Generating functions

[Ralf Stephan]

> f(x) being a rational function :
> 
>   (a x^k + b x^(k-1) + ...) / (e x^l + f x^(l-1) + ... )
>[...]
>    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.

This is the easiest:

The denominator polynomial must have roots such that the
inverse root having the largest absolute value is positive. 
Then the sequence eventually becomes monotonous, the direction
being only dependent on the coefficient signs of the monomials 
with the largest exponent.

Example: 1 / [(1/2-x)*(1/3-x)] = 6 / (1-5x+6x^2) 
Sequence: 6, 30, 114, 390...
Example: 1 / [(1/2-x)*(1/3+x)] = 6 / (1+x-6x^2) 
Sequence: 6, -6, 42, -78, + - + - + -

This is because the abs. largest inv. root dominates the sum
of powers of all inv. roots eventually, regardless of factor.
It does NOT suffice to look at the signs of the largest monomials:

Example: 1 / [(1/2+x)*(1/3-x)] = 6 / (1-x-6x^2) 
Sequence: 6, 6, 42, 78 (same as above but positive)


ralf