Generating functions

[Thomas Baruchel]

Hi,

f(x) being a rational function :

  (a x^k + b x^(k-1) + ...) / (e x^l + f x^(l-1) + ... )

the expansion of f(x) is very easy to compute. I have the following
questions concerning the expansion of such a function :
  a) is there a sufficient criterion for having no null coefficient ?
     Of course in the definition of f(x), the numerator must have
     a term of degree 0 being not null (if this term is null, initial
     terms in the expansion will also be null), but it is not sufficient:
     1/(-x^2 + 1) has [1,0,1,0,1,0,1,0,1,...] in its expansion.
   b) is there a sufficient criterion for having an increasing sequence
     in the expansion (initial terms don't matter) ?
     Same question with the absolute value of the terms.
   c) is there a sufficient criterion for having some control on
     the sign of the terms (all positive, alternate, etc.) ?

Regards,

-- 
Thomas Baruchel
  Home Page: http://baruchel.free.fr/~thomas/