Symmetric polynomials in the symplectic alphabet and the change of variables z_j = x_j + x_j ^-1

Given a symmetric polynomial P in 2n variables, there exists a unique symmetric polynomial Q in n variables such that P(x₁,…,x_n, x^-1 ₁,….,x^-1 _n) = Q(x₁ + x^-1 ₁,…, x_n + x^-1 _n). We denote this polynomial Q by Φ_n(P) and show that Φ_n is an epimorphism of algebras. We compute Φ_n(P) for several families of symmetric polynomials P: symplectic and orthogonal Schur polynomials, elementary symmetric polynomials, complete homogeneous polynomials, and power sums. Some of these formulas were already found by Elouafi (2014) and Lachaud (2016). The polynomials of the form Φ_n(s⁽²ⁿ⁾ _λ/µ), where s⁽²ⁿ⁾ _λ/µ is a skew Schur polynomial in 2n variables, arise naturally in the study of the minors of symmetric banded Toeplitz matrices, when the generating symbol is a palindromic Laurent polynomial, and its roots can be written as x1,…, x_n, x^-1 ₁,. …x^-1 _n. Trench (1987) and Elouafi (2014) found efficient formulas for the determinants of symmetric banded Toeplitz matrices. We show that these formulas are equivalent to the result of Ciucu and Krattenthaler (2009) about the factorization of the characters of classical groups.