qu’en développant la quantité
![{\displaystyle {\begin{aligned}c\mathrm {Z} _{i-n+1}^{(0)}&+\ldots \\&+\ldots \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e25ec5c603dc06d2644acf7af54c698daf4547a)
on ait
![{\displaystyle \mathrm {M} _{2}+\mathrm {N} _{2}z+\ldots +{\frac {1}{t_{1}}}\left(\mathrm {M} _{2}^{(1)}+\mathrm {N} _{2}^{(1)}z+\ldots \right)+\ldots +\ldots +{\frac {1}{t_{1}^{i-2}}}\mathrm {M} _{2}^{(i-2)},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/461eba96af2fcf922abb80ad1609318789498429)
et ainsi de suite ; on aura
![{\displaystyle {\frac {1}{t^{i}}}=\mathrm {M} +\mathrm {N} z+\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0d720a07fa68878fc17cb321719bcceed688446)
![{\displaystyle {\begin{aligned}&+{\frac {1}{t_{1}}}\left(\mathrm {M} ^{(1)}+\mathrm {N} ^{(1)}z+\ldots \right)+{\frac {1}{t_{1}^{2}}}\left(\mathrm {M} ^{(2)}+\mathrm {N} ^{(2)}z+\ldots \right)+\ldots +{\frac {1}{t_{1}^{i}}}\mathrm {M} ^{(i)}\\&+{\frac {1}{t}}\left[\mathrm {M} _{1}+\mathrm {N} _{1}z+\ldots +{\frac {1}{t_{1}}}\left(\mathrm {M} _{1}^{(1)}+\mathrm {N} _{1}^{(1)}z+\ldots \right)\right.\\&\qquad \qquad \qquad \qquad \qquad \left.+{\frac {1}{t_{1}^{2}}}\left(\mathrm {M} _{1}^{(2)}+\mathrm {N} _{1}^{(2)}z+\ldots \right)+\ldots +{\frac {1}{t_{1}^{i-1}}}\mathrm {M} _{1}^{({i-1})}\right]\\&+{\frac {1}{t^{2}}}\left[\mathrm {M} _{2}+\mathrm {N} _{2}z+\ldots +{\frac {1}{t_{1}}}\left(\mathrm {M} _{2}^{(1)}+\mathrm {N} _{2}^{(1)}z+\ldots \right)+\ldots +{\frac {1}{t_{1}^{i-2}}}\mathrm {M} _{2}^{({i-2})}\right]\\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&+{\frac {1}{t^{n-1}}}\left[\mathrm {M} _{n-1}+\mathrm {N} _{n-1}z+\ldots +{\frac {1}{t_{1}}}\left(\mathrm {M} _{n-1}^{(1)}+\mathrm {N} _{n-1}^{(1)}z+\ldots \right)+\ldots \right.\\&\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left.+{\frac {1}{t_{1}^{i-n+1}}}\mathrm {M} _{n-1}^{({i-n+1})}\right].\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfb56afb3d63e54918ad898415609b3761409120)
Cela posé, si l’on nomme
la quantité
![{\displaystyle {\begin{aligned}\mathrm {A} y_{x,x_{1}}+\mathrm {B} \,\ y_{x+1,x_{1}}+\mathrm {C} \,\ y_{x+2,x_{1}}\quad &+\ldots \\+\mathrm {B} _{1}y_{x,x_{1}+1}+\mathrm {C} _{1}y_{x+1,x_{1}+1}&+\ldots \\+\mathrm {C} _{2}y_{x,x_{1}+2}\quad &+\ldots \\&+\ldots ,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb986d8334c974b02fbda05b80555915473e7af9)
le coefficient de
dans le développement de la quantité
sera, par l’article XIII,
l’équation précédente donnera conséquemment, en la multipliant par
et en passant des fonctions génératrices aux variables correspondantes,
![{\displaystyle {\begin{aligned}y_{x+i,x_{1}}=&\ \quad \mathrm {M} \quad y_{x,x_{1}}\quad +\mathrm {N} \quad \nabla y_{x,x_{1}}\ \ \ +\ldots \\&+\mathrm {M} ^{(1)}y_{x,x_{1}+1}+\mathrm {N} ^{(1)}\nabla y_{x,x_{1}+1}+\ldots \\&+\mathrm {M} ^{(2)}y_{x,x_{1}+2}+\mathrm {N} ^{(2)}\nabla y_{x,x_{1}+2}+\ldots \\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&+\mathrm {M} ^{(i)}y_{x,x_{1}+i}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3caac3414d044a77b8459ba3217c42c7d540b7f9)