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„µ¦¡· ¼‹œr Á°„¨´„¬–rŸ¨‡¼–°œ»„¦¤ q
Proofs of Some q-Series-Product Identities
Œ´˜¦´¥ ®œ¼Á¡¦1 ¨³«¦µ´¥ „o°Š«· ¦·ªŠ«r2
Chutchai Nupet and Sarachai Kongsiriwong
š‡´—¥n °
Šµœª·‹´¥œ¸Ê¤¸ª´˜™»ž¦³Š‡r Á¡ºÉ°¡· ¼‹œr Á°„¨´„¬–rŸ¨‡¼ –®oµ·É Š°´ œ—´  Á°„¨´„¬–rŸ¨‡¼ –Á‹È—·ÉŠ°´ œ—´  ¨³
Á°„¨´„¬–rŸ¨‡¼–°œ»„¦¤ q °ºÉœÇ ץčo„¦³ªœ„µ¦¡· ¼‹œrÁ—¸¥ª„´œ Ž¹ÉŠÄo‡»–¤´˜·°Š¦µ„ °Š®œ¹ÉŠÂ¨³
Á°„¨´„¬–rŸ¨‡¼–µ¤·ÉŠ°´œ—´ °Š Jacobi
‡ÎµÎµ‡´ : Á°„¨´„¬–rŸ¨‡¼–µ¤·ÉŠ°´œ—´ °Š Jacobi, Á°„¨´„¬–rŸ¨‡¼–®oµ·ÉŠ°´œ—´, Á°„¨´„¬–rŸ¨‡¼–Á‹È—·ÉŠ°´ œ—´,
Á°„¨´„¬–r°ŠWinquist
Abstract
In this paper, we gives new uniform proofs of the quintuple product identity, the septuple product
identity and other q-series-product identities. Some properties of roots of unity and the Jacobi triple product
identity are used to complete the proofs.
Keywords: Jacobi triple product identity, Quintuple product identity, Septuple product identity, Winquist’s
identity
1. Introduction
A q-series is a series involving coefficients of the formሺܽǢ ݍ ሻ௡ ൌෑሺͳ െ ƒݍ௞ ሻ ୬ିଵ
୩ୀ଴
for ݊ ൒ ͳ , where ሺܽǢ ݍሻஶ defined as ሺܽǢ ݍሻஶ ൌෑሺͳ െ ƒݍ௞ ሻ ஶ
୩ୀ଴ Ǥ
1 œ· ·˜ž¦·µÃš £µ‡ª·µ‡–·˜«µ˜¦r ‡–³ª·š¥µ«µ˜¦r ¤®µª· š¥µ¨´¥Š ¨µœ‡¦· œš¦r Š ¨µ 90112
Master's Degree Student, Department of Mathematics, Faculty of Science, Prince of Songkla University, Songkhla, 90112.
Corresponding author : Ú¦«´¡šr 0 84197 6426 E-mail:por075@hotmai.com
2 Ÿ«. —¦. £µ‡ª·µ‡–· ˜«µ˜¦r ‡–³ª·š¥µ«µ˜¦r ¤®µª·š¥µ¨´¥Š ¨µœ‡¦·œš¦r Š ¨µ 90112
Assist. Prof. Dr., Department of Mathematics, Faculty of Science, Prince of Songkla University, Songkhla, 90112.