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3. The septuple product identity
In this section, we prove the septuple product identity by using some properties of fifth roots of unity
and the Jacobi triple product identity.
Theorem 3.1. For any complex numbers z and q, with ݖ ് Ͳ and ȁݍȁ ൏ ͳ,
ሺݍସǡ ݍ଺ǡ ݍଵ଴Ǣ ݍଵ଴ሻஶ ෍ ሺെͳሻ௡ ሺݍହ௡మାଷ௡ݖ ହ௡ାଷ ൅ ݍହ௡మିଷ௡ݖ ହ௡ ሻ ஶ
௡ୀିஶ
െሺ ݍଶǡ ݍ଼ǡ ݍଵ଴Ǣ ݍଵ଴ሻஶ ෍ ሺെͳሻ௡ ሺݍହ௡ మା௡ݖ ହ௡ାଶ ൅ ݍହ௡మି௡ ݖହ௡ାଵ ሻ ஶ
௡ୀିஶ ൌ ሺݍଶǡ ݍ ଶǡ ݖǡ ݖ ିଵݍ ଶǡ ݖ ଶǡ ݖ ିଶ ݍଶǢ ݍଶሻஶǤ ሺ͵Ǥͳሻ
Proof. Let f(z) denote the right hand side of (3.1). Since f(z) is analytic on Ͳ ൏ ȁݖ ȁ ൏ λ, we can write f as a Laurent
series ݂ሺݖ ሻ ൌ ෍ ܽ௡ݖ ௡ ஶ
௡ୀିஶ Ǥ ሺ͵Ǥʹሻ
From the definition of f, we find that݂ሺݍݖ ଶሻ ൌ ሺݍଶǡ ݍ ଶǡ ݍݖ ଶǡ ݖ ିଵ ǡ ݖ ଶݍସǡ ݖ ିଶ ݍିଶ Ǣ ݍଶሻஶ
ൌ ሺͳ െݖ ିଵ ሻሺͳ െ ݖ ିଶ ݍିଶ ሻሺͳ െ ݖ ିଶ ሻ
ሺͳ െ ݖሻሺͳ െ ݖ ଶሻሺͳ െ ݖ ଶݍଶሻ ሺݍଶǡ ݍଶǡ ݖǡ ݖିଵ ݍଶǡ ݖ ଶǡ ݖ ିଶ ݍଶǢ ݍଶሻஶ
ൌ െ ͳݖ ହݍଶ ݂ሺݖሻ
and hence ݂ሺݖ ሻ ൌ െݖ ହݍଶ݂ሺݍݖ ଶሻǤ Thus, from (3.2), we have
෍ ܽ ௡ ݖ௡ ஶ
௡ୀିஶ ൌ െݖ ହݍଶ ෍ ܽ௡ ݍଶ௡ ݖ ௡ ஶ
௡ୀିஶ ൌ െ ෍ ܽ௡ ݍଶ௡ାଶݖ ௡ାହ ஶ
௡ୀିஶ ൌ െ ෍ ܽ௡ିହ ݍଶ௡ି଼ ݖ௡ ஶ
௡ୀିஶ Ǥ
Equating coefficients of ݖ ௡ on both sides, we find that, for each n, ܽ௡ ൌ െݍ ଶ௡ି଼ܽ ௡ିହǤ ሺ͵Ǥ͵ሻ
By iteration, we find that, for each integer n, ܽହ௡ ൌ ሺെͳሻ ௡ݍହ௡ మିଷ௡ܽ଴ǡ
ܽହ௡ାଵ ൌ ሺെͳሻ௡ ݍହ௡మି௡ ܽଵǡ
ܽହ௡ାଶ ൌ ሺെͳሻ௡ ݍହ௡మା௡ ܽଶǡ
ܽହ௡ାଷ ൌ ሺെͳሻ௡ ݍହ௡మାଷ௡ ܽଷǡ
ܽହ௡ାସ ൌ ሺെͳሻ௡ ݍହ௡మାହ௡ ܽସǤ
From (3.2), we therefore have݂ሺݖ ሻ ൌ ܽ଴ ෍ ሺെͳሻ௡ݍ ହ௡మିଷ௡ݖ ହ௡ ஶ
௡ୀିஶ ൅ ܽଵ ෍ ሺെͳሻ௡ ݍହ௡మ ି௡ݖ ହ௡ାଵ ஶ
௡ୀିஶ
൅ܽଶ ෍ ሺെͳሻ௡ ݍହ௡మା௡ ݖ ହ௡ାଶ ஶ
௡ୀିஶ ൅ ܽଷ ෍ ሺെͳሻ௡ ݍହ௡మାଷ௡ݖ ହ௡ାଷ ஶ
௡ୀିஶ
൅ܽସ ෍ ሺെͳሻ௡ ݍହ௡మାହ௡ ݖହ௡ାସ Ǥ ஶ
௡ୀିஶ ሺ͵ǤͶሻ
From the definition of f, we also find that݂ሺݖ ିଵ ሻ ൌ ሺݍଶǡ ݍଶ ǡ ݖ ିଵ ǡ ݍݖଶǡ ݖ ିଶ ǡ ݖ ଶݍଶǢ ݍଶሻஶ
ൌ ሺͳ െݖ ିଵ ሻሺͳ െ ݖ ିଶ ሻ
ሺͳ െ ݖ ሻሺͳ െ ݖ ଶሻ ሺݍଶǡ ݍ ଶǡ ݖ ିଵ ݍଶǡ ݖ ǡ ݖ ିଶ ݍଶǡ ݖ ଶǢ ݍଶሻஶ
ൌ ͳݖ ଷ ݂ሺݖ ሻ
and hence ݂ሺݖ ሻ ൌ ݖ ଷ݂ሺݖ ିଵ ሻǤ Thus, from (3.2), we have