Those of us who use q-series in our mathematical research are often asked the
question, "What is a q-series?" The quickest and simplest (but not so accurate or
informative) answer is:
is a series with q's in the summands." More informa-
tively, we might say q-series contain products (a; q)n, where
(0.1) (a;q)o:=1, (a;q)n:=(1-a)(1-aq) .. ·(1-aqn-
This is not entirely accurate, because in such series one often lets parameters tend
to 0 or oo, and so products of the type (0.1) may no longer appear. Theta functions
frequently arise and so are also thought of as q-series, even though they contain
no products of the form (0.1). Lambert series, or generalized Lambert series, of-
ten make appearances, especially in applications to number theory, and are also
regarded as part of the subject of q-series. In arithmetic applications of modular
forms, which include theta functions, one often needs their q-expansions. Thus, a
component of the vast theory of modular forms also has a home in the theory of
q-series. In conclusion, to paraphrase a senator who once claimed that he could
not define pornography, but he knew it when he saw it, most of us working with
q-series cannot give a good definition of a q-series, but we know a q-series when we
see it.
The subject of q-series can be said to begin with Euler and his pentagonal num-
ber theorem. In fact, q-series are sometimes called Eulerian series. Contributions
were made by Gauss, Jacobi, and Cauchy, but the first attempt at a systematic
development, especially from the point of view of studying series with the products
(0.1) in the summands, was made by E. Heine in 1847. In the latter part of the
nineteenth and in the early portions of the twentieth centuries, two English mathe-
maticians, L. J. Rogers and F. H. Jackson, made fundamental contributions. Their
work was largely ignored by the mathematical community, and so for many years the
subject of q-series was considered to be an unimportant, obscure topic on the fringes
of respectable mathematics. To illustrate the humble position occupied by the sub-
ject for several years, we offer two testimonies. In 1940, G. H. Hardy, on page 222
of his famous book, Ramanujan, described what we now call Ramanujan's famous
1'lj.;1 summation theorem as "a remarkable formula with many parameters." This is
now one of the fundamental theorems of the subject, but Hardy, as well as other
mathematicians during his time, could not foresee its importance. T. W. Chaundy,
in his obituary ofF. H. Jackson
London Math. Soc. 37 (1942) 126-128), uncivilly
claimed, "The problem [q-series] is very much that of unscrambling an egg."
Despite such humble beginnings, the subject of q-series has flourished in the
past three decades. There are several reasons for this.
took mathematicians
several years before most of Rogers and Ramanujan's contributions to q-series were
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