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

Let A=(aij)A=(aij) be an n×nn×n complex matrix. For any real μμ, define the polynomialPμ(A)=∑σ∈Sna1σ(1)⋯anσ(n)μℓ(σ),where ℓ(σ)ℓ(σ) is the number of inversions of the permutation σσ in the symmetric group SnSn. We analyze and establish a conjecture on the location of the zeros of Pμ(A)Pμ(A), when AA is a non-diagonal positive definite matrix. We prove the conjecture for the particular case when AA is a Jacobi matrix. Our proof is independent from known results, and uses a connection with orthogonal polynomials and chain sequences.