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Question Number 162825 by KONE last updated on 01/Jan/22

Answered by mr W last updated on 01/Jan/22

$$2.$$$$P_{n+2}-2{XP}_{n+1}+P_n=0$$$$t^2-2Xt+1=0(char.eqn.)$$$$t=X\pm \sqrt{X^2-1}$$$$P_n=A{(X+\sqrt{X^2-1})}^n+B{(X-\sqrt{X^2-1})}^n$$$$P_0=A+B=1$$$$P_1=A(X+\sqrt{X^2-1})+B(X-\sqrt{X^2-1})=X$$$$\Rightarrow A=B=\frac{1}{2}$$$$P_n=\frac{{(X+\sqrt{X^2-1})}^n+{(X-\sqrt{X^2-1})}^n}{2}$$$$P_n=\frac{{(X+\sqrt{X^2-1})}^n+{(X+\sqrt{X^2-1})}^{-n}}{2}$$$$P_n=\frac{e^{n\ln (X+\sqrt{X^2-1})}+e^{-n\ln (X+\sqrt{X^2-1})}}{2}$$$$P_n=\cosh \left[n\ln (X+\sqrt{X^2-1})\right]$$$$P_n=\cosh \left(n{\cosh }^{-1}X\right)$$

Commented by KONE last updated on 02/Jan/22

$$merciavous$$

Commented by Tawa11 last updated on 02/Jan/22

$$\mathrm{Great}\mathrm{sir}$$

Answered by KONE last updated on 01/Jan/22

$$besoin{d}^{\prime }aide$$

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