Transformation and summation formulas for Kampé de Fériet series \(F^{0:3}_{1:1} (1,1)\) (Q1922984)

By series manipulations and utilization of some classical results it is shown that the series \[ \sum^\infty_{m,n=0} {{(a)_m (b)_m (c)_m (a')_n (b')_n (c')_n} \over {(d)_{m+n} (e)_m (e')_n m!n!}} \] reduces to a gamma fraction multiplied by a \({}_4 F_3[1]\), if \(a'=d-a\), and one of the conditions (i) \(s=e'\), (ii) \(s=1\wedge -a\in\mathbb{N}\), (iii) \(s=1\wedge -c\in\mathbb{N}\wedge -c'\in\mathbb{N}\), where \(s=d+e-a-b-c\), is satisfied. Some particular cases of the reduction formulas are studied.