Page:Annales de mathématiques pures et appliquées, 1819-1820, Tome 10.djvu/377

 Les corrections sont expliquées en page de discussion

367

DES ÉQUATIONS DIFFÉRENTIELLES.

2.^o Coefficient B.

${\begin{aligned}B=\Delta q&-(1+2n)\Delta ^{2}q+n\left(2+6n+6n^{2}\right)\Delta ^{3}q-\left(6+22n+36n^{2}+24n^{3}\right)\Delta ^{4}q\\&+\left(24+100n+210n^{2}+240n^{3}+120n^{4}\right)\Delta ^{5}q\\&-\left(120+548n+1350n^{2}+2040n^{3}+1800n^{4}+720n^{5}\right)\Delta ^{6}q\\&+\left(720+3528n+9744n^{2}+17640n^{3}+21000n^{4}+15120n^{5}+5040n^{6}\right)\Delta ^{7}q\\&-\left(5040+26136n+78792n^{2}+162456n^{3}+2325200n^{4}\right.\\&\qquad \left.+231840n^{5}+741120n^{6}+40320n^{7}\right)\Delta ^{8}q\\&+\left(40320+219168n+708744n^{2}+161481n^{3}+2693880n^{4}\right.\\&\qquad \left.+3265920n^{5}+2731840n^{6}+1451520n^{7}+362880n^{8}\right)\Delta ^{9}q\\&-\left(362880+2053152n+7036200n^{2}+17368320n^{3}+32319000n^{4}\right.\\&\qquad +45554560n^{5}+47628000n^{6}+35078400n^{7}\\&\qquad \left.+16329600n^{8}+3628800n^{9}\right)\Delta ^{10}q\\&+\left(3628800+21257280n+76521456n^{2}+201828000n^{3}\right.\\&\qquad +410031600n^{4}+649479600n^{5}+795175920n^{6}\\&\qquad +731808000n^{7}+479001600n^{8}+199584000n^{9}\\&\qquad \left.+39916800n^{10}\right)\Delta ^{11}q\\&-\left(39916800+241087680n+905507856n^{2}+2526193824n^{3}\right.\\&\qquad +5519487600n^{4}+9604465200n^{5}+13293292320n^{6}\\&\qquad +14411295360n^{7}+11855289600n^{8}+6985440000n^{9}\\&\qquad \left.+2634508800n^{10}+479001600n^{11}\right)\Delta ^{12}q\\\end{aligned}}$