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let-P-x-1-n-x-2-1-n-calculate-P-n-x-and-P-n-0-

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Question Number 74501 by mathmax by abdo last updated on 25/Nov/19
let P(x)=(1/(n!))(x^2 −1)^n calculate P^((n)) (x) and P^( (n)) (0)
$$\:{let}\:{P}\left({x}\right)=\frac{\mathrm{1}}{{n}!}\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{{n}} \\ $$$${calculate}\:{P}^{\left({n}\right)} \left({x}\right)\:\:{and}\:{P}^{\:\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$
Commented by mathmax by abdo last updated on 28/Nov/19
we have P(x)=(1/(n!))Σ_(k=0) ^n C_n ^k x^(2k) (−1)^(n+k) =(((−1)^n )/(n!)) Σ_(k=0) ^n (−1)^k C_n ^k x^(2k) ⇒P^((n)) (x)= (((−1)^n )/(n!))+(((−1)^n )/(n!))Σ_(k=1) ^n (−1)^k C_n ^k (x^(2k) )^()n)) (x^(2k) )^((1)) =(2k)x^(2k−1) ,(x^(2k) )^((2)) =(2k)(2k−1) x^(2k−2) .... (x^(2k) )^((n)) =(2k)(2k−1)....(2k−n+1)x^(2k−1) =(((2k)!)/((2k−n)!)) x^(2k−1) ⇒ P^((n)) (x)=(((−1)^n )/(n!))+(((−1)^n )/(n!)) Σ_(k=1) ^n (−1)^k C_n ^k (((2k)!)/((2k−n)!)) x^(2k−1)
$${we}\:{have}\:{P}\left({x}\right)=\frac{\mathrm{1}}{{n}!}\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:{x}^{\mathrm{2}{k}} \left(−\mathrm{1}\right)^{{n}+{k}} \\ $$$$=\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\left(−\mathrm{1}\right)^{{k}} \:{C}_{{n}} ^{{k}} \:{x}^{\mathrm{2}{k}} \:\Rightarrow{P}^{\left({n}\right)} \left({x}\right)=\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!}+\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!}\sum_{{k}=\mathrm{1}} ^{{n}} \:\left(−\mathrm{1}\right)^{{k}} \:{C}_{{n}} ^{{k}} \:\left({x}^{\mathrm{2}{k}} \right)^{\left.\right)\left.{n}\right)} \\ $$$$\left({x}^{\mathrm{2}{k}} \right)^{\left(\mathrm{1}\right)} =\left(\mathrm{2}{k}\right){x}^{\mathrm{2}{k}−\mathrm{1}} \:,\left({x}^{\mathrm{2}{k}} \right)^{\left(\mathrm{2}\right)} =\left(\mathrm{2}{k}\right)\left(\mathrm{2}{k}−\mathrm{1}\right)\:{x}^{\mathrm{2}{k}−\mathrm{2}} \:…. \\ $$$$\left({x}^{\mathrm{2}{k}} \right)^{\left({n}\right)} =\left(\mathrm{2}{k}\right)\left(\mathrm{2}{k}−\mathrm{1}\right)….\left(\mathrm{2}{k}−{n}+\mathrm{1}\right){x}^{\mathrm{2}{k}−\mathrm{1}} \\ $$$$=\frac{\left(\mathrm{2}{k}\right)!}{\left(\mathrm{2}{k}−{n}\right)!}\:{x}^{\mathrm{2}{k}−\mathrm{1}} \:\Rightarrow \\ $$$${P}^{\left({n}\right)} \left({x}\right)=\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!}+\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}!}\:\sum_{{k}=\mathrm{1}} ^{{n}} \left(−\mathrm{1}\right)^{{k}} \:{C}_{{n}} ^{{k}} \:\:\frac{\left(\mathrm{2}{k}\right)!}{\left(\mathrm{2}{k}−{n}\right)!}\:{x}^{\mathrm{2}{k}−\mathrm{1}} \\ $$

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