"Trivializing" Generalizations of some Izergin-Korepin-type Determinants

We generalize (and hence trivialize and routinize) numerous explicit evaluations of determinants and pfaffians due to Kuperberg, as well as a determinant of Tsuchiya. The level of generality of our statements render their proofs easy and routine, by using Dodgson Condensation and/or Krattenthaler’s factor exhaustion method. All our matrices will be assumed to be embedded inside an infinite matrix. The first theorem adds parameters to the determinant formulas found in Kuperberg [Ku] (Theorem 15), as well as older determinants, mentioned there, due to Cauchy, Stembridge, Laksov-Lascoux-Thorup, and Tsuchiya [T]. This way, the formulation is suited to the method of [AZ]. Our proofs are much more succinct and automatable, since their generality enables an easy induction using Dodgson’s rule [D, AZ], or by employing Krattenthaler’s elegant factor exhaustion method [Kr1]. Relevant background for this paper can found in [Ku], and references thereof. Theorem 1: det 1 xi + yj + Axiyj 1,n...

• Type: Journal
• Year: 2007
• Where: DMTCS
• Authors: Tewodros Amdeberhan, Doron Zeilberger
• Keywords: Determinants, DMTCS 2003, DMTCS 2007, Factor Exhaustion, Numerous Explicit Evaluations