reciprocal polynomial

Definition [1]Let $p:ℂ\to ℂ$be a polynomial of degree $n$ with complex (or real)coefficients. Then $p$ is a reciprocal polynomial if

$$p(z)=\pm z^{n}p(1/z)$$

for all $z\in ℂ$.

Examples of reciprocal polynomials are Gaussian polynomials, as well as the characteristic polynomials of orthogonal matrices (including the identity matrix as a special case), symplectic matrices, involution matrices (http://planetmath.org/LinearInvolution), and the Pascal matrices [2].

It is clear that if $z$ is a zero for a reciprocal polynomial, then$1/z$ is also a zero. This property motivates the name. Thismeans that the spectra of matrices of above type is symmetricwith respect to the unit circle in $ℂ$; if $\lambda \in ℂ$ is aneigenvalue, so is $1/\lambda$.

The sum, difference, and product of two reciprocal polynomials is again a reciprocal polynomial. Hence, reciprocal polynomials form an algebra over the complex numbers.