On the eigenvalue problem for a particular class of finite Jacobi matrices

A function F with simple and nice algebraic properties is defined on a subset of the space of complex sequences. Some special functions are expressible in terms of F, first of all the Bessel functions of first kind. A compact formula in terms of the function F is given for the determinant of a Jacobi matrix. Further we focus on the particular class of Jacobi matrices of odd dimension whose parallels to the diagonal are constant and whose diagonal depends linearly on the index. A formula is derived for the characteristic function. Yet another formula is presented in which the characteristic function is expressed in terms of the function F in a simple and compact manner. A special basis is constructed in which the Jacobi matrix becomes a sum of a diagonal matrix and a rank-one matrix operator. A vector-valued function on the complex plain is constructed having the property that its values on spectral points of the Jacobi matrix are equal to corresponding eigenvectors. (C) 2010 Elsevier Inc. All rights reserved.