Factorization of the characteristic function of a Jacobi matrix

In a recent paper a class of infinite Jacobi matrices with discrete character of spectra has been introduced. With each Jacobi matrix from this class an analytic function is associated, called the characteristic function, whose zero set coincides with the point spectrum of the corresponding Jacobi operator. Here it is shown that the characteristic function admits Hadamard's factorization in two possible ways -- either in the spectral parameter or in an auxiliary parameter which may be called the coupling constant. As an intermediate result, an explicit expression for the power series expansion of the logarithm of the characteristic function is obtained.