full2010.pdf - page 426

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From (2.3) and the definition of f, we therefore have ሺݍଶǡ ݍݖ ǡ ݖ ିଵ ݍǢ ݍଶሻஶሺݖ ଶǡ ݖ ିଶ ݍସǢ ݍସሻஶ
ൌ ܽ ଴ ෍ ݍଷ௡మିଶ௡ݖ ଷ௡ ஶ
௡ୀିஶ ൅ ܽଵ ෍ ݍଷ௡మ ݖ ଷ௡ାଵ ஶ
௡ୀିஶ ൅ ܽଶ ෍ ݍଷ௡మ ାଶ௡ݖ ଷ௡ାଶ ஶ
௡ୀିஶ
ൌ ܽ ଴ ෍ ൫ݍଷ௡మ ିଶ௡ݖ ଷ௡ െ ݍଷ௡మାଶ௡ݖ ଷ௡ାଶ ൯Ǥ ሺʹǤͶሻ ஶ
௡ୀିஶ
Next, we calculate ܽ଴. Setting ݖ ൌ ݍ߱ in (2.4), we obtain
ሺݍଶǡ ݍ ଶ߱ǡ ߱ଶ Ǣ ݍ ଶሻஶ ሺݍ ଶ߱ଶ ǡ ݍଶ߱Ǣ ݍସሻஶ ൌ ܽ଴ ෍ ൫ݍଷ௡ మା௡ െ ݍଷ௡మାହ௡ାଶ ߱ ଶ൯Ǥ ሺʹǤͷሻ ஶ
௡ୀିஶ
From left hand side of (2.5), using (1.4), we haveሺݍଶǡ ݍ ଶ߱ǡ ߱ଶ Ǣ ݍ ଶሻஶ ሺݍ ଶ߱ଶ ǡ ݍଶ߱Ǣ ݍସሻஶ ൌ ሺͳ െ ߱ଶሻሺݍଶǡ ݍଶ߱ǡ ݍଶ߱ ଶǢ ݍଶሻஶሺݍଶ߱ ଶǡ ݍଶ߱Ǣ ݍସሻஶ
ൌ ሺͳ െ ߱ଶሻሺݍ଺Ǣ ݍ଺ሻஶ ሺݍ ଶǡ ݍଶ߱ ଶǡ ݍ ଶ߱Ǣ ݍସ ሻஶ ሺݍଶǢ ݍସሻஶ
ൌ ሺͳ െ ߱ଶሻ ሺݍ଺Ǣ ݍ଺ ሻஶ ሺݍ ଺Ǣ ݍଵଶሻஶ ሺݍଶǢ ݍସሻஶ Ǥ ሺʹǤ͸ሻ
From right hand side of (2.5), using the Jacobi triple product identity, we have
ܽ଴ ෍ ൫ݍଷ௡మା௡ െ ݍ ଷ௡మାହ௡ାଶ ߱ଶ൯ ஶ
௡ୀିஶ ൌ ܽ଴ ෍ ൫ݍଷ௡మା௡ െ ݍଷሺ௡ିଵሻమାହሺ௡ିଵሻାଶ ߱ ଶ൯ ஶ
௡ୀିஶ
ൌ ܽ଴ ෍ ൫ݍଷ௡మା௡ െ ݍ ଷ௡మା௡߱ ଶ൯ ஶ
௡ୀିஶ
ൌ ܽ଴ሺͳ െ ߱ ଶሻ ෍ ݍଷ௡ మା௡ ஶ
௡ୀିஶ ൌ ܽ଴ሺͳ െ ߱ ଶሻሺݍ଺ǡ െ ݍଶǡ െݍସǢ ݍ଺ሻஶ
ൌ ܽ଴ሺͳ െ ߱ ଶሻ ሺݍ଺ǡ െݍଶǡ െݍ ସǡ െݍ଺Ǣ ݍ଺ሻஶ ሺെݍ଺Ǣ ݍ଺ሻஶ
ൌ ܽ଴ሺͳ െ ߱ ଶሻ ሺݍ଺Ǣ ݍ଺ሻஶሺെ ݍଶǢ ݍଶሻஶ ሺെ ݍ଺Ǣ ݍ଺ሻஶ Ǥ ሺʹǤ͹ሻ
From (2.6), (2.7) and using Euler’s identity, ሺെݍǢ ݍሻஶ ൌ ͳȀሺݍǢ ݍଶሻஶ , we have ܽ଴ ൌ ͳǤ Substituting ܽ଴ ൌ ͳ in (2.4),
we get ݂ሺݖ ሻ ൌ ෍ ൫ݍଷ௡మିଶ௡ ݖ ଷ௡ െ ݍଷ௡మାଶ௡ݖ ଷ௡ାଶ ൯ ஶ
௡ୀିஶ
ൌ ෍ ൫ݍଷ௡మ ିଶ௡ݖ ଷ௡ െ ݍଷሺି௡ିଵሻమାଶሺି௡ିଵሻ ݖଷሺି௡ିଵሻାଶ ൯ ஶ
௡ୀିஶ
ൌ ෍ ൫ݍଷ௡మ ିଶ௡ݖ ଷ௡ െ ݍଷ௡మାସ௡ାଵ ݖ ିଷ௡ିଵ൯ ஶ
௡ୀିஶ
ൌ ෍ ݍଷ௡మା௡ ሺݖ ଷ௡ݍ ିଷ௡ െ ݖିଷ௡ିଵ ݍଷ௡ାଵ ሻ ஶ
௡ୀିஶ Ǥ
Hence we have finished the proof.