Asymptotics of instability zones of the Hill operator with a two term potential

Let γn denote the length of the nth zone of instability of the Hill operator Ly =−y − [4tα cos 2x + 2α2 cos 4x]y, where α = 0, and either both α, t are real, or both are pure imaginary numbers. For even n we prove: if t , n are fixed, then for α→0 γn = 8αn 2n[(n−1)!]2 n/2 k=1 t2 −(2k −1)2 1+ O(α) , and if α, t are fixed, then for n→∞ γn = 8|α/2|n [2 · 4 ···(n−2)]2 cos π 2 t 1+O log n n . The asymptotics for α→0, for n = 2m, imply the following identities for squares of integers: k s=1 m2 − i2 s = 1 j1<···