The Determinant of an Elliptic Sylvesteresque Matrix

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.