Rodrigues' formula

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. 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. Rodrigues' type formula—If w ( x ) = W ( x ) / B ( x ) , W ′ ( x ) W ( x ) = A ( x ) B ( x ) , {\displaystyle w(x)=W(x)/B(x),\quad {\frac {W'(x)}{W(x)}}={\frac {A(x)}{B(x)}},} where A ( x ) {\displaystyle A(x)} is a polynomial with degree at most 1 and B ( x ) {\displaystyle B(x)} is a polynomial with degree at most 2, and lim x → a x k W ( x ) = 0 , lim x → b x k W ( x ) = 0. {\displaystyle \lim _{x\to a}x^{k}W(x)=0,\qquad \lim _{x\to b}x^{k}W(x)=0.} for any k = 0 , 1 , 2 , … {\displaystyle k=0,1,2,\dots } . Then, if d n d x n [ B ( x ) n w ( x ) ] ≠ 0 {\displaystyle {\frac {d^{n}}{dx^{n}}}\!\left[B(x)^{n}w(x)\right]\neq 0} for all n = 0 , 1 , 2 , … {\displaystyle n=0,1,2,\dots } , then P n ( x ) = c n w ( x ) d n d x n [ B ( x ) n w ( x ) ] , {\displaystyle P_{n}(x)={\frac {c_{n}}{w(x)}}{\frac {d^{n}}{dx^{n}}}\!\left[B(x)^{n}w(x)\right],} for some constants c n {\displaystyle c_{n}} .