cette formule, les valeurs de
lorsque celles de
seront connues.
Nommons, pour abréger,
la fonction
Si l’on différentie, par rapport à
l’équation
![{\displaystyle \lambda ^{-s}={\frac {1}{2}}b_{s}^{(0)}+b_{s}^{(1)}\cos \theta +b_{s}^{(2)}\cos 2\theta +\ldots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95f8ce3758a7d31e2f0d0f71ce9bf6140f24015f)
on aura
![{\displaystyle -2s(\alpha -\cos \theta )\lambda ^{-s-1}={\frac {1}{2}}{\frac {db_{s}^{(0)}}{d\alpha }}+{\frac {db_{s}^{(1)}}{d\alpha }}\cos \theta +{\frac {db_{s}^{(2)}}{d\alpha }}\cos 2\theta +\ldots \,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2175c70938b609a9688be8cc51f074296100e28)
mais on a
![{\displaystyle -\alpha +\cos \theta ={\frac {1-\alpha ^{2}-\lambda }{2\alpha }}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dd2f02abef2a6e54149678948d113e166422846)
on aura donc
![{\displaystyle {\frac {s\left(1-\alpha ^{2}\right)}{\alpha }}\lambda ^{-s-1}-{\frac {s\lambda ^{-s}}{\alpha }}={\frac {1}{2}}{\frac {db_{s}^{(i)}}{d\alpha }}={\frac {s\left(1-\alpha ^{2}\right)}{\alpha }}b_{s+1}^{(i)}-{\frac {db_{s}^{(i)}}{d\alpha }}\cos \theta +\ldots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fe2234e2c724e8b487aadfcde76ce5812043ab5)
d’où l’on tire généralement
![{\displaystyle {\frac {db_{s}^{(i)}}{d\alpha }}={\frac {s\left(1-\alpha ^{2}\right)}{\alpha }}b_{s+1}^{(i)}-{\frac {sb_{s}^{(i)}}{d\alpha }}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/350dd7da876142b1190b67601fa766b728d7124b)
En substituant au lieu de
sa valeur donnée par la formule (b), on aura
![{\displaystyle {\frac {db_{s}^{(i)}}{d\alpha }}={\frac {i+(i+2s)\alpha ^{2}}{\alpha \left(1-\alpha ^{2}\right)}}b_{s}^{(i)}-{\frac {2(i-s+1)}{1-\alpha ^{2}}}b_{s}^{(i+1)}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7b8593157cc8fbaef69c63ea1c48efd185fe23c)
Si l’on différentie cette équation, on aura
![{\displaystyle {\frac {d^{2}b_{s}^{(i)}}{d\alpha ^{2}}}={\frac {i+(i+2s)\alpha ^{2}}{\alpha \left(1-\alpha ^{2}\right)}}{\frac {db_{s}^{(i)}}{d\alpha }}+\left({\frac {2(i+s)\left(1+\alpha ^{2}\right)}{1-\alpha ^{2}}}-{\frac {i}{\alpha ^{2}}}\right)b_{s}^{(i)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/185ff1a2355903895ee34ae94281956c5986a847)
![{\displaystyle -{\frac {2(i-s+1)}{1-\alpha ^{2}}}{\frac {db_{s}^{(i+1)}}{d\alpha }}-{\frac {4(i-s+1)\alpha }{\left(1-\alpha ^{2}\right)^{2}}}b_{s}^{(i+1)}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/461800fee56b67d17b4724e18096a8a290096b80)
En différentiant encore, on aura
![{\displaystyle {\frac {d^{3}b_{s}^{(i)}}{d\alpha ^{3}}}={\frac {i+(i+2s)\alpha ^{2}}{\alpha \left(1-\alpha ^{2}\right)}}{\frac {d^{2}b_{s}^{(i)}}{d\alpha ^{2}}}+2\left({\frac {2(i+s)\left(1+\alpha ^{2}\right)}{\left(1-\alpha ^{2}\right)^{2}}}-{\frac {i}{\alpha ^{2}}}\right){\frac {db_{s}^{(i)}}{d\alpha }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d34cb4c8407c6a7fe751ac9556b7203acc62f4f)
![{\displaystyle +\left({\frac {4(i+s)\alpha \left(3+\alpha ^{2}\right)}{\left(1-\alpha ^{2}\right)^{3}}}-{\frac {2i}{\alpha ^{3}}}\right)b_{s}^{(i)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26188c99b3dd442a38b3c940c661eabd344203ac)
![{\displaystyle -{\frac {2(i-s+1)}{1-\alpha ^{2}}}{\frac {d^{2}b_{s}^{(i+1)}}{d\alpha ^{2}}}-{\frac {8(i-s+1)\alpha }{\left(1-\alpha ^{2}\right)^{2}}}{\frac {db_{s}^{(i+1)}}{d\alpha }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17fbbc5488055554daa8301f46fff1da8c0b0c71)
![{\displaystyle -{\frac {4(i-s+1)\left(1+3\alpha ^{2}\right)}{\left(1-\alpha ^{2}\right)^{3}}}b_{s}^{(i+1)}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91f7240b5c9f262152842d7b3c158574aba8e23c)