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
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