {t^{b - 1}}{(1 - t)^{c - b - 1}}{e^{zt}}\left[ {\int_0^\infty {p^{s - 1}}_\xi {\Psi _\eta }\left( {{{ - p} \over {t(1 - t)}}} \right)dp} \right]dt. \cr} Substituting $u = \frac{p}{t (1 - t)}$ u = {p \over {t(1 - t)}} in the above equation we get $\int_{0}^{\infty} p^{s - 1}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) dp = t^{s} {(1 - t)}^{s}^{Ψ} \hat{Γ} (s) .$ \int_0^\infty {p^{s - 1}}_\xi {\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right)dp = {t^s}{(1 - t)^s}{ ^\Psi }\hat \Gamma (s). Thus, we have $ℳ [^{Ψ} {\hat{Φ}}_{p} (b; c; z) : s] = \frac{^{Ψ} \hat{Γ} (s) B (b + s, c + s - b)}{B (b, c - b)} Φ (b + s; c + 2 s; z) .$ {\cal M}\left[ {^\Psi {{\hat \Phi }_p}(b;c;z):s} \right] = {{^\Psi \hat \Gamma (s)B(b + s,c + s - b)} \over {B(b,c - b)}}\Phi (b + s;c + 2s;z).

Corollary 18

For Re(s) > 0, we have the following equality: $^{Ψ} {\hat{Φ}}_{p} (b; c; z) = \frac{1}{2 π i} \int_{- i \infty}^{+ i \infty} \frac{^{Ψ} \hat{Γ} (s) B (b + s, c + s - b)}{B (b, c - b)} Φ (b + s; c + 2 s; z) p^{- s} ds .$ ^\Psi {\hat \Phi _p}(b;c;z) = {1 \over {2\pi i}}\int_{ - i\infty }^{ + i\infty } {{^\Psi \hat \Gamma (s)B(b + s,c + s - b)} \over {B(b,c - b)}}\Phi (b + s;c + 2s;z){p^{ - s}}ds.

The following two theorems are about the transformation formulas of _ξ Ψ_η-Gauss and _ξΨ_η-confluent hypergeometric functions.

Theorem 19

The following equality holds true: $^{Ψ} {\hat{F}}_{p} (a, b; c; z) = (1 - z)^{- a} [^{Ψ} {\hat{F}}_{p} (a, c - b; b; \frac{z}{z - 1})] .$ ^\Psi {\hat F_p}(a,b;c;z) = (1 - z{)^{ - a}}\left[ {^\Psi {{\hat F}_p}\left( {a,c - b;b;{z \over {z - 1}}} \right)} \right].

Proof

By writing ${[1 - z (1 - t)]}^{- a} = (1 - z)^{- a} {(1 + \frac{zt}{1 - z})}^{- a}$ {\left[ {1 - z(1 - t)} \right]^{ - a}} = (1 - z{)^{ - a}}{\left( {1 + {{zt} \over {1 - z}}} \right)^{ - a}} and replacing t → 1 − t in (9), we obtain $^{Ψ} {\hat{F}}_{p} (a, b; c; z) = \frac{{(1 - z)}^{- a}}{B (b, c - b)} \int_{0}^{1} t^{c - b - 1} {(1 - t)}^{b - 1} {(1 - \frac{zt}{z - 1})}^{- a}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) dt .$ ^\Psi {\hat F_p}(a,b;c;z) = {{{{(1 - z)}^{ - a}}} \over {B(b,c - b)}}\int_0^1 {t^{c - b - 1}}{(1 - t)^{b - 1}}{\left( {1 - {{zt} \over {z - 1}}} \right)^{ - a}}_\xi {\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right)dt. Then we have $^{Ψ} {\hat{F}}_{p} (a, b; c; z) = (1 - z)^{- a} [^{Ψ} {\hat{F}}_{p} (a, c - b; b; \frac{z}{z - 1})],$ ^\Psi {\hat F_p}(a,b;c;z) = (1 - z{)^{ - a}}\left[ {^\Psi {{\hat F}_p}\left( {a,c - b;b;{z \over {z - 1}}} \right)} \right], which is the result.

Theorem 20

The following equality holds true: $^{Ψ} {\hat{Φ}}_{p} (b; c; z) = e^{z} [^{Ψ} {\hat{Φ}}_{p} (c - b; b; - z)] .$ ^\Psi {\hat \Phi _p}(b;c;z) = {e^z}\left[ {^\Psi {{\hat \Phi }_p}(c - b;b; - z)} \right].

Proof

From the definition of confluent hypergeometric function, we have (14) $\begin{array}{l} ^{Ψ} {\hat{Φ}}_{p} (b; c; z) = \sum_{n = 0}^{\infty} \frac{^{Ψ} {\hat{B}}_{p} (b + n, c - b)}{B (b, c - b)} \frac{z^{n}}{n!} \\ = \frac{1}{B (b, c - b)} \int_{0}^{1} t^{b + n - 1} {(1 - t)}^{c - b - 1} e^{zt}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) dt . \end{array}$ \eqalign{& ^\Psi {{\hat \Phi }_p}(b;c;z) = \sum\limits_{n = 0}^\infty {{^\Psi {{\hat B}_p}(b + n,c - b)} \over {B(b,c - b)}}{{{z^n}} \over {n!}} \cr & = {1 \over {B(b,c - b)}}\int_0^1 {t^{b + n - 1}}{(1 - t)^{c - b - 1}}{e^{zt}}_\xi {\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right)dt. \cr} Replacing t = 1 − u in (14), we obtain $\begin{array}{l} ^{Ψ} {\hat{Φ}}_{p} (b; c; z) = \frac{1}{B (b, c - b)} \int_{0}^{1} u^{c - b - 1} {(1 - u)}^{b + n - 1} e^{z (1 - u)}_{ξ} Ψ_{η} (\frac{- p}{u (1 - u)}) du \\ = e^{z} [^{Ψ} {\hat{Φ}}_{p} (c - b; b; - z)], \end{array}$ \eqalign{& ^\Psi {{\hat \Phi }_p}(b;c;z) = {1 \over {B(b,c - b)}}\int_0^1 {u^{c - b - 1}}{(1 - u)^{b + n - 1}}{e^{z(1 - u)}}_\xi {\Psi _\eta }\left( {{{ - p} \over {u(1 - u)}}} \right)du \cr & = {e^z}\left[ {^\Psi {{\hat \Phi }_p}(c - b;b; - z)} \right], \cr} which gives the result.

The following theorems are about the differential and difference relations for _ξ Ψ_η-Gauss hypergeometric and _ξ Ψ_η-confluent hypergeometric functions.

Theorem 21

The following relations hold true:(15) $Δ_{a} [^{Ψ} {\hat{F}}_{p} (a, b; c; z)] = z \frac{b}{c}^{Ψ} {\hat{F}}_{p} (a + 1, b + 1; c + 1; z)$ \quad {\Delta _a}\left[ {^\Psi {{\hat F}_p}(a,b;c;z)} \right] = z{b \over c}{ ^\Psi }{\hat F_p}(a + 1,b + 1;c + 1;z)(16) $a Δ_{a} [^{Ψ} {\hat{F}}_{p} (a, b; c; z)] = z \frac{d}{dz} {^{Ψ} {\hat{F}}_{p} (a, b; c; z)}$ \quad a{\Delta _a}\left[ {^\Psi {{\hat F}_p}(a,b;c;z)} \right] = z{d \over {dz}}\left\{ {^\Psi {{\hat F}_p}(a,b;c;z)} \right\}(17) $b Δ_{b} [^{Ψ} {\hat{Φ}}_{p} (b; c + 1; z)] = - c Δ_{c} [^{Ψ} {\hat{Φ}}_{p} (b; c; z)]$ \quad b{\Delta _b}\left[ {^\Psi {{\hat \Phi }_p}(b;c + 1;z)} \right] = - c{\Delta _c}\left[ {^\Psi {{\hat \Phi }_p}(b;c;z)} \right](18) $\frac{d}{dz} {^{Ψ} {\hat{Φ}}_{p} (b; c; z)} = \frac{b}{c}^{Ψ} {\hat{Φ}}_{p} (b; c + 1; z) - Δ_{c} [^{Ψ} {\hat{Φ}}_{p} (b; c; z)]$ \quad {d \over {dz}}\left\{ {^\Psi {{\hat \Phi }_p}(b;c;z)} \right\} = {b \over c}{ ^\Psi }{\hat \Phi _p}(b;c + 1;z) - {\Delta _c}\left[ {^\Psi {{\hat \Phi }_p}(b;c;z)} \right]where Δ_αdenotes the difference operator defined by $Δ_{α} f (α, \dots) = f (α + 1, \dots) - f (α, \dots) .$ {\Delta _\alpha }f(\alpha , \ldots ) = f(\alpha + 1, \ldots ) - f(\alpha , \ldots ).

Proof

It is seen from (9) and the difference operator Δ_a that (19) $\begin{array}{l} Δ_{a}^{Ψ} \hat{F} (a, b; c; z) =^{Ψ} \hat{F} (a + 1, b; c; z) -^{Ψ} \hat{F} (a, b; c; z) \\ = \frac{z}{B (b, c - b)} \int_{0}^{1} t^{b} {(1 - t)}^{c - b - 1} {(1 - zt)}^{- a - 1}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) dt . \end{array}$ \eqalign{& {\Delta _a}^\Psi \hat F(a,b;c;z{) = ^\Psi }\hat F(a + 1,b;c;z){ - ^\Psi }\hat F(a,b;c;z) \cr & = {z \over {B(b,c - b)}}\int_0^1 {t^b}{(1 - t)^{c - b - 1}}{(1 - zt)^{ - a - 1}}_\xi {\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right)dt. \cr} If we write a + 1,b + 1 and c + 1 instead of a,b and c in equation (9), we get the following equation: (20) $^{Ψ} \hat{F} (a + 1, b + 1; c + 1; z) = \frac{1}{B (b + 1, c - b)} \int_{0}^{1} t^{b} {(1 - t)}^{c - b - 1} {(1 - zt)}^{- a - 1}_{ξ} Ψ_{η} (- \frac{p}{t (1 - t)}) dt .$ ^\Psi \hat F(a + 1,b + 1;c + 1;z) = {1 \over {B(b + 1,c - b)}}\int_0^1 {t^b}{(1 - t)^{c - b - 1}}{(1 - zt)^{ - a - 1}}_\xi {\Psi _\eta }\left( { - {p \over {t(1 - t)}}} \right)dt. Now using (11) and (20) in (19) we get (15). Using differentiation formula (12) proves (16). Using the difference operator and (10), we obtain (17). Using differentiation formula (13) with n = 1, and considering (17) gives us (18).

Results and Recommendations

In this study, we introduced new generalizations of gamma, beta, Gauss and confluent hypergeometric functions with the help of Fox-Wright function. We also obtained some of their integral representations, Mellin transformations, derivative formulas, transformation formulas and reduction relations.

When the special cases of these functions are examined, it is seen that these functions are the generalizations of the following predefined functions which can be found in the literature:

For p = 0: $\begin{array}{l} Γ (x) =^{Ψ} Γ_{0} [ \begin{matrix} {(1,0)}_{1,1} \\ {(1,0)}_{1,1} \end{matrix} | x], \\ B (x, y) =^{Ψ} B_{0} [ \begin{matrix} {(1,0)}_{1,1} \\ {(1,0)}_{1,1} \end{matrix} | x, y], \\ F (a, b; c; z) =^{Ψ} F_{0} [ \begin{matrix} {(1,0)}_{1,1} \\ {(1,0)}_{1,1} \end{matrix} | a, b; c; z], \\ Φ (b; c; z) =^{Ψ} Φ_{0} [ \begin{matrix} {(1,0)}_{1,1} \\ {(1,0)}_{1,1} \end{matrix} | b; c; z] . \end{array}$ \eqalign{& \Gamma (x{) = ^\Psi }{\Gamma _0}\left[ {\matrix{{{{(1,0)}_{1,1}}} \cr {{{(1,0)}_{1,1}}} \cr } |x} \right], \cr & B(x,y{) = ^\Psi }{B_0}\left[ {\matrix{{{{(1,0)}_{1,1}}} & {} & {} \cr {{{(1,0)}_{1,1}}} & {} & {} \cr } |x,y} \right], \cr & F(a,b;c;z{) = ^\Psi }{F_0}\left[ {\matrix{{{{(1,0)}_{1,1}}} \cr {{{(1,0)}_{1,1}}} \cr } |a,b;c;z} \right], \cr & \Phi (b;c;z{) = ^\Psi }{\Phi _0}\left[ {\matrix{{{{(1,0)}_{1,1}}} \cr {{{(1,0)}_{1,1}}} \cr } |b;c;z} \right]. \cr} For p ≠ 0: $\begin{array}{l} Γ_{p} (x) =^{Ψ} Γ_{p} [ \begin{matrix} {(1,0)}_{1,1} \\ {(1,0)}_{1,1} \end{matrix} | x], \\ B_{p} (x, y) =^{Ψ} B_{p} [ \begin{matrix} {(1,0)}_{1,1} \\ {(1,0)}_{1,1} \end{matrix} | x, y], \\ F_{p} (a, b; c; z) =^{Ψ} F_{p} [ \begin{matrix} {(1,0)}_{1,1} \\ {(1,0)}_{1,1} \end{matrix} | a, b; c; z], \\ Φ_{p} (b; c; z) =^{Ψ} Φ_{p} [ \begin{matrix} {(1,0)}_{1,1} \\ {(1,0)}_{1,1} \end{matrix} | b; c; z] . \end{array}$ \eqalign{& {\Gamma _p}(x{) = ^\Psi }{\Gamma _p}\left[ {\matrix{{{{(1,0)}_{1,1}}} \cr {{{(1,0)}_{1,1}}} \cr } |x} \right], \cr & {B_p}(x,y{) = ^\Psi }{B_p}\left[ {\matrix{{{{(1,0)}_{1,1}}} & {} & {} \cr {{{(1,0)}_{1,1}}} & {} & {} \cr } |x,y} \right], \cr & {F_p}(a,b;c;z{) = ^\Psi }{F_p}\left[ {\matrix{{{{(1,0)}_{1,1}}} \cr {{{(1,0)}_{1,1}}} \cr } |a,b;c;z} \right], \cr & {\Phi _p}(b;c;z{) = ^\Psi }{\Phi _p}\left[ {\matrix{{{{(1,0)}_{1,1}}} \cr {{{(1,0)}_{1,1}}} \cr } |b;c;z} \right]. \cr} And also for p ≠ 0: $\begin{array}{l} Γ_{p}^{(α, β)} (x) = {\frac{Γ (β)}{Γ (α)}}^{Ψ} Γ_{p} [ \begin{matrix} {(α, 1)}_{1,1} \\ {(β, 1)}_{1,1} \end{matrix} | x], \\ B_{p}^{(α, β)} (x, y) = {\frac{Γ (β)}{Γ (α)}}^{Ψ} B_{p} [ \begin{matrix} {(α, 1)}_{1,1} \\ {(β, 1)}_{1,1} \end{matrix} | x, y], \\ F_{p}^{(α, β)} (a, b; c; z) = {\frac{Γ (β)}{Γ (α)}}^{Ψ} F_{p} [ \begin{matrix} {(α, 1)}_{1,1} \\ {(β, 1)}_{1,1} \end{matrix} | a, b; c; z], \\ Φ_{p}^{(α, β)} (b; c; z) = {\frac{Γ (β)}{Γ (α)}}^{Ψ} Φ_{p} [ \begin{matrix} {(α, 1)}_{1,1} \\ {(β, 1)}_{1,1} \end{matrix} | b; c; z], \end{array}$ \eqalign{& \Gamma _p^{(\alpha ,\beta )}(x) = {{{\Gamma (\beta )} \over {\Gamma (\alpha )}}^\Psi }{\Gamma _p}\left[ {\matrix{{{{(\alpha ,1)}_{1,1}}} \cr {{{(\beta ,1)}_{1,1}}} \cr } |x} \right], \cr & B_p^{(\alpha ,\beta )}(x,y) = {{{\Gamma (\beta )} \over {\Gamma (\alpha )}}^\Psi }{B_p}\left[ {\matrix{{{{(\alpha ,1)}_{1,1}}} \cr {{{(\beta ,1)}_{1,1}}} \cr } |x,y} \right], \cr & F_p^{(\alpha ,\beta )}(a,b;c;z) = {{{\Gamma (\beta )} \over {\Gamma (\alpha )}}^\Psi }{F_p}\left[ {\matrix{{{{(\alpha ,1)}_{1,1}}} \cr {{{(\beta ,1)}_{1,1}}} \cr } |a,b;c;z} \right], \cr & \Phi _p^{(\alpha ,\beta )}(b;c;z) = {{{\Gamma (\beta )} \over {\Gamma (\alpha )}}^\Psi }{\Phi _p}\left[ {\matrix{{{{(\alpha ,1)}_{1,1}}} \cr {{{(\beta ,1)}_{1,1}}} \cr } |b;c;z} \right], \cr} where Γ,B,F and Φ are the classic gamma, beta, Gauss and confluent hypergeometric functions; Γ_p, B_p, F_p and Φ_p are the functions defined in [7, 8, 10]; $Γ_{p}^{(α, β)}$ \Gamma _p^{(\alpha ,\beta )} , $B_{p}^{(α, β)}$ B_p^{(\alpha ,\beta )} , $F_{p}^{(α, β)}$ F_p^{(\alpha ,\beta )} and $Φ_{p}^{(α, β)}$ \Phi _p^{(\alpha ,\beta )} are the functions defined in [25].

Besides, the generalized beta function described in this study can be used to define similar generalizations of multivariate hypergeometric functions, which also known as Appell, Lauricella, Horn and Srivastava functions (see [3, 5] and the references therein). Further properties of these functions can be examined and they can also be used in fractional theory (see for example [2, 4, 30] and the references therein).

eISSN:: 2444-8656
Language:: English

Publication timeframe:: Volume Open
Journal Subjects:: Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics

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A study on certain properties of generalized special functions defined by Fox-Wright function

Published Online: Mar 31, 2020

Page range: 147 - 162

Received: Mar 26, 2019

Accepted: Apr 08, 2019

DOI: https://doi.org/10.2478/amns.2020.1.00014

KeywordsGamma function, Beta function, Fox-Wright function, Gauss hypergeometric function, Confluent hypergeometric function

© 2019 Enes Ata et al., published by Sciendo

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

Keywords
Gamma function, Beta function, Fox-Wright function, Gauss hypergeometric function, Confluent hypergeometric function