Reduction formulae for Karlsson-Minton type hypergeometric functions

We prove a master theorem for hypergeometric functions of Karlsson-Minton type, stating that a very general multilateral U(n) Karlsson-Minton type hypergeometric series may be reduced to a finite sum. This identity contains the Karlsson-Minton summation formula and many of its known generalizations as special cases, and it also implies several "Bailey-type" identities for U(n) hypergeometric series, including multivariable 10-W-9 transformations of Milne and Newcomb and of Kajihara. Even in the one-variable case our identity is new, and even in this case its proof depends on the theory of multivariable hypergeometric series.