Reference request for anti-palindromic polynomials.

Question

I have come across a lot of papers that are written about the palindromic polynomials, however, I am recently interested in polynomials satisfying $$f(-x) = x^nf(1/x)$$ for $n\geq 1$ and for all $x\in \mathbb{R}$ except $0.$ Is there any reference where the roots of such polynomials are studied?

Answer 1

Assuming $n=\deg f$, it is easy to see that there is no such polynomial if $n$ is odd. So let us assume $n$ is even from now on. We may also assume $f$ is monic.

Let $S$ denote the multiset of roots of $f$ in $\mathbb{C}$ (note that $\# S$ is even). Then $f$ satisfies your condition if and only if $$\prod_{\alpha \in S} (x+\alpha) = \prod_{\alpha \in S} (1-\alpha x) = \bigl(\prod_{\alpha \in S} \alpha\bigr) \cdot \prod_{\alpha \in S} (x-1/\alpha).$$ This is equivalent to $\prod_{\alpha \in S} \alpha = 1$ and $S$ is invariant under the involution $\iota : z \mapsto -1/z$ (note that $S$ is contained in $\mathbb{C}^\times$). Now $\iota$ has 2 fixed points, namely $i$ and $-i$. If $S$ contains $\{i,-i\}$ then $f$ is divisible by $x^2+1$, and since $x^2+1$ satisfies your condition, the polynomial $f(x)/(x^2+1)$ also does. Dividing by $x^2+1$ as many times as needed, we may assume that $S$ does not contain $\{i,-i\}$. There are three cases:

1. $S \cap \{i,-i\}=\emptyset$. In this case $f(x) = \prod_{j=1}^{n/2} (x-z_j)(x+1/z_j)$ for some non necessarily distinct $z_j \in \mathbb{C}^\times \backslash \{\pm i\}$. Since the product of the roots is equal to $1$, we must have $n \equiv 0 \textrm{ mod } 4$.
2. $S$ contains $i$. Then the multiplicity of $i$ as a root of $f$ is even, so we may write $f(x) = \prod_{j=1}^{n/2} (x-z_j)(x+1/z_j)$ for some non necessarily distinct $z_j \in \mathbb{C}^\times \backslash \{-i\}$, and again $n \equiv 0 \textrm{ mod } 4$.
3. $S$ contains $-i$. Similarly $f(x) = \prod_{j=1}^{n/2} (x-z_j)(x+1/z_j)$ with $z_j \in \mathbb{C}^\times \backslash \{i\}$ and $n \equiv 0 \textrm{ mod } 4$.

In conclusion, the polynomials $f$ satisfying your condition are those of the form $$\lambda (x^2+1)^h \prod_{j=1}^{2m} (x-z_j)(x+1/z_j)$$ for some non necessarily distinct $z_j \in \mathbb{C}^\times$ and $\lambda \in \mathbb{C}^\times$, $h \in \{0,1\}$. Equivalently, a polynomial $f$ satisfies your condition if and only if its multiset of roots is the union of an even number of multisets $\{z,-1/z\}$ with $z \in \mathbb{C}^\times$ plus possibly one copy of $\{i,-i\}$.