# The determinant of an elliptic Sylvesteresque matrix

### (16 pages)

**Abstract.**
We evaluate the determinant of a matrix whose entries are
elliptic hypergeometric terms and whose form is reminiscent of
Sylvester matrices. A hypergeometric determinant
evaluation of a matrix of this type has appeared in the context of
approximation theory, in the work of Feng, Krattenthaler and Xu.
Our determinant evaluation is
an elliptic extension of their evaluation, which has two
additional parameters (in addition to the base *q* and nome *p* found
in elliptic hypergeometric terms). We also extend the evaluation to
a formula transforming an elliptic determinant into a multiple of
another elliptic determinant. This transformation has two further
parameters. The proofs of the determinant evaluation and the transformation formula
require an elliptic determinant lemma due to Warnaar, and the
application of two *C*_{n} elliptic formulas that extend Frenkel and
Turaev's _{10}*V*_{9} summation formula and _{12}*V*_{11}
transformation
formula, results due to Warnaar, Rosengren, Rains, and Coskun and
Gustafson.

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