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4. Other q-series-product identities
By the same method that we used to prove the quintuple product identity and the septuple product identity, we can
prove further product identities by using cube roots of unity.
Example 4.1 (Winquist’s identity). For any complex number q, with ȁݍȁ ൏ ͳ , and any nonzero complex numbers
a and b, ෍ ෍ ሺെͳሻ௠ା௡ݍ ଷ௠మ ାଷ௡మାଷ௠ା௡ ଶ ሺܽିଷ௠ ܾିଷ௡ െ ܽିଷ௠ ܾଷ௡ାଵ െ ܽିଷ௡ାଵ ܾିଷ௠ିଵ ൅ ܽଷ௡ାଶ ܾିଷ௠ିଵሻ ஶ
௡ୀିஶ
ஶ
௠ୀିஶ ൌ ሺݍǡ ݍǡ ܽǡ ܽ ିଵݍǡ ܾǡ ܾିଵ ݍǡ ܾܽǡ ܽିଵ ܾିଵ ݍǡ ܾܽ ିଵ ǡ ܽିଵ ܾݍǢ ݍሻஶǤ
Example 4.2 (Kongsiriwong-Liu, 2003). For any complex numbers z and q, with ݖ ് Ͳ and ȁݍȁ ൏ ͳ,
ሺݍ଺Ǣ ݍ଺ሻஶ ሺݍ଺ Ǣ ݍଵଶሻஶ ෍ ݍଷ௡మ ஶ
௡ୀିஶ ݖ ଷ௡
െݍሺݍଶǡ ݍଵ଴ǡ ݍଵଶǢ ݍଵଶሻஶ ෍ ሺݍଷ௡ మାଶ௡ ஶ
௡ୀିஶ ݖ ଷ௡ାଵ ൅ ݍଷ௡మିଶ௡ ݖଷ௡ିଵ ሻ
ൌ ሺݍଶǡ ݍݖǡ ݖିଵ ݍǢ ݍଶሻஶ ሺݍସ ǡ ݖ ଶݍଶǡ ݖ ିଶ ݍଶǢ ݍସሻஶǤ
Using fifth roots of unity, we can prove further identity.
Example 4.3 (Kongsiriwong-Liu, 2003). For any complex numbers z and q, with ݖ ് Ͳ and ȁݍȁ ൏ ͳ, ሺݍǢ ݍଶሻହ ஶ൅ ͶሺݍହǢ ݍଵ଴ሻஶ ͷሺݍହǡ ݍହǡ ݍଵ଴Ǣ ݍଵ଴ሻஶ ෍ ሺെͳሻ ௡ݍହ௡మ ஶ
௡ୀିஶ ݖହ௡
൅ ሺݍǢ ݍଶ ሻହ ஶെ ሺݍହǢ ݍଵ଴ሻஶ ͷሺݍଷǡ ݍ ଻ǡ ݍଵ଴Ǣ ݍଵ଴ሻஶ ෍ ሺെͳሻ௡ݍ ହ௡మݖ ହ௡ ሺݍଶ௡ ஶ
௡ୀିஶ ݖ ൅ ݍିଶ௡ݖ ିଵ ሻ
൅ ሺݍǢ ݍଶ ሻହ ஶെ ሺݍହǢ ݍଵ଴ሻஶ ͷሺݍǡ ݍ ଽǡ ݍଵ଴Ǣ ݍଵ଴ሻஶ ෍ ሺെͳሻ௡ݍ ହ௡మݖ ହ௡ ሺݍସ௡ ஶ
௡ୀିஶ ݖ ଶ ൅ ݍିସ௡ݖ ିଶ ሻ
ൌ ሺݍǡ ݍݖǡ ݖ ିଵ ݍǡ ݖଶݍ ǡ ݖ ିଶ ݍǢ ݍଶሻஶ Ǥ
Using fourth roots of unity, we can prove further identities.
Example 4.4 (Kongsiriwong-Liu, 2003). For any complex numbers z and q, with ݖ ് Ͳ and ȁݍȁ ൏ ͳ, ሺെ ݍଶǢ ݍଶሻଶ ஶሺെ ݍ଺Ǣ ݍ଺ሻଶ ஶ൅ ሺെ ݍସǢ ݍସሻஶሺെ ݍଵଶǢ ݍଵଶሻஶ ʹሺݍ଼ǡ െݍ ଼ǡ െݍ଼Ǣ ݍ଼ሻஶ ෍ ݍସ௡మ ஶ
௡ୀିஶ ݖ ସ௡
െݍ ሺെݍଶǢ ݍଶ ሻଶ ஶሺെݍ଺ Ǣ ݍ଺ ሻଶ ஶ
ሺݍ଼ǡ െݍ ଺ǡ െݍଶǢ ݍ଼ሻஶ ෍ ݍସ௡మݖ ସ௡ ሺݍ ଶ௡ ஶ
௡ୀିஶ ݖ ൅ ݍିଶ௡ݖ ିଵ ሻ
൅ݍଶ ሺെݍଶ Ǣ ݍ ଶሻଶ ஶሺെݍ଺ Ǣ ݍ ଺ሻଶ ஶെ ሺെݍସǢ ݍସ ሻஶ ሺെݍଵଶǢ ݍଵଶሻஶ ሺݍ଼ǡ െݍ ସǡ െݍସǢ ݍ଼ሻஶ ෍ ݍ ସ௡మାସ௡ݖ ସ௡ାଶ ஶ
௡ୀିஶ ൌ ሺݍݖ ǡ ݖ ିଵ ݍǢ ݍଶሻஶሺݖଷ ݍଷǡ ݖ ିଷ ݍଷǢ ݍ଺ሻஶǤ
Example 4.5 (Kongsiriwong-Liu, 2003). For any complex numbers z and q, with ݖ ് Ͳ and ȁݍȁ ൏ ͳ, ሺെ ݍସǢ ݍସሻஶሺെ ݍ଺Ǣ ݍଵଶሻஶ ൅ ሺെݍଶǢ ݍଶሻଶ ஶሺെݍଷǢ ݍ଺ሻஶ ʹሺݍ଼ǡ െ ݍହǡ െݍଷǢ ݍ଼ሻஶ ෍ ݍସ௡మ ஶ
௡ୀିஶ ݖସ௡ ሺݍିଷ௡ െ ݍଷ௡ݖ ଷሻ
൅ ሺെݍସǢ ݍସሻஶ ሺെݍ଺Ǣ ݍଵଶሻஶ െ ሺെݍଶ Ǣ ݍଶ ሻଶ ஶሺെݍଷ Ǣ ݍ ଺ሻஶ ʹݍሺݍ଼ǡ െݍ ǡ െݍ଻Ǣ ݍ଼ሻஶ ෍ ݍସ௡మݖ ସ௡ ሺݍ ି௡ ஶ
௡ୀିஶ ݖ ൅ ݍ௡ݖ ଶሻ
ൌ ሺݍݖ ǡ ݖ ିଵ ݍǢ ݍଶሻஶሺݖଷ ǡ ݖ ିଷ ݍ଺Ǣ ݍ଺ሻஶǤ