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Important q-series-product identities are Jacobi triple product identity, quintuple product identity,
septuple product identity and Winquist's identity.
The quintuple product identity was first published in Schwarz's book (Schwarz, 1893) in 1893.
G. N. Watson (Watson, 1929) gave a proof in 1929 and consequently the identity is sometimes known as Watson's
quintuple product identity. Since Watson's proof, many proofs of the identity has been given. A simple proof was
given by L. Carlitz and M. V. Subbarao (Carlitz-Subbarao, 1972). R. J. Evans (Evans, 1990) gave a proof using
complex function theory. For a history on the identity, see B. C. Berndt's book (Berndt, 1998).
The septuple product identity was first discovered by H. M. Farkas and I. Kra (Farkas-Kra, 1999) in
1999. D. Foata and G.-N. Han (Foata-Han, 2001) gave another proof of the identity and called the identity the
septuple product identity. In this paper we prove the quintuple product identity and septuple product identity. By
the same method, we can proof Winquist's identity and the other product identity.
We recall some simple facts about roots of unity and use the notation of infinite product. Let ߱ ൌ
‡š’ ሺʹߨ ݅Ȁ͵ሻ. Then we have ͳ ൅ ߱ ௡ ൅ ߱ଶ௡ ൌ ൜ ͵ ǡ ݊ ؠ Ͳሺ‘†͵ሻǡ
Ͳ ǡ ݊ ء Ͳሺ‘†͵ሻǡ
and, for any complex number x, ሺͳ െ ݔሻሺͳ െ ݔ߱ሻሺͳ െ ݔ߱ ଶሻ ൌ ͳ െ ݔଷ . (1.1)
Similarly, let ߫ ൌ ‡š’ ሺʹߨ݅Ȁͷሻ. Then we have, for any complex number x,
ሺͳ െ ݔሻሺͳ െ ݔ߫ሻሺͳ െ ݔ߫ ଶሻሺͳ െ ݔ߫ଷ ሻሺͳ െ ݔ߫ସሻ ൌ ͳ െ ݔ ହ. (1.2)
Let a, ߙǡ ߚǡ ߛ and q be complex numbers with ȁݍȁ ൏ ͳ. Set
ሺܽǢ ݍ ሻஶ ൌෑሺͳ െ ƒݍ௡ ሻ ஶ
୬ୀ଴ ǡ ሺͳǤ͵ሻ
and ሺߙ ǡ ߚǡ ڮ ǡ ߛǢ ݍሻஶ ൌ ሺߙǢ ݍሻஶሺߚǢ ݍሻஶ ڮ ሺߛǢ ݍሻஶ.
Using (1.1) and (1.3), we obtain ሺܽǢ ݍሻஶሺܽ߱Ǣ ݍ ሻஶ ሺܽ߱ଶ Ǣ ݍ ሻஶ ൌ ሺܽଷǢ ݍଷ ሻஶ . (1.4)
Using (1.2) and (1.3), we obtainሺܽǢ ݍሻஶሺܽ߫Ǣ ݍሻஶሺܽ߫ ଶǢ ݍሻஶ ሺܽ߫ଷǢ ݍሻஶሺܽ߫ ସǢ ݍሻஶ ൌ ሺܽହǢ ݍହሻஶ. (1.5)
In this paper, we will prove some q-series-product identities using the Jacobi triple product identity: for
each pair of complex numbers z and q, with ݖ ് Ͳ and ȁݍȁ ൏ ͳ,
෍ ሺെͳሻ௡ ݍሺ௡మି௡ሻȀଶݖ ௡ ஶ
௡ୀିஶ ൌ ሺݍǡ ݖ ǡ ݖ ିଵ ݍǢ ݍሻஶǤ