An Automatic Approach to the Generating Functions of Some Special Sequences

Abstract

In this paper, we give explicit algorithms to compute generating functions of some special sequences, based on the operations of differential operators and shift operators in the non-commutative context and Zeilberger’s holonomic algorithm.
It can be found that not only ordinary generating functions and exponential generating functions but also generating functions of the general form \(\sum_{n} a_n(x)w(y, n)\) can now be computed automatically. Moreover, we generalize this approach and present explicit algorithms to compute \(2\)-variable ordinary power series generating functions and mixed-type generating functions. As applications, various examples are given in the paper.