On Warnaar's elliptic matrix inversion and
Karlsson-Minton-type elliptic hypergeometric series
operator method, we give a new proof of
recent elliptic extension of Krattenthaler's matrix inversion.
Further, using a theta function identity closely related to Warnaar's
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.
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