In the usual notation, let ℂ denote the set of *complex* numbers. For

{\alpha}_{j}\in \mathbb{C}\phantom{\rule{1em}{0ex}}(j=1,\dots ,p)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\beta}_{j}\in \mathbb{C}\setminus {\mathbb{Z}}_{0}^{-}\phantom{\rule{1em}{0ex}}({\mathbb{Z}}_{0}^{-}:={\mathbb{Z}}^{-}\cup \{0\}=\{0,-1,-2,\dots \}),

the *generalized hypergeometric series* {}_{p}F_{q} with *p* numerator parameters {\alpha}_{1},\dots ,{\alpha}_{p} and *q* denominator parameters {\beta}_{1},\dots ,{\beta}_{q} is defined by (see, for example, [[1], Chapter 4]; see also [[2], p.73]):

where

\omega :=\sum _{j=1}^{q}{\beta}_{j}-\sum _{j=1}^{p}{\alpha}_{j}\phantom{\rule{1em}{0ex}}({\alpha}_{j}\in \mathbb{C}\phantom{\rule{0.25em}{0ex}}(j=1,\dots ,p);{\beta}_{j}\in \mathbb{C}\setminus {\mathbb{Z}}_{0}^{-}\phantom{\rule{0.25em}{0ex}}(j=1,\dots ,q))

(1.2)

and {(\lambda )}_{\nu} is the Pochhammer symbol or the *shifted factorial* since

{(1)}_{n}=n!\phantom{\rule{1em}{0ex}}(n\in {\mathbb{N}}_{0}),

which is defined (for \lambda ,\nu \in \mathbb{C}), in terms of the familiar gamma function Γ, by

{(\lambda )}_{\nu}:=\frac{\mathrm{\Gamma}(\lambda +\nu )}{\mathrm{\Gamma}(\lambda )}=\{\begin{array}{cc}1\hfill & (\nu =0;\lambda \in \mathbb{C}\setminus \{0\}),\hfill \\ \lambda (\lambda +1)\cdots (\lambda +n-1)\hfill & (\nu =n\in \mathbb{N};\lambda \in \mathbb{C}),\hfill \end{array}

(1.3)

it being *understood conventionally* that {(0)}_{0}:=1.

By using the notations as described in [[3], p.33, Equation 1.4(1)] (with n=3) (see also [[4], p.114, Equation (1)], [[5], Equation (2.1)] and [[6], p.60, Equation 1.7(1)]), Lauricella’s triple hypergeometric function {F}_{A}^{(3)}(x,y,z) is defined by

\begin{array}{rcl}{F}_{A}^{(3)}(x,y,z)& =& {F}_{A}^{(3)}(a,{b}_{1},{b}_{2},{b}_{3};{c}_{1},{c}_{2},{c}_{3};x,y,z)\\ :=& \sum _{m,n,p=0}^{\mathrm{\infty}}\frac{{(a)}_{m+n+p}{({b}_{1})}_{m}{({b}_{2})}_{n}{({b}_{3})}_{p}}{{({c}_{1})}_{m}{({c}_{2})}_{n}{({c}_{3})}_{p}}\frac{{x}^{m}}{m!}\frac{{y}^{n}}{n!}\frac{{z}^{p}}{p!}\\ (|x|+|y|+|z|<1;{c}_{j}\in \mathbb{C}\setminus {\mathbb{Z}}_{0}^{-}\phantom{\rule{0.25em}{0ex}}(j=1,2,3)).\end{array}

(1.4)

Motivated essentially by the works by Lardner [7] and Carlson [8], by simply splitting Lauricella’s triple hypergeometric function {F}_{A}^{(3)}(x,y,z) into eight parts, very recently Choi *et al.* [5] presented several relationships between {F}_{A}^{(3)}(x,y,z) and the Srivastava function {F}^{(3)}[x,y,z] (see also [9]). The widely-investigated Srivastava’s triple hypergeometric function {F}^{(3)}[x,y,z], which was introduced over four decades ago by Srivastava [[10], p.428] (see also [5], [[3], p.44, Equation 1.5(14)] and [[6], p.69, Equation 1.7(39)]), provides an interesting unification (and generalization) of Lauricella’s 14 triple hypergeometric functions {F}_{1},\dots ,{F}_{14} (see [11], [[12], pp.113-114]) and Srivastava’s three functions {H}_{A}, {H}_{B} and {H}_{C} (see [[13], pp.99-100]; see also [14–16], [[3], p.43] and [[6], pp.60-68]).

Exton’s triple hypergeometric function {X}_{8} (see [17]; see also [[3], p.84, Entry 41a and p.101, List 41a]) is defined by

In fact, in 1982 Exton [17] published a very interesting and useful research paper in which he encountered a number of triple hypergeometric functions of second order whose series representations involve such products as {(a)}_{2m+2n+p} and {(a)}_{2m+n+p} and introduced a set of 20 distinct triple hypergeometric functions {X}_{1} to {X}_{20} and also gave their integral representations of Laplacian type which include the confluent hypergeometric functions {}_{0}F_{1}, {}_{1}F_{1}, a Humbert function {\psi}_{1} and a Humbert function {\varphi}_{2} in their kernels. It is not out of place to mention here that Exton’s functions {X}_{1} to {X}_{20} have been studied a lot until today; see, for example, the works [12, 18–23] and [24]. Moreover, Exton [17] presented a large number of very interesting transformation formulas and reducible cases with the help of two known results which are called in the literature Kummer’s first and second transformations or theorems.

Here, in this paper, we aim at establishing eleven new and interesting transformations between Lauricella’s triple hypergeometric function {F}_{A}^{(3)}(x,y,z) and Exton’s function {X}_{8} in the form of a single result. Our results presented here are derived with the help of two general summation formulae for the terminating {}_{2}F_{1}(2) series which were very recently obtained by Kim *et al.* [25] and also include the relationship between {F}_{A}^{(3)}(x,y,z) and {X}_{8} due to Exton [17].