On Warnaarʹs elliptic matrix inversion and Karlsson–Minton-type elliptic hypergeometric series

Using Krattenthalerʹs operator method, we give a new proof of Warnaarʹs recent elliptic extension of Krattenthalerʹs matrix inversion. Further, using a theta function identity closely related to Warnaarʹs inversion, we derive summation and transformation formulas for elliptic hypergeometric series of Karlsson–Minton type. A special case yields a particular summation that was used by Warnaar to derive quadratic, cubic and quartic transformations for elliptic hypergeometric series. Starting from another theta function identity, we derive yet different summation and transformation formulas for elliptic hypergeometric series of Karlsson–Minton type. These latter identities seem quite unusual and appear to be new already in the trigonometric (i.e., p = 0 ) case.