Even solutions of the Ince equation
where are real with , and is regarded as a spectral parameter are considered. This equation contains the Mathieu equation, the Whittaker-Hill equation, and the Lamé equation. Let , , for , where , and be the tridiagonal matrix . Consider the polynomials . Under some assumptions, the sequence is orthogonal in some sense (Theorem 1). For the polynomials , two types of results are obtained. First, if , , denote the zeros of (), then the sequence converges to – the th eigenvalue of (*) (), as (Theorem 2). Second, the interlacing properties of the zeros are discussed (Theorem 4). The lower and upper bounds for the eigenvalues of the Mathieu equation (Theorem 5), the Whittaker–Hill equation (Theorem 6), and the Lamé equation (Theorems 7,8) are given.