From: spellucci@mathematik.th-darmstadt.de (Peter Spellucci)
Newsgroups: sci.math.research
Subject: Re: difference equations
Date: 1 Apr 1998 10:18:09 GMT
Keywords: difference equations

In article <bpcEqnv3o.Dnq@netcom.com>, bpc@netcom.com (Benjamin P. Carter) writes:
snip snip |> 
|> What I am interested in is the theory of linear difference equations
|> with constant coefficients, which ought to be very similar to the
|> better known theory of linear (ordinary) differential equations with
|> constant coefficients. 
|> 
snip  
the book : 
Saber N. Elaydi: an introduction to difference equations ,
  springer publ. 1996, ISBN 0-387-94582-2
has a complete treatment of that. your solution idea is completely 
correct and the proof goes by tranforming the difference equation into a 
first order one for a vector, with the companion matrix of your polynomial 
as its matrix, and then using the theory of the jordan normal form of
that matrix. since it is a so called frobenius matrix , it has only one linear
ly independent eigenvector for every eigenvalue (zero of the polynomial)
hence exactly one jordan block associated with any eigenvalue (zero).
hope this helps
peter.

==============================================================================

From: bpc@netcom.com (Benjamin P. Carter)
Newsgroups: sci.math.research
Subject: Re: difference equations
Date: Sun, 5 Apr 1998 07:28:22 GMT

Following is a summary of responses (both news and e-mail) to my
previous query about difference equations.

Ilya Zakharevich and Peter Spellucci suggested using Jordan form to prove
the basic theorem I had asked about.  Use of a generating function was
suggested by Brian Stewart, David Wagner, Iain Davidson, and Robin
Chapman. Other methods or references were suggested by Stephen
Montgomery-Smith, Andrej Bauer, Keith Briggs, L. Andrew Campbell, and
Robert Low.  Thanks to all of these people and anyone I forgot for their 
many helpful suggestions.  The following books were mentioned:

Agarwal, Ravi P.:  "Difference equations and inequalities: theory, 
methods, and applications"
 
Behnke, Bachmann, Fladt, Suess: "Fundamentals of Mathematics, Volume III: 
Analysis"   
 
Biggs: "Discrete Mathematics"
 
Boole: "Finite Differences"
 
Saber N. Elaydi: "An introduction to difference equations"
 
Samuel I. Goldberg:  "Introduction to Difference Equations"   .
 
Graham, Knuth and Patashnik: `Concrete Math' 
 
H. Levy and F. Lessman: "Finite Difference Equations"
 
M. Petkovsek, H. Wilf, D. Zeilberger: "A = B"
 
R.P. Stanley: "Enumerative Combinatorics, vol. I"
 
Wilf:  "Generatingfunctionology".

I don't promise to read all of these books from cover to cover, but I will
look for some of them in libraries.  Thanks again for all replies. 

-- 
    Ben Carter