j’ai
![{\displaystyle {\begin{aligned}&\left[1-q\left(\cos \theta \pm \sin \theta {\sqrt {-1}}\right)\right]^{-\lambda }=\\&\qquad \qquad \qquad \qquad 1+\lambda q\left(\cos \theta \pm \sin \theta {\sqrt {-1}}\right)\\&\qquad \qquad \qquad \qquad +{\frac {\lambda (\lambda +1)}{2}}q^{2}\left(\cos 2\theta \pm \sin 2\theta {\sqrt {-1}}\right)\\&\qquad \qquad \qquad \qquad +{\frac {\lambda (\lambda +1)(\lambda +2)}{2.3}}q^{3}\left(\cos 3\theta \pm \sin 3\theta {\sqrt {-1}}\right)+\ldots .\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19fdbc14712a8eb25b8a8b13f4e010ffa5b69e3e)
Soit, pour abréger,
![{\displaystyle {\begin{aligned}1+&\lambda q\cos \theta +{\frac {\lambda (\lambda +1)}{2}}q^{2}\cos 2\theta +{\frac {\lambda (\lambda +1)(\lambda +2)}{2.3}}q^{3}\cos 3\theta +\ldots =\mathrm {M} ,\\&\lambda q\sin \,\theta +{\frac {\lambda (\lambda +1)}{2}}q^{2}\sin \,2\theta +{\frac {\lambda (\lambda +1)(\lambda +2)}{2.3}}q^{3}\sin \,3\theta +\ldots =\mathrm {N} \,;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7063fb4d662ca9f7a76126f2b9318a8c7cdf218f)
on aura
![{\displaystyle {\begin{aligned}&\left[1-q\left(\cos \theta +\sin \theta {\sqrt {-1}}\right)\right]^{-\lambda }=\mathrm {M+N} {\sqrt {-1}},\\&\left[1-q\left(\cos \theta -\sin \theta {\sqrt {-1}}\right)\right]^{-\lambda }=\mathrm {M-N} {\sqrt {-1}}\,;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be56ef8a6d1781a80980d8a2142fb1b793913587)
donc
![{\displaystyle \left(1-2q\cos \theta +q^{2}\right)^{-\lambda }=\mathrm {\left(\mathrm {M+N} {\sqrt {-1}}\right)\left(\mathrm {M-N} {\sqrt {-1}}\right)=M^{2}+N^{2}} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c04d61ebaad97394d0e0465fd92e84bdda4a0cda)
Or, si l’on fait les carrés des deux séries
et
qu’on ajoute ensemble les termes qui ont le même coefficient, et qu’on remarque que
![{\displaystyle \cos m\theta \times \cos n\theta +\sin m\theta \times \sin \theta =\cos(m-n)\theta ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d41495237ecd3a72bf4608641131c30d7862d59)
et
étant des nombres quelconques, on trouvera
![{\displaystyle \left(1-2q\cos \theta +q^{2}\right)^{-\lambda }=\mathrm {A+B\cos \theta +C\cos 2\theta +D} \cos 3\theta +\ldots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5dfd3c29532119485e04992c40c2aab2d7b04d0)
Et les coefficients
seront exprimés de la manière suivante
![{\displaystyle {\begin{aligned}\mathrm {A} =&1+\lambda ^{2}q^{2}+{\frac {\lambda ^{2}(\lambda +1)^{2}}{2^{2}}}q^{4}+{\frac {\lambda ^{2}(\lambda +1)^{2}(\lambda +2)^{2}}{2^{2}.3^{2}}}q^{6}+\ldots ,\\\mathrm {B} =&2\lambda q+2\lambda {\frac {\lambda (\lambda +1)}{2}}q^{3}+2{\frac {\lambda (\lambda +1)}{2}}{\frac {\lambda (\lambda +1)(\lambda +2)}{2.3}}q^{5}+\ldots ,\\&+2{\frac {\lambda (\lambda +1)(\lambda +2)}{2.3}}{\frac {\lambda (\lambda +1)(\lambda +2)(\lambda +3)}{2.3.4}}q^{7}+\ldots ,\\\mathrm {C} =&2{\frac {\lambda (\lambda +1)}{2}}q^{2}+2\lambda {\frac {\lambda (\lambda +1)(\lambda +2)}{2.3}}q^{4}+\ldots \,;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b03d1f8bd9ab0e97eca1751720a328d464b76f9)
et ainsi de suite.