Page:Joseph Louis de Lagrange - Œuvres, Tome 3.djvu/36

${\begin{aligned}+&\mathrm {E} \alpha ^{p+4q-1}y^{p+4q}\psi '(y)+\mathrm {E} _{1}\alpha ^{2p+3q-2}.{\frac {1}{2}}{\frac {d\left[y^{2p+3q}\psi '(y)\right]}{dy}}\\&+\mathrm {E} _{2}\alpha ^{3p+2q-3}.{\frac {1}{2.3}}{\frac {d^{2}\left[y^{3p+2q}\psi '(y)\right]}{dy^{2}}}+\mathrm {E} _{3}\alpha ^{4p+q-4}.{\frac {1}{2.3.4}}{\frac {d^{3}\left[y^{4p+q}\psi '(y)\right]}{dy^{3}}}\\&+\mathrm {E} _{4}\alpha ^{5p-5}.{\frac {1}{2.3.4.5}}{\frac {d^{4}\left[y^{5p}\psi '(y)\right]}{dy^{4}}}\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\end{aligned}}$

Si l’on fait $\psi (y)=y^{m},$ et par conséquent $\psi '(y)=my^{m-1},$ on trouvera, en faisant $y=1,$

${\begin{aligned}{\frac {x^{m}}{\alpha ^{m}}}=&1\\&+m\mathrm {A} \alpha ^{p-1}\\&+m\mathrm {B} \alpha ^{p+q-1}+{\frac {m(m+2p-1)}{2}}\mathrm {B} _{1}\alpha ^{2p-2}+\\&+m\mathrm {C} \alpha ^{p+2q-1}+{\frac {m(m+2p+q-1)}{2}}\mathrm {C} _{1}\alpha ^{2p+q-2}\\&\quad +{\frac {m(m+3p-1)(m+3p-2)}{2.3}}\mathrm {C} _{2}\alpha ^{3p+q-3}\\\\&+m\mathrm {D} \alpha ^{p+3q-1}+{\frac {m(m+2p+2q-1)}{2}}\mathrm {D} _{1}\alpha ^{2p+2q-2}\\&\quad +{\frac {m(m+3p+q-1)(m+3p+q-2)}{2.3}}\mathrm {D} _{2}\alpha ^{3p+q-3}\\&\quad +{\frac {m(m+4p+q-1)(m+4p+q-2)(m+4p+q-3)}{2.3.4}}\mathrm {D} _{3}\alpha ^{4p-4}\\\\&+m\mathrm {E} \alpha ^{p+4q-1}+{\frac {m(m+2p+3q-1)}{2}}\mathrm {E} _{1}\alpha ^{2p+3q-2}\\&\quad +{\frac {m(m+3p+2q-1)(m+3p+2q-2)}{2.3}}\mathrm {E} _{2}\alpha ^{3p+2q-3}\\&\quad +{\frac {m(m+4p+q-1)(m+4p+q-2)(m+4p+q-3)}{2.3.4}}\mathrm {E} _{3}\alpha ^{4p-4}\\&\quad +{\frac {m(m+5p-1)(m+5p-2)(m+5p-3)(m+5p-4)}{2.3.4.5}}\mathrm {E} _{4}\alpha ^{5p-5}\\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\end{aligned}}$