Generalized Chebychev polynomials connected with a point interaction for the discrete Schrödinger equation

We consider a semi-infinite Jacobi matrix corresponding to a point interaction for the discrete Schrödinger operator. The entries of this matrix with indices (i, i - 1) and (i - 1, i) are equal to 1 if i ≠ k or to some positive number a if i = k. All other entries of this matrix are equal to zero. The spectral analysis of this problem (for k = 1) leads to a new class of polynomials generalizing the classical Chebyshev polynomials [1]. We will extend this class from the case k = 1 to the case of any integer k ≥ 1. Namely, we find for these polynomials expressions in terms of the classical Chebyshev polynomials. We obtain also a measure of orthogonality for generalized Chebyshev polynomials. This report is a continuation of authors' work [2].