Transformation formula for a double Clausenian hypergeometric series, its q-analogue, and its invariance group

A transformation formula for a double basic hypergeometric series of type Φ0:2;21:2;2 is derived. This transformation yields a double series analogue of Sears’ transformation for a terminating 3Φ2 series. In the limit q→1, the formula reduces to a transformation for a terminating double Clausenian hypergeometric series of unit argument (one of the proper Kampé de Fériet series, F0:2;21:2;2(1,1)). This formula is a double series analogue of Whippleʹs terminating 3F2 transformation. This transformation gives rise to a transformation group (the invariance group) acting on the parameters of the double series. The invariance group is examined and shown to be a subgroup of a double copy of the symmetries of the square.