Ordinary and basic bivariate hypergeometric transformations associated with the Appell and Kampé de Fériet functions (Q1398710)

Some transformation formulas for bivariate hypergeometric functions and their \(q\)-analogues are given. An example is \[ \sum_{i=0}^m \sum_{j=0}^n \frac{(a)_{i+j}(b)_{i+j}(c+c'-1)_{i+j}}{(a+b)_{i+j}(e)_{i+j}(e')_{i+j}} \frac{(-m)_i(e'+n)_i}{i!(c)_i}\frac{(-n)_j(e+m)_j}{j!(c')_j} \] \[ =\sum_{i=0}^m \sum_{j=0}^n \frac{(c+c'-1)_{i+j}}{(a+b)_{i+j}} \frac{(-m)_i(a)_i(b)_i}{i!(c)_i(e)_i}\frac{(-n)_j(a)_j(b)_j}{j!(c')_j(e')_j}. \]