Pascal-like determinants are recursive

Abstract

AbstractLet P=[pi,j]i,j⩾0 be an infinite matrix whose entries satisfy pi,j=λpi−1,j+μpi,j−1+νpi−1,j−1 for i,j⩾1, and whose first column respectively row satisfy linear recurrences with constant coefficients of orders ρ respectively σ. Then we show that its principal minors dn satisfy dn=∑j=1δcjωjndn−j where cj are constants, ω=λμ+ν, and δ=ρ+σ−2ρ−1. This implies a recent conjecture of Bacher [J. Théor. Nombres Bordeaux 14 (2002) 19–41]. The tools that we use are a formula for the generating function of a matrix product, and the fact that the principal minors of every band-diagonal matrix satisfy a nontrivial linear recurrence