The Determinant of an Elliptic Sylvesteresque Matrix

TL;DR

This paper extends the evaluation of determinants with elliptic hypergeometric entries, generalizing previous work by adding parameters and establishing transformation formulas using advanced elliptic hypergeometric identities.

Contribution

It provides an elliptic extension of a known hypergeometric determinant evaluation and introduces a transformation formula between elliptic determinants.

Findings

Derived an elliptic determinant evaluation with two additional parameters.

Established a transformation formula relating two elliptic determinants.

Utilized elliptic hypergeometric identities and lemmas for proofs.

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.