Title: Explicit solutions of linear recurrence equations with polynomial coefficients
Speaker: Prof. Dr. Marko Petkovsek
(Faculty of Mathematics and Physics, University of Ljubljana, Slovenia)
Time and Location: Wednesday, January 15, 2020, 1 p.m.
RISC Seminarroom, Hagenberg castle.
Abstract: 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.