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I am going to solve a partial difference equation, firstly to transform it into a partial differential equation by using generating function method. Then, the partial differential equation can be solved by using method of characteristics.
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In many cases, the recurrence equations that people are solving involves index of only non-negative values. Here I have a recurrence equation which arises from transport of light in an infinite 1D chain:
where is the Kronecker delta function. i.e.:
...
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In the above website, you can download the ebook "generatingfunctionology". In short, it teaches the technique of generating function, and different usage of it in mathematics.
Generating function is a very useful technique in discrete mathematics. And we can use it to solve recurrence equations easily. For example:
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