2 KEVIN W. 3. KADELL

Euler [9, Chapter 16] gave the power series expansions (1.2) and (1.5)

of these classical partition generating functions. Setting x = -t(1-q)

or y = t(1-q) and letting q 1, we find that (1.2) and (1.5) are

q-analogs of the Taylor series for exp(t). Sylvester [15] developed a

constructive theory of partitions based largely upon the Durfee square.

From this he obtained (1.3), due to Cauchy [8, p. 4-8], and (1.6). Setting

x = 1 in (1.6) yields Euler's [9, p. 270] pentagonal number theorem

(1.8) (q)^ = 1 + 1 (-1)" q 2 d+q n ).

n = 1

See Franklin [11, pp. 448-50] for an elegant combinatorial proof. All of

our results so far fit into the theory of basic hypergeometric series as

special cases of the limiting form of Jackson's theorem (see Slater [14-,

(3.3.1.3)]). Andrews [1] generalized Sylvester's work by replacing the

square with a rectangle of arbitrary dimensions. The 2 by 1 rectangle

yielded (1.4-) and (1.7) which are not limiting cases of any of the

standard formulas for basic hypergeometric series. We say that a series

of the general form

(1.9)

2

m(—)

1 + I q 2 (...)

n = 1

has rate of convergence m. Thus the expansions (1.2), (1.3) and (1.4-)

1

of

(yq)

have rates of convergence 0, 2 and 4 - while (1.5), (1.6)

and (1.7) have rates of convergence 1, 3 and 5, respectively. For

1

k 2. 1 Andrews' k by 1 rectangle gives expansions of

( y q

and

(xq) with rates of convergence 2k and 2k + 1, respectively. This

increased rapidity of convergence is accompanied by increased complexity -

more factors from the infinite product and a polynomial of greater degree

with more terms.

We replace the rectangle with the more flexible path function.

Section 2 gives the basic notation and properties enjoyed by our path

functions. In Section 3, we give a constructive derivation of the path

1

function and related expansions of

ya)

Section 4 - gives the