Page:Annales de mathématiques pures et appliquées, 1824-1825, Tome 15.djvu/203

 Les corrections sont expliquées en page de discussion

$\Sigma (n,r)=M_{r}\left\{a^{r}+{\frac {r}{m-r+1}}\varphi (n,2)a^{r-1}b+{\frac {r}{m-r+1}}.{\frac {r-1}{m-r+2}}\varphi (n,3)a^{r-2}b^{2}\right.$

$+{\frac {r}{m-r+1}}.{\frac {r-1}{m-r+2}}.{\frac {r-2}{m-r+3}}\varphi (n,4)a^{r-3}b^{3}+\ldots$

$+{\frac {r}{m-r+1}}.{\frac {r-1}{m-r+2}}.{\frac {r-2}{m-r+3}}\ldots {\frac {r-i+1}{m-r+i}}\varphi (n,i+1)a^{r-i}b^{i}+\ldots$

$\left.+{\frac {\varphi (n,r+1)}{M_{r}}}b^{r}\right\}.$(Y)

Mais, à cause de la relation connue $\varphi (p,q)=\varphi (q,p),$ on a

${\begin{aligned}&m-r+1=\varphi (2,m-r+1)=\varphi (m-r+1,2),\\&(m-r+1)(m-r+2)=1.2\varphi (3,m-r+1)=1.2\varphi (m-r+1,3),\\&(m-r+1)(m-r+2)(m-r+3)=1.2.3\varphi (4,m-r+1)\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad =1.2.3\varphi (m-r+1,4),\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&(m-r+1)(m-r+2)\ldots (m-r+i)=1.2\ldots i\varphi (i+1,m-r+1)\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad =1.2\ldots i\varphi (m-r+1,i+1)\,;\end{aligned}}$

on a de plus

$M_{r}={\frac {m}{1}}.{\frac {m-1}{2}}.{\frac {m-2}{3}}\ldots {\frac {m-r+1}{r}}=\varphi (r+1,m-r+1)$

$=\varphi (m-r+1,r+1)$

Au moyen de ces valeurs, la série (Y) deviendra

$\Sigma (n,r)=M_{r}\left\{a^{r}+{\frac {r}{1}}{\frac {\varphi (n,2)}{\varphi (m-r+1,2)}}a^{r-1}b+{\frac {r}{1}}.{\frac {r-1}{2}}.{\frac {\varphi (n,3)}{\varphi (m-r+1,3)}}a^{r-2}b^{2}+\right.$

$+{\frac {r}{1}}.{\frac {r-1}{2}}.{\frac {r-2}{3}}{\frac {\varphi (n,4)}{\varphi (m-r+1,4)}}a^{r-3}b^{3}+\ldots$

$+{\frac {r}{1}}.{\frac {r-1}{2}}\ldots {\frac {r-i+1}{i}}.{\frac {\varphi (n,i+1)}{\varphi (m-r+1,i+1)}}a^{r-i}b^{i}+\ldots$

$\left.+{\frac {\varphi (n,r+1)}{\varphi (m-r+1,r+1)}}b^{r}\right\}.$(Z)