Real zeros of random cosine polynomials with palindromic blocks of coefficients

It is well known that a random cosine polynomial $V_n(x) = \sum_ {j=0} ^{n} a_j \cos (j x) , \ x \in (0,2 \pi)$, with the coefficients being independent and identically distributed (i.i.d.) real-valued standard Gaussian random variables (asymptotically) has $2n / \sqrt{3}$ expected real roots. On the other hand, out of many ways to construct a dependent random polynomial, one is to force the coefficients to be palindromic. Hence, it makes sense to ask how many real zeros a random cosine polynomial (of degree $n$) with identically and normally distributed coefficients possesses if the coefficients are sorted in palindromic blocks of a fixed length $\ell.$ In this paper, we show that the asymptotics of the expected number of real roots of such a polynomial is $\mathrm{K}_\ell \cdot 2n / \sqrt{3}$, where the constant $\mathrm{K}_\ell$ (depending only on $\ell$) is greater than 1, and can be explicitly represented by a double integral formula. That is to say, such polynomials have slightly more expected real zeros compared with the classical case with i.i.d. coefficients.