Prof. Marko Petkovsek: "Explicit solutions of linear recurrence equations with polynomial coefficients"
Sprache des Titels:
When solving functional equations, one tends to look first for an explicit representation of the solution, i.e., for an expression built from the independent variable and the constants by means of various
admissible basic operations. Here we consider the problem of finding explicit solutions of homogeneous linear recurrence equations with polynomial coefficients (LRE).
Historically, the design of algorithms for finding explicit solutions proceeded by admitting more and more basic operations. Algorithms are known for finding, e.g., polynomial, rational, hypergeometric, d'Alembertian, and Liouvillian solutions. Alas, often no such non-zero
solutions exist, so it is natural to think of classes of explicitly representable sequences which properly contain the Liouvillian sequences.
One operation under which Liouvillian sequences are not closed is the Cauchy product or convolution of sequences. A convolution has the form of a definite sum, so we can ask more generally:
How to find solutions of LRE represented as (nested) definite sums of simpler sequences?
We will make a (tiny) step towards answering this question.