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 -