Asymptotics of a ${}_3F_2$ hypergeometric function with four large parameters

We consider the asymptotic behaviour of the generalised hypergeometric function $${}_3F_2\bl(\!\!\begin{array}{c} 1, (1+t)k/2, (1+t)k/2+1/2\\tk+1, k+1\end{array}\!\!; x\br),\qquad 0<x,t\leq 1$$ as the parameter $k\to+\infty$. Numerical results illustrating the accuracy of the resulting expansion are given.