The μ-permanent of a tridiagonal matrix, orthogonal polynomials, and chain sequences

Let A = (a(ij)) bean n x n complex matrix. For any real mu, define the polynomial P(mu)(A) = Sigma(sigma epsilon Sn) a(1 sigma(1)) ... a(n sigma(n))mu(l(sigma)), where l(sigma) is the number of inversions of the permutation a in the symmetric group S(n). We analyze and establish a conjecture on the location of the zeros of P(mu) (A), when A is a non-diagonal positive definite matrix. We prove the conjecture for the particular case when A is a Jacobi matrix. Our proof is independent from known results, and uses a connection with orthogonal polynomials and chain sequences. (C) 2009 Elsevier Inc. All rights reserved.