Page:Annales de mathématiques pures et appliquées, 1819-1820, Tome 10.djvu/386

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INTÉGRATION APPROCHÉE

${\begin{array}{lr}A=&q-2\Delta q+10\Delta ^{2}q-76\Delta ^{3}q+772\Delta ^{4}q-9808\Delta ^{5}q\qquad \qquad \qquad \\&+149552\Delta ^{6}q-2660544\Delta ^{7}q\\B=&\Delta q-5\Delta ^{2}q+38\Delta ^{3}q-386\Delta ^{4}q+4904\Delta ^{5}q\qquad \qquad \qquad \qquad \\&-74776\Delta ^{6}q+1330272\Delta ^{7}q\\C=&\Delta ^{2}q-9\Delta ^{3}q+95\Delta ^{4}q-1220\Delta ^{5}q+18664\Delta ^{6}q-332388\Delta ^{7}q\\D=&\Delta ^{3}q-14\Delta ^{4}q+195\Delta ^{5}q-3065\Delta ^{6}q+55104\Delta ^{7}q\\E=&\Delta ^{4}q-20\Delta ^{5}q+355\Delta ^{6}q-6685\Delta ^{7}q\\F=&\Delta ^{5}q-27\Delta ^{6}q+595\Delta ^{7}q\\G=&\Delta ^{6}q-35\Delta ^{7}q\\H=&\Delta ^{7}q\\\end{array}}$

d’où on conclura

${\begin{aligned}y=&q+(x-2)\Delta q+\left(x^{2}-5x+10\right)\Delta ^{2}q+\left(x^{3}-9x^{2}+38x-76\right)\Delta ^{3}q\\&+\left(x^{4}-14x^{3}+95x^{2}-386x+772\right)\Delta ^{4}q\\&+\left(x^{5}-20x^{4}+195x^{3}-1220x^{2}+4904x-9808\right)\Delta ^{5}q\\&+\left(x^{6}-27x^{5}+355x^{4}-3065x^{3}+18664x^{2}-74776x+149652\right)\Delta ^{6}q\\&+\left(x^{7}-35x^{6}+595x^{5}-6685x^{4}+55104x^{3}-332388x^{2}+1330272x\right.\\&\qquad \left.-2660544\right)\Delta ^{7}q+\ldots \end{aligned}}$

On aura ainsi

${\begin{array}{ll}{\text{Pour }}x=&3\ldots y=q+\Delta q+4\Delta ^{2}q-16\Delta ^{3}q\\{\text{Pour }}x=&4\ldots y=q+2\Delta q+6\Delta ^{2}q-4\Delta ^{3}q+108\Delta ^{4}q\\{\text{Pour }}x=&5\ldots y=q+3\Delta q+10\Delta ^{2}q+14\Delta ^{3}q+92\Delta ^{4}q-788\Delta ^{5}q\\{\text{Pour }}x=&6\ldots y=q+4\Delta q+16\Delta ^{2}q+44\Delta ^{3}q+148\Delta ^{4}q-328\Delta ^{5}q\\&\qquad \qquad \qquad +7644\Delta ^{6}q\\{\text{Pour }}x=&7\ldots y=q+5\Delta q+24\Delta ^{2}q+92\Delta ^{3}q+324\Delta ^{4}q-6588\Delta ^{5}q\\&\qquad \qquad \qquad -233324\Delta ^{6}q-79672\Delta ^{7}\\\end{array}}$

et ainsi de suite.