In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) generates the Legendre polynomials. It was independently introduced by Olinde Rodrigues (1816), Sir James Ivory (1824) and Carl Gustav Jacobi (1827). The name "Rodrigues formula" was introduced by Heine in 1878, after Hermite pointed out in 1865 that Rodrigues was the first to discover it. The term is also used to describe similar formulas for other orthogonal polynomials. Askey (2005) describes the history of the Rodrigues formula in detail.
Statement Let ( P n ( x ) ) n = 0 ∞ {\displaystyle (P_{n}(x))_{n=0}^{\infty }} be a sequence of orthogonal polynomials on the interval [ a , b ] {\displaystyle [a,b]} with respect to weight function w ( x ) {\displaystyle w(x)} . That is, they have degrees deg ( P n ) = n {\displaystyle \deg(P_{n})=n} , satisfy the orthogonality condition
∫ a b P m ( x ) P n ( x ) w ( x ) d x = K n δ m , n {\displaystyle \int _{a}^{b}P_{m}(x)P_{n}(x)w(x)\,dx=K_{n}\delta _{m,n}}
where K n {\displaystyle K_{n}} are nonzero constants depending on n {\displaystyle n} , and δ m , n {\displaystyle \delta _{m,n}} is the Kronecker delta. The interval [ a , b ] {\displaystyle [a,b]} may be infinite in one or both ends.
More abstractly, this can be viewed through Sturm–Liouville theory. Define an operator L f := − 1 w ( W f ′ ) ′ {\displaystyle Lf:=-{\frac {1}{w}}(Wf')'} , then the differential equation is equivalent to L P n = λ n P n {\displaystyle LP_{n}=\lambda _{n}P_{n}} . Define the functional space X = L 2 ( [ a , b ] , w ( x ) d x ) {\displaystyle X=L^{2}([a,b],w(x)dx)} as the Hilbert space of functions over [ a , b ] {\displaystyle [a,b]} , such that ⟨ f , g ⟩ := ∫ a b f g w {\displaystyle \langle f,g\rangle :=\int _{a}^{b}fgw} . Then the operator L {\displaystyle L} is self-adjoint on functions satisfying certain boundary conditions, allowing us to apply the spectral theorem.
Generating function A simple argument using Cauchy's integral formula shows that the orthogonal polynomials obtained from the Rodrigues formula have a generating function of the form
G ( x , u ) = ∑ n = 0 ∞ u n P n ( x ) G(x,u)=\sum _{n=0}^{\infty }u^{n}P_{n}(x)
The P n ( x ) {\displaystyle P_{n}(x)} functions here may not have the standard normalizations. But we can write this equivalently as
G ( x , u ) = ∑ n = 0 ∞ u n N n N n P n ( x ) G(x,u)=\sum _{n=0}^{\infty }{\frac {u^{n}}{N_{n}}}N_{n}P_{n}(x)
where the N n {\displaystyle N_{n}} are chosen according to the application so as to give the desired normalizations. The variable u may be replaced by a constant multiple of u so that
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