Page:Annales de mathématiques pures et appliquées, 1819-1820, Tome 10.djvu/47

43

DES FONCTIONS POLYNOMES.

${\begin{array}{rlr}rA_{r,s}=&\Sigma \left\{m\left(a_{m,n}{\frac {\operatorname {d} A_{r+h,s+k}}{\operatorname {d} a_{m+h,n+k}}}+b_{m,n}{\frac {\operatorname {d} A_{r+h,s+k}}{\operatorname {d} b_{m+h,n+k}}}\right)\right\},&(15)\\\\sA_{r,s}=&\Sigma \left\{n\left(a_{m,n}{\frac {\operatorname {d} A_{r+h,s+k}}{\operatorname {d} a_{m+h,n+k}}}+b_{m,n}{\frac {\operatorname {d} A_{r+h,s+k}}{\operatorname {d} b_{m+h,n+k}}}\right)\right\},&(16)\\\\rB_{r}=&\Sigma '\left\{m\left(a'_{m}{\frac {\operatorname {d} B_{r-1}}{\operatorname {d} a_{m-1,0}}}+b'_{m}{\frac {\operatorname {d} B_{r-1}}{\operatorname {d} b_{m-1,0}}}\right)\right\},&(17)\\\\rB_{r}=&\Sigma '\left\{m\left(a'_{m}{\frac {\operatorname {d} B_{r-m}}{\operatorname {d} a_{0,0}}}+b'_{m}{\frac {\operatorname {d} B_{r-m}}{\operatorname {d} b_{0,0}}}\right)\right\},&(18)\\\\rB_{r}=&\Sigma '\left\{m\left(a'_{m}{\frac {\operatorname {d} B_{r+h}}{\operatorname {d} a_{m+h,0}}}+b'_{m}{\frac {\operatorname {d} B_{r+h}}{\operatorname {d} b_{m+h,0}}}\right)\right\}.&(19)\end{array}}$

Dans ces trois dernières formules, le signe $\Sigma '$ ne se rapporte qu’à $m$ ; et nous avons fait, pour abréger,

${\begin{aligned}&B_{r}=A_{r,0}+{\frac {z}{x}}A_{r-1,1}+{\frac {z^{2}}{x^{2}}}A_{r-2,2}+\ldots {\frac {z^{r}}{x^{r}}}A_{0,r},\\\\&a'_{m}=a_{m,0}+{\frac {z}{x}}a_{m-1,1}+{\frac {z^{2}}{x^{2}}}a_{m-2,2}+\ldots {\frac {z^{m}}{x^{m}}}a_{0,m},\\\\&b'_{m}=b_{m,0}+{\frac {z}{x}}b_{m-1,1}+{\frac {z^{2}}{x^{2}}}b_{m-2,2}+\ldots {\frac {z^{m}}{x^{m}}}b_{0,m}.\end{aligned}}$

Quant à la manière d’y parvenir, voici pour cela une méthode que je crois plus simple que l’emploi des formules (9), (10), (11), (12).

Les équations (1), (2) donnent

${\frac {\operatorname {d} u}{\operatorname {d} x}}+{\frac {z}{x}}{\frac {\operatorname {d} u}{\operatorname {d} z}}=\Sigma \left\{(m+n)\left(x^{m-1}z^{n}\right)\left(a_{m,n}{\frac {\operatorname {d} u}{\operatorname {d} y}}+b_{m,n}{\frac {\operatorname {d} u}{\operatorname {d} y'}}\right)\right\}$

ou, en faisant $m+n=\alpha ,$