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Or we can form an ordinary generating function
Then using the recurrence for and initial conditions we get
that
. We wish to find the coefficient of in
the expansion of (which is denoted ). We use partial
fractions and the binomial expansion to obtain the same result as before.
In general, the ordinary generating function associated with the sequence
is
, a ``formal
power series''. It is deduced from the recurrence and the initial
conditions.
Addition, subtraction, scalar multiplication, differentiation and
integration work as expected. The new thing is the ``product'' of
two such series:
where

is the ``convolution'' of the sequences
and
. Some functional
substitution also works.
Any identities give information about the
coefficients. We are not concered about convergence, but within the
radius of convergence we get extra information about values.

root
2002-06-10