full2010.pdf - page 428

390
෍ ܽ௡ ݖ ௡ ஶ
௡ୀିஶ ൌ ݖ ଷ ෍ ܽ௡ ݖ ି௡ ஶ
௡ୀିஶ ൌ ෍ ܽ௡ ݖ ି௡ାଷ ஶ
௡ୀିஶ ൌ ෍ ܽି௡ାଷݖ ௡ ஶ
௡ୀିஶ Ǥ
It follows that ܽସ ൌ ܽିଵ ǡ ܽଷ ൌ ܽ଴ and ܽଶ ൌ ܽଵǤ However, from (3.3), we see that ܽସ ൌ െܽିଵ Ǥ Thus ܽସ ൌ0.
Substituting ܽଷ ൌ ܽ଴ǡ ܽଶ ൌ ܽଵ and ܽସ ൌ Ͳ in (3.4) and using the definition of f, we therefore have
ሺݍଶǡ ݍଶǡ ݖ ǡ ݖ ିଵ ݍଶǡ ݖ ଶǡ ݖ ିଶݍଶ Ǣ ݍ ଶሻஶ ൌ ܽ ଴ ෍ ሺെͳሻ௡ ݍହ௡మ ሺݍ ିଷ௡ݖ ହ௡ ൅ ݍଷ௡ݖ ହ௡ାଷ ሻ ஶ
௡ୀିஶ
൅ܽଵ ෍ ሺെͳሻ௡ ݍହ௡మ ሺݍି௡ݖ ହ௡ାଵ ൅ ݍ௡ݖ ହ௡ାଶ ሻǤ ሺ͵Ǥͷሻ ஶ
௡ୀିஶ
Next, we calculate ܽ଴ and ܽଵ. Setting ݖ ൌ ͳ in (3.5), we have
Ͳ ൌ ܽ଴ ෍ ሺെͳሻ ௡ݍହ௡మ ሺݍିଷ௡ ൅ ݍଷ௡ሻ ஶ
௡ୀିஶ ൅ ܽଵ ෍ ሺെͳሻ௡ ݍହ௡మ ሺݍ ି௡ ൅ ݍ௡ ሻǤ ሺ͵Ǥ͸ሻ ஶ
௡ୀିஶ
Setting ݖ ൌ ߫ in (3.5), we obtainሺݍଶǡ ݍଶǡ ߫ǡ ߫ିଵ ݍଶǡ ߫ ଶǡ ߫ ିଶ ݍଶǢ ݍଶሻஶ ൌ ܽ଴ ෍ ሺെͳሻ௡ݍହ௡మ ሺݍିଷ௡ ൅ ݍଷ௡߫ ଷሻ ஶ
௡ୀିஶ
൅ܽଵ ෍ ሺെͳሻ௡ ݍହ௡మ ሺݍି௡߫ ൅ ݍ௡ ߫ଶሻǤ ሺ͵Ǥ͹ሻ ஶ
௡ୀିஶ
From left hand side of (3.7) and using (1.5), we haveሺݍଶ ǡ ݍଶǡ ߫ǡ ߫ିଵ ݍଶǡ ߫ ଶǡ ߫ ିଶ ݍଶǢ ݍଶሻஶ ൌ ሺݍଶǡ ݍଶǡ ߫ǡ ߫ସݍଶǡ ߫ ଶǡ ߫ ଷݍଶǢ ݍଶ ሻஶ ൌ ሺͳ െ ߫ሻሺͳ െ ߫ ଶሻሺݍ ଶǡ ݍଶǡ ߫ ݍଶǡ ߫ ସݍଶǡ ߫ ଶݍଶǡ ߫ ଷݍଶǢ ݍଶሻஶ ൌ ሺͳ െ ߫ െ ߫ ଶ ൅ ߫ ଷሻሺݍଶǢ ݍଶ ሻஶ ሺݍଵ଴Ǣ ݍଵ଴ሻஶ ൌ ሺͳ െ ߫ െ ߫ ଶ ൅ ߫ ଷሻሺݍଶǡ ݍସǡ ݍ଺ ǡ ݍ଼ǡ ݍଵ଴Ǣ ݍଵ଴ሻஶሺݍଵ଴Ǣ ݍଵ଴ሻஶ Ǥ ሺ͵Ǥͺሻ
From right hand side of (3.7), we use (3.6) and the Jacobi triple product identity, we haveܽ ଴ ෍ ሺെͳሻ௡ ݍହ௡మ ሺݍ ିଷ௡ ൅ ݍ ଷ௡߫ଷ ሻ ஶ
௡ୀିஶ ൅ ܽଵ ෍ ሺെͳሻ௡ ݍହ௡మ ሺݍି௡߫ ൅ ݍ௡ ߫ଶሻ ஶ
௡ୀିஶ
ൌ ܽ ଴ ෍ ሺെͳሻ௡ ݍହ௡మିଷ௡ሺͳ ൅ ߫ ଷሻ ஶ
௡ୀିஶ ൅ ܽଵ ෍ ሺെͳሻ௡ ݍହ௡మି௡ ሺ߫ ൅ ߫ ଶሻ ஶ
௡ୀିஶ
ൌ ܽ ଴ ෍ ሺെͳሻ௡ ݍହ௡మିଷ௡ሺͳ ൅ ߫ ଷሻ ஶ
௡ୀିஶ െ ܽ଴ ෍ ሺെͳሻ௡ ݍହ௡మ ିଷ௡ሺ߫ ൅ ߫ ଶሻ ஶ
௡ୀିஶ
ൌ ܽ ଴ሺͳ െ ߫ െ ߫ଶ ൅ ߫ ଷሻ ෍ ሺെͳሻ௡ ݍହ௡మିଷ௡ ஶ
௡ୀିஶ ൌ ܽ ଴ሺͳ െ ߫ െ ߫ଶ ൅ ߫ ଷሻሺݍଶǡ ݍ଼ǡ ݍଵ଴Ǣ ݍଵ଴ሻஶǤ ሺ͵Ǥͻሻ
Hence, we have ܽ଴ ൌ ሺݍସǡ ݍ଺ǡ ݍଵ଴Ǣ ݍଵ଴ሻஶ. Substituting ܽ଴ in (3.6) and using the Jacobi triple product identity, we
have ܽଵ ൌ െሺݍସǡ ݍ଺ǡ ݍଵ଴Ǣ ݍଵ଴ሻஶ ሺݍଶǡ ݍ଼ǡ ݍଵ଴Ǣ ݍଵ଴ሻஶ ሺݍସǡ ݍ଺ǡ ݍଵ଴Ǣ ݍଵ଴ሻஶ ൌ െሺݍ ଶǡ ݍ଼ǡ ݍଵ଴Ǣ ݍଵ଴ሻஶǤ
Substituting ܽ଴ ൌ ሺݍସǡ ݍ଺ǡ ݍଵ଴Ǣ ݍଵ଴ሻஶ and ܽଵ ൌ െሺݍ ଶǡ ݍ଼ǡ ݍଵ଴Ǣ ݍଵ଴ሻஶ in (3.5), we therefore have finished the proof.