Page:Annales de mathématiques pures et appliquées, 1825-1826, Tome 16.djvu/341

${\begin{aligned}\operatorname {f} _{n}(a,b,c,\ldots p,r,s)&={\frac {\operatorname {f} _{n-1}(a,b,c,\ldots p,q,s)-\operatorname {f} _{n-1}(a,b,c,\ldots p,q,r)}{s-r}},\\\\&={\frac {\operatorname {f} _{n-1}(a,b,c,\ldots p,q,r)}{r-s}}+{\frac {\operatorname {f} _{n-1}(a,b,c,\ldots p,q,s)}{s-r}}\,;\end{aligned}}$

il viendra donc, en substituant,

$\operatorname {f} _{n}(a,b,c,\ldots q,r,s)$

$=\left\{{\begin{aligned}&{\frac {\operatorname {f} a}{(a-b)(a-c)\ldots (a-q)(a-r)}}+{\frac {\operatorname {f} a}{(a-b)(a-c)\ldots (a-q)(a-s)}}\\\\+&{\frac {\operatorname {f} b}{(b-a)(b-c)\ldots (b-q)(b-r)}}+{\frac {\operatorname {f} b}{(b-a)(b-c)\ldots (b-q)(b-s)}}\\+&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots +\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\+&{\frac {\operatorname {f} q}{(q-a)(q-b)\ldots (q-p)(q-r)}}+{\frac {\operatorname {f} q}{(q-a)(q-b)\ldots (q-p)(q-s)}}\\\\+&{\frac {\operatorname {f} r}{(r-a)(r-b)\ldots (r-p)(r-q)}}+{\frac {\operatorname {f} s}{(s-a)(s-b)\ldots (s-p)(s-q)}}\end{aligned}}\right\}\,:$

où, en réduisant à une seule les fractions de même numérateur,

$\operatorname {f} _{n}(a,b,c,\ldots p,q,s)=\left\{{\begin{aligned}&{\frac {\operatorname {f} a}{(a-b)(a-c)\ldots (a-r)(a-s)}}\\\\+&{\frac {\operatorname {f} b}{(b-a)(b-c)\ldots (b-r)(b-s)}}\\&+\ldots \ldots \ldots \ldots \ldots \ldots \\+&{\frac {\operatorname {f} r}{(r-a)(r-b)\ldots (r-q)(r-s)}}\\\\+&{\frac {\operatorname {f} r}{(s-a)(s-b)\ldots (s-q)(s-r)}}\end{aligned}}\right\}\,;$