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2. The quintuple product identity
In this section, we prove the quintuple product identity by using basic properties of cube roots of unity
and the Jacobi triple product identity.
Theorem 2.1. For any complex numbers z and q, with ݖ ് Ͳ and ȁݍȁ ൏ ͳ,
෍ ݍଷ௡మା௡ ሺݖଷ௡ ݍିଷ௡ െ ݖ ିଷ௡ିଵݍ ଷ௡ାଵሻ ஶ
௡ୀିஶ ൌ ሺݍଶǡ ݍݖ ǡ ݖ ିଵ ݍǢ ݍଶሻஶሺݖ ଶǡ ݖ ିଶ ݍସǢ ݍସሻஶǤ ሺʹǤͳሻ
Proof. Let f(z) denote the right hand side of (2.1). Since f(z) is analytic on Ͳ ൏ ȁݖ ȁ ൏ λ, we can write f as a Laurent
series ݂ሺݖ ሻ ൌ ෍ ܽ௡ݖ ௡ ஶ
௡ୀିஶ Ǥ ሺʹǤʹሻ
From the definition of f, we find that݂ሺݍݖଶ ሻ ൌ ሺݍଶǡ ݍݖ ଷǡ ݖ ିଵ ݍିଵ Ǣ ݍଶሻஶሺݖ ଶݍସǡ ݖିଶ Ǣ ݍସሻஶ
ൌ ሺͳ െݖ ିଵ ݍିଵ ሻሺͳ െ ݖ ିଶ ሻ
ሺͳ െ ݍݖሻሺͳ െ ݖ ଶሻ ሺݍଶǡ ݍݖ ǡ ݖ ିଵ ݍǢ ݍଶሻஶሺݖ ଶǡ ݖ ିଶ ݍସǢ ݍସሻஶ
ൌ ݂ሺݖሻ
ݖ ଷݍ
and hence ݂ሺݖ ሻ ൌ ݖ ଷݍ݂ሺݍݖ ଶሻǤ Thus, from (2.2), we have
෍ ܽ௡ ݖ ௡ ஶ
௡ୀିஶ ൌ ݖ ଷݍ ෍ ܽ௡ ݍଶ௡ݖ ௡ ஶ
௡ୀିஶ ൌ ෍ ܽ௡ ݍଶ௡ାଵ ݖ ௡ାଷ ஶ
௡ୀିஶ ൌ ෍ ܽ௡ିଷݍଶ௡ିହ ݖ ௡ ஶ
௡ୀିஶ Ǥ
Equating coefficients of ݖ ௡ on both sides, we find that, for each n, ܽ௡ ൌ ݍଶ௡ିହ ܽ௡ିଷǤ
By iteration, we find that, for each integer n, ܽଷ௡ ൌ ݍ ଷ௡మିଶ௡ܽ଴ǡ
ܽଷ௡ାଵ ൌ ݍଷ௡మ ܽଵǡ
ܽଷ௡ାଶ ൌ ݍଷ௡మାଶ௡ ܽଶǤ
From (2.2), we therefore have݂ሺݖ ሻ ൌ ෍ ܽଷ௡ ݖ ଷ௡ ஶ
௡ୀିஶ ൅ ෍ ܽଷ௡ାଵݖ ଷ௡ାଵ ஶ
௡ୀିஶ ൅ ෍ ܽଷ௡ାଶ ݖ ଷ௡ାଶ ஶ
௡ୀିஶ
ൌ ܽ଴ ෍ ݍଷ௡మ ିଶ௡ݖ ଷ௡ ஶ
௡ୀିஶ ൅ ܽଵ ෍ ݍ ଷ௡మݖ ଷ௡ାଵ ஶ
௡ୀିஶ ൅ ܽଶ ෍ ݍଷ௡మାଶ௡ݖ ଷ௡ାଶ ஶ
௡ୀିஶ Ǥ ሺʹǤ͵ሻ
From the definition of f, we also find that݂ሺݖ ିଵሻ ൌ ሺ ݍଶǡ ݖ ିଵ ݍǡ ݍݖ Ǣ ݍଶሻஶሺݖ ିଶǡ ݖ ଶݍସǢ ݍସሻஶ
ൌ ሺͳ െݖ ିଶ ሻ
ሺͳ െ ݖଶሻ ሺݍଶǡ ݖ ିଵ ݍǡ ݍݖǢ ݍଶሻஶሺݖ ିଶ ǡ ݖ ଶݍସǢ ݍସሻஶ
ൌ െ ͳݖ ଶ ݂ሺݖ ሻ
and hence ݂ሺݖ ሻ ൌ െݖ ଶ݂ሺݖ ିଵ ሻǤ Thus, from (2.2), we have
෍ ܽ௡ ݖ ௡ ஶ
௡ୀିஶ ൌ െݖ ଶ ෍ ܽ ௡ ݖି௡ ஶ
௡ୀିஶ ൌ െ ෍ ܽ௡ ݖ ି௡ାଶ ஶ
௡ୀିஶ ൌ െ ෍ ܽି௡ାଶݖ ௡ ஶ
௡ୀିஶ Ǥ
It follows that ܽଶ ൌ െܽ଴ and ܽଵ ൌ Ͳ.