Petkovsek.txt

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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.