The authors’ main result is a generating relation involving Srivastava’s general triple hypergeometric function \(F^{(3)}:\) \[ \sum^{\infty}_{n=0}\frac{(1+\alpha +m)_ n}{n!}F^{(3)}\left[ \begin{matrix} a::b;-;-:-n;-m-n;c-b;\\ c::-;-;-:1+\beta;1+\alpha;-;\end{matrix} x,y,z \right]t^ n= \]
\[ =(1-y)^{-a}(1-t)^{-1-\alpha -m} F^{(3)}\left[ \begin{matrix} a::b,1+\alpha +m;-;-:-;-;\\ c::-;-;- :1+\beta;1+\alpha;\end{matrix} X,Y,Z \right], \] where \[ X=(-xt)/((1-y(1- t)),\quad Y=(-y)/((1-y)(1-t)),\quad Z=(z-y)/(1-y). \] A number of particular cases and related results are discussed.