Real Zeros of Random Cosine Polynomials with Palindromic Blocks of Coefficients

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