### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 052, 15 pages      arXiv:1802.09885      https://doi.org/10.3842/SIGMA.2018.052
Contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications

### The Determinant of an Elliptic Sylvesteresque Matrix

Gaurav Bhatnagar and Christian Krattenthaler
Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

Received February 28, 2018, in final form May 27, 2018; Published online May 30, 2018

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.

Key words: determinant; $C_n$ elliptic hypergeometric series; Sylvester matrix.

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