On some transformations of weighted second-order difference equations

This Sturm-Liouville problems often appear in mathematics, mechanics, physics. Spectral analysis of Sturm-Liouville operators plays on important role in knowing behaviour of operators. There are two methods to do this analysis. One of these methods is classical method that uses the arguments of differential calculus and other is difference equation form. In this from, continuous Sturm-Liouville operators are converted to discrete form and obtained a difference such an equation is called discrete Sturm-Liouville operator. Sometimes this method could be useful because of using simple operations instead of complicated operations of calculus, [1]. Most of this conversion of continuous Sturm-Liouville operator to discrete Sturm-Liouville difference equation gives a general weighted second-order difference equation.