推 Frobenius:想不到為什麼會多了1/2？ 02/24 23:11
→ akrsw:對，我不知道為何會多出 1/2。 02/24 23:16
※ 編輯: akrsw 來自: 218.165.184.61 (02/24 23:22)
※ 引述《Frobenius (▽．(▽×▽φ)=0)》之銘言：
: Show that
: n-s
: (s - n)! (-1) (2n - 2s)!
: ────── = ────────
: (2s - 2n)! (n - s)!
: Here s and n are integers with s < n. This result can be used to avoid
: negative factorials such as in the series representations of the spherical
: Neumann funtions and the Legendre functions of the second kind.
(s-n)! = (s-n)(s-n-1)!
= (s-n)(s-n-1)...(2s-2n+1)(2s-2n)!
(s-n)!/(2s-2n)! = (-1)^(n-s) (n-s)(n-s+1)...(2n-2s-1)
= (-1)^(n-s) (2n-2s-1)!/(n-s-1)!
= (-1)^(n-s) [(2n-2s)!/(2n-2s)]/[(n-s)!/(n-s)]
= (-1)^(n-s) (2n-2s)!/[2*(n-s)!]
請問哪裡錯？請指正，謝謝。
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