GENERALIZED HYPERGEOMETRIC SERIES
corresponding results for (xq) . When the number of free parameters is
large, the constructive approach no longer works. We rely upon
q-difference equations to develop path functions. In Section 5, we give
the infinite products nN, N _ 0, (which we wish to expand) and discuss
the q-difference equations they satisfy (which form a group). Section 6
develops the theory of separations from which we obtain path functions.
In Section 7, we derive infinite families of expansions. For 0 _ N _ 3
and m _ 0, n has an expansion with rate of convergence m. For
example, IU defined by (1.10) has the expansion (1.11) with rate of
convergence 0.
We will show that (1.11) includes the expansions (1.2) through (1.6)
(which have rates of convergence 0 through 4) as limiting cases. As
generally occurs when several expansions are combined into a single
expansion with more free parameters, (1.11) inherits the worst properties
of its constituents. It has the complexity of (1.4-) but, like (1.2), it
only has rate of convergence 0. We discover that the same complexity
which results from increasing the rate of convergence also occurs when
free parameters are added. All of our results are expressed in one
formula which reflects the common influence of the rate of convergence and
the number of free parameters. In Section 7, we discuss the form required
for expansions of nN for N M -
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