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In this paper, a method for calculating powers of general tridiagonal matrices is introduced. This method employs the close relationship among tridiagonal matrices, second-order linear homogeneous difference equations, and orthogonal polynomials. Some examples are included to demonstrate the implementation of the method.Mathematics Subject Classification: 65Q05; 39A05

The comparison of differential operators is a problem of the theory of partial differential operators with constant coefficients. This problem up to now doesn't have a complete solution. It was formulated in the sixties by Lars Hormander in his monograph "The Analysis of Linear Partial Differential Operators". Many facts of the theory of partial differential equations can be formulated by using the concept of pre-order relation over the set of differential operators, however it is too complicated to check the comparability condition of two differential operators. In this paper we get some sufficient conditions for the comparability of two differential operators.

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