Page:Annales de mathématiques pures et appliquées, 1822-1823, Tome 13.djvu/171

165

RACINES DES NOMBRES.

${\begin{array}{lll}A'_{m-2}+bA'_{m-3}&=A''_{m-2}&,\end{array}}$

on aura

$A''_{m-2}={\frac {m}{1}}.{\frac {m-1}{2}}(a+b)^{m-2}\,;$

qu’en posant encore

${\begin{array}{lll}A''_{1}+b&=A'''_{1}&,\\A''_{2}+bA'''_{1}&=A'''_{2}&,\\A''_{3}+bA'''_{2}&=A'''_{3}&,\\\ldots \ldots &\ldots &\\A''_{m-3}+bA'''_{m-4}&=A'''_{m-3}&,\end{array}}$

on aura

$A'''_{m-3}={\frac {m}{1}}.{\frac {m-1}{2}}.{\frac {m-2}{3}}(a+b)^{m-3},$

et ainsi de suite. On a donc ainsi

${\begin{array}{ll}A_{m}&=(a+b)^{m},\\A'_{m-1}&={\frac {m}{1}}(a+b)^{m-1},\\\\A''_{m-2}&={\frac {m}{1}}.{\frac {m-1}{2}}(a+b)^{m-2},\\\\\ldots &\ldots \ldots \\A_{1}^{(m-1)}&={\frac {m}{1}}(a+b),\\\\A_{0}^{(m)}&=1\,;\end{array}}$

d’où