Page:Annales de mathématiques pures et appliquées, 1819-1820, Tome 10.djvu/381

371

DES ÉQUATIONS DIFFÉRENTIELLES.

7.^o Coefficient G.

${\begin{aligned}G&=\Delta ^{6}q--(21+7n)\Delta ^{7}q+\left(322+169n+56n^{2}\right)\Delta ^{8}q\\&-\left(4536+3822n+2016n^{2}+504n^{3}\right)\Delta ^{9}q\\&+\left(63273+16150n+48720n^{2}+22680n^{3}+5040n^{4}\right)\Delta ^{10}q\\&-\left(902055+1104411n+1016400n^{2}+665280n^{3}+277200n^{4}\right.\\&\qquad \left.+55440n^{5}\right)\Delta ^{11}q\\&+\left(13339535+18462906n+20015688n^{2}+16465680n^{3}\right.\\&\qquad \left.+9702000n^{4}+3659040n^{5}+665280n^{6}\right)\Delta ^{12}q\\\end{aligned}}$

8.^o Coefficient H.

${\begin{aligned}H&=\Delta ^{7}q-(28+8n)\Delta ^{8}q+\left(546+288n+72n^{2}\right)\Delta ^{9}q\\&+\left(9450+6960n+3240n^{2}+720n^{3}\right)\Delta ^{10}q\\&+\left(157773+145200n+95040n^{2}+39600n^{3}+7920n^{4}\right)\Delta ^{11}q\\&-\left(2637558+2859384n+2352241n^{2}+1386000n^{3}\right.\\&\qquad \left.+322720n^{4}+95040n^{5}\right)\Delta ^{12}q\\\end{aligned}}$

9.^o Coefficient I.

${\begin{aligned}I&=\Delta ^{8}q-(36+9n)\Delta ^{9}q+\left(870+405n+90n^{2}\right)\Delta ^{10}q\\&-\left(18150+11880n+4950n^{2}+990n^{3}\right)\Delta ^{11}q\\&+\left(357423+294030n+173250n^{2}+65340n^{3}+11880n^{4}\right)\Delta ^{12}q\\\end{aligned}}$

10.^o Coefficient K.

${\begin{aligned}K&=\Delta ^{9}q-(45+10n)\Delta ^{10}q+\left(1320+550n+110n^{2}\right)\Delta ^{11}q\\&\qquad +\left(32670+19250n+7260n^{2}+1320n^{3}\right)\Delta ^{12}q\\\end{aligned}}$