Dual form of combinatorial problems and Laplace techniques (Q2707837)

The authors investigate methods to calculate generating functions from recurrence relations. They mention three well-known commonly used methods being: (1) The \(z\)-transform function i.e. multiplying both sides of the relation by \(z^n\) and summing over \(n\) gives an algebraic equation for the generating function. (2) The use of Pólya's index theorem, from which one directly gets relations for the generating function. (3) Using difference equations, exploiting them by using continued fractions or Laplace transforms.NEWLINENEWLINENEWLINEThe emphasis of the paper is on using Laplace transforms. In this respect a useful technique is to look at a kind of dual for linear difference equations, especially in case the Laplace transform on the original difference equations does not lead to the desired information, application of this transform on the dual equations sometimes does the job. The usefulness of the theory presented in Section 2, is demonstrated in Section 3, where a lot of nice examples are worked out. This paper is not an easy one, but I think it is very worthwhile to study it carefully.