Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 8.djvu/179

Or on a

$\sideset {_{1}}{_{n}}{\mathrm {N} }={\frac {q^{n-1}}{r^{n-1}}}\,;$

donc

$\sideset {_{2}}{_{n}}{\mathrm {N} }={\frac {p}{r}}{\frac {q^{n-1}}{r^{n-1}}}+{\frac {q}{r}}\sideset {_{2}}{_{n-1}}{\mathrm {N} },$

partant

$\sideset {_{2}}{_{n}}{\mathrm {N} }={\frac {q^{n-1}}{r^{n-1}}}{\frac {p}{r}}(n+\mathrm {C} )\,;$

or, posant $n=1,\ \sideset {_{2}}{_{1}}{\mathrm {N} }={\frac {p}{r}}\,;$ donc

$\mathrm {C} =0.$

Ensuite

$\sideset {_{3}}{_{n}}{\mathrm {N} }={\frac {p^{2}}{r^{2}}}{\frac {q^{n-1}}{r^{n-1}}}n+{\frac {q}{r}}\sideset {_{3}}{_{n-1}}{\mathrm {N} }\,;$

donc

$\sideset {_{3}}{_{n}}{\mathrm {N} }={\frac {q^{n-1}}{r^{n-1}}}\left[{\frac {p^{2}}{r^{2}}}{\frac {n(n+1)}{1.2}}+\mathrm {C} )\right]\,;$

or, posant $n=1,\ \sideset {_{3}}{_{1}}{\mathrm {N} }={\frac {p^{2}}{r^{2}}}\,;$ donc

$\mathrm {C} =0,$

et généralement

$\sideset {_{m}}{_{n}}{\mathrm {N} }={\frac {p^{m-1}q^{n-1}}{r^{m+n-2}}}{\frac {n(n+1)\ldots (n+m-2)}{1.2.3\ldots (m-1)}}.$

On a ensuite

$\sideset {_{1}}{_{n}}{\mathrm {M} }={\frac {1-q}{r}}{\frac {q^{n-2}}{r^{n-2}}}\,;$

donc

$\sideset {_{2}}{_{n}}{\mathrm {M} }={\frac {q}{r}}\sideset {_{2}}{_{n-1}}{\mathrm {M} }+{\frac {p}{r}}{\frac {1-q}{r}}{\frac {q^{n-2}}{r^{n-2}}}\,;$

partant,

$\sideset {_{2}}{_{n}}{\mathrm {M} }={\frac {q^{n-2}}{r^{n-2}}}{\frac {p}{r}}{\frac {1-q}{r}}(n-1)+\mathrm {C} {\frac {q^{n-1}}{r^{n-1}}}\,;$

or, posant $n=1,\ \sideset {_{2}}{_{n}}{\mathrm {M} }={\frac {1-p}{r}}\,;$ donc

et

$\sideset {_{2}}{_{n}}{\mathrm {M} }={\frac {q^{n-2}}{r^{n-2}}}{\frac {p}{r}}{\frac {1-q}{r}}(n-1)+{\frac {q^{n-1}}{r^{n-1}}}{\frac {1-p}{r}}.$