Editing Rodrigues' formula (section)

=== Legendre ===
Rodrigues stated his formula for Legendre polynomials <math>P_n</math>:
<math display="block">P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} \!\left[ (x^2 -1)^n \right]\!.</math><math display="block">(1 - x^2) P_n''(x) - 2 x P_n'(x) + n (n + 1) P_n(x) = 0</math>For Legendre polynomials, the generating function is defined as <math display="”block”">G(x,u)=\sum_{n=0}^\infty u^nP_n(x)</math>.

The contour integral gives the '''Schläfli integral'''<ref>{{Citation |last=Schläfli |first=Ludwig |title=Über die zwei Heineschen Kugelfunktionen mit beliebigem Parameter und ihre ausnahmslose Darstellung durch bestimmte Integrale |date=1881 |work=Gesammelte Mathematische Abhandlungen |pages=317–392 |url=https://doi.org/10.1007/978-3-0348-4116-0_27 |place=Basel |publisher=Springer Basel |isbn=978-3-0348-4044-6}}</ref> for Legendre polynomials:<math display="block">P_n(x) = \frac{1}{2\pi i 2^n} \oint_C \frac{(t^2-1)^n}{(t-x)^{n+1}} dt</math> Summing up the integrand,<math display="block">G(x,u) = \frac{1}{\sqrt{1 - 2ux + u^2}} \frac{1}{2\pi i} \oint_C \left(\frac{1}{t - t_-} - \frac{1}{t - t_+}\right) dt</math>where <math>t_\pm = \frac{1}{u} (1 \pm \sqrt{1 - 2ux + u^2})</math>. For small <math>u</math>, we have <math>t_- \approx x, t_+ \to \infty</math>, which heuristically suggests that the integral should be the residue around <math>t_-</math>, thus giving<math display="block">G(x,u) = \frac{1}{\sqrt{1 - 2ux + u^2}}</math>