Page:Annales de mathématiques pures et appliquées, 1819-1820, Tome 10.djvu/379

 Les corrections sont expliquées en page de discussion

369

DES ÉQUATIONS DIFFÉRENTIELLES.

4.^o Coefficient D.

${\begin{aligned}D&=\Delta ^{3}q-(6+4n)\Delta ^{4}q+\left(35+40n+20n^{2}\right)\Delta ^{5}q\\&\qquad -\left(225+340n+300n^{2}+120n^{3}\right)\Delta ^{6}q\\&+\left(1624+2940n+3500n^{2}+2520n^{3}+840n^{4}\right)\Delta ^{7}q\\&-\left(13132+27076n+39200n^{2}+38640n^{2}+23520n^{4}+6720n^{5}\right)\Delta ^{8}q\\&+\left(118124+269136n+448980n^{2}+544320n^{3}+458640n^{4}\right.\\&\qquad \left.+241920n^{5}+60480n^{6}\right)\Delta ^{9}q\\&-\left(1172700+2894720n+5386500n^{2}+7592760n^{3}\right.\\&\qquad \left.+7938000n^{4}+5846400n^{5}+2721600n^{6}+604800n^{7}\right)\Delta ^{10}q\\&+\left(12753576+33638000n+683386040n^{2}+108246600n^{3}\right.\\&\qquad +132529320n^{4}+121968000n^{5}+79833600n^{6}\\&\qquad \left.+33264000n^{7}+6652800n^{8}\right)\Delta ^{11}q\\&-\left(150917976+421032304n+919914600n^{2}\right.\\&\qquad +1600744200n^{3}+2215548720n^{4}\\&\qquad +2401882560n^{5}+1975881600n^{6}\\&\qquad +1164240000n^{7}+439084800n^{8}\\&\qquad \left.+79833600n^{9}\right)\Delta ^{12}q\\\end{aligned}}$

5.^o Coefficient E.

${\begin{aligned}E&=\Delta ^{4}q-(10+5n)\Delta ^{5}q+\left(86+75n+30n^{2}\right)\Delta ^{6}q\\&-\left(735+875n+630n^{2}+210n^{3}\right)\Delta ^{7}q\\&+\left(6769+9800n+9660n^{2}+5880n^{3}+1680n^{4}\right)\Delta ^{8}q\\&-\left(67284+112245n+136080n^{2}+114660n^{3}+6048n^{4}\right.\\&\qquad \left.+13720n^{5}\right)\Delta ^{9}q\\\end{aligned}}$