The generalized hypergeometric function with r numerator and s denominator parameters, as the series, reads where ${\left(\mu \right)}_n=\mu (\mu +1)\cdots (\mu +n-1)$$(n\in \mathbb{N})$ and ${\left(\mu \right)}_0=1$ signify the Pochhammer symbol, ${\tau }_j\in \mathbb{C}$, and ${\upsilon }_j\in \mathbb{C}\setminus {\mathbb{Z}}_{⩽0},j\in \overline{1,s}:=\{1,2,\dots ,s\}$ for some $s\in \mathbb{N}$. The symbol ${\left(\mu \right)}_m$ is represented by ${\left(\mu \right)}_m=\mathsf{\Gamma }(\mu +m)/\mathsf{\Gamma }\left(\mu \right)$ $\left(\mu \in \mathbb{C}\setminus {\mathbb{Z}}_{⩽0},m\in {\mathbb{N}}_0\right)$, $\mathsf{\Gamma }$ being the familiar Gamma function whose acquainted integral is In this and other instances, the sets of positive integers, integers, real numbers, and complex numbers will be denoted by $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{R}$, and $\mathbb{C}$, respectively. Also let ${\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$ and ${\mathbb{Z}}_{⩽0}:=\mathbb{Z}\setminus \mathbb{N}$. The series in (1) converges for all $z\in \mathbb{C}$ if $r⩽s$. It is divergent for all $z\in \mathbb{C}\setminus \left\{0\right\}$ when $r>s+1$, unless at least one numerator parameter is in ${\mathbb{Z}}_{⩽0}$ in which case (1) is a polynomial. For the remaining case $r=s+1$, the series converges on the unit circle $\left|z\right|=1$ under the constraints ${\tau }_j\in \mathbb{C}\setminus {\mathbb{Z}}_{⩽0}$$(j\in \overline{1,r})$ and For the noted particular cases, ${}_2{F}_1$ is called the (Gauss’s) hypergeometric function and ${}_1{F}_1$, which is also denoted by $\mathsf{\Phi }$, is referred to as the confluent (Kummer’s) hypergeometric function.

In 1997, Chaudhry et al. [1, p. 20, Eq. (1.7) ] introduced and explored the p-extended Beta integral: from which follows a series of investigations of generalized incomplete gamma functions and their applications (see [2,3,4]; see also [5]). The p-extended Beta integral in (3) is turned out to be connected to the Macdonald, error and Whittaker functions. The case $p=0$ of (3) becomes the classical Beta function given by (see, for example, [6, p. 8, Eq. (43)]):

Making use of the subsequent transformation in which the numerator Beta function is replaced by the p-extended Beta function in (3), Chaudhry et al. [7] introduced the p-Gauss’s hypergeometric function and the p-Kummer’s confluent hypergeometric function which are, respectively, given as follows: and The functions were studied by Chaudhry et al. [7], who revealed numerous intriguing identities and formulas. These include integral representations, differentiation properties, Mellin transforms, transformations, recurrence relations, summation formulas, and asymptotic formulas.

Özarslan and Özergin [8] introduced and investigated the p-extensions of two variable Appell’s hypergeometric functions $F_1$ and $F_2$ and three variable Lauricella’s hypergeometric function $F_D^{\left(3\right)}$ (see, for example, [9, Chapter 1]), among which, the p-extended $F_2$ function is recalled: They [8] also introduced a new extended Riemann–Liouville fractional derivative to present several intriguing generating relations for the p-Gauss’s hypergeometric function (6).

Like (3), the p-extensions (6), (7) and (8) when $p=0$ return to Gauss’s hypergeometric function ${}_2{F}_1$, the confluent hypergeometric function ${}_1{F}_1$ and Appell’s hypergeometric function $F_2$ of two variables, respectively.

Our investigation is primarily motivated by the vast range of potential applications of extended Gauss’s hypergeometric, confluent hypergeometric, and Appell functions in various fields of mathematical, physical, engineering, and statistical sciences (as detailed in [1,7,8]) and the references therein). In this study, we undertake a systematic exploration of the extended Horn’s double hypergeometric function $H_{4,p}$ and extended Exton’s triple hypergeometric functions $X_{8,p}$. Specifically, we aim to present various integral representations of Euler and Laplace type, as well as certain integral representations involving Bessel and modified Bessel functions, Mellin transform, Laguerre polynomial representation, transformation formula, and recurrence relation. Additionally, we provide a generating function for the $X_{8,p}$ and several bounding inequalities for the$H_{4,p}$ and $X_{8,p}$.

In terms of the extended Beta function $B(\eta ,\xi ;p)$ in (3), this section introduces the following extended Horn’s double hypergeometric function $H_{4,p}$: For $\tau ,\upsilon \in \mathbb{C}$ and $\varphi$, ${\varphi }^{\prime }\in \mathbb{C}\setminus {\mathbb{Z}}_{⩽0}$, where $\Re \left({\varphi }^{\prime }\right)>\Re \left(\upsilon \right)>0$. In light of (5), the case $p=0$ in (9) gives the classical Horn’s double hypergeometric function $H_4$ (see, for example, [9, p. 24 and p. 59], [10]): For $\tau ,\upsilon \in \mathbb{C}$ and $\varphi$, ${\varphi }^{\prime }\in \mathbb{C}\setminus {\mathbb{Z}}_{⩽0}$,

In terms of the extended beta function $\mathrm{B}(x,y;p)$ in (3), this section introduces the following extended Exton’s triple hypergeometric function $X_{8,p}$: For $\tau ,\upsilon ,{\upsilon }^{\prime }\in \mathbb{C}$ and ${\varphi }_1,{\varphi }_2,{\varphi }_3\in \mathbb{C}\setminus {\mathbb{Z}}_{⩽0}$,

Setting $p=0$ in (41) yields the Exton’s triple hypergeometric function $X_8$ (see, for example, [9, p.84, 41a and p. 101]): For $\tau ,\upsilon ,{\upsilon }^{\prime }\in \mathbb{C}$ and ${\varphi }_1,{\varphi }_2,{\varphi }_3\in \mathbb{C}\setminus {\mathbb{Z}}_{⩽0}$,