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About Rodrigues' formula Also known as '''Rodrigues' formula'''
About Rodrigues' formula In mathematics, Rodrigues' formula generates the Legendre polynomials. It was independently introduced by Olinde Rodrigues, Sir James Ivory and Carl Gustav Jacobi. 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.
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In mathematics, Rodrigues' formula generates the Legendre polynomials. It was independently introduced by Olinde Rodrigues, Sir James Ivory and Carl Gustav Jacobi. 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.
Article 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.
Contents 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
Examples These formulae
are for the classical orthogonal polynomials. Similar formulae hold for many other sequences of orthogonal functions arising from Sturm–Liouville equations, and these are also called the Rodrigues formula (or Rodrigues' type formula), especially when the resulting sequence is polynomial.
Legendre Source:
Rodrigues stated his formula for Legendre polynomials
P
n
{\displaystyle P_{n}}
:
P
n
(
x
)
=
1
2
n
n
!
d
n
d
x
n
[
(
x
2
−
1
)
n
]
.
{\displaystyle P_{n}(x)={\frac {1}{2^{n}n!}}{\frac {d^{n}}{dx^{n}}}\!\left[(x^{2}-1)^{n}\right]\!.}
(
1
Hermite Source:
Physicist's Hermite polynomials:
H
n
(
x
)
=
(
−
1
)
n
e
x
2
d
n
d
x
n
[
e
−
x
2
]
=
(
2
x
−
d
d
x
)
n
⋅
1.
Laguerre Source:
For associated Laguerre polynomials
L
n
(
α
)
(
x
)
=
x
−
α
e
x
n
!
d
n
d
x
n
(
e
−
x
x
n
+
α
)
=
x
−
α
n
!
Jacobi Source:
P
n
(
α
,
β
)
(
x
)
=
(
−
1
)
n
2
n
n
!
(
1
−
x
)
−
α
(
1
+
x
)
−
β
d
n
d
x
Article from Wikipedia (CC BY-SA 4.0), where it is maintained by volunteer editors.
G(x,u)=\sum _{n=0}^{\infty }{\frac {u^{n}}{N_{n}}}N_{n}P_{n}(x)
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
G(x,\alpha u)=\sum _{n=0}^{\infty }{\frac {\alpha ^{n}u^{n}}{N_{n}}}N_{n}P_{n}(x)
This gives an alternate form of the generating function.
By Cauchy's integral formula, Rodrigues’ formula is equivalent to
{\displaystyle P_{n}(x)={\frac {n!}{2\pi i}}{\frac {c_{n}}{w(x)}}\oint _{C}{\frac {B^{n}(t)w(t)}{(t-x)^{n+1}}}\,dt}
where the integral is along a counterclockwise closed loop around
Then the complex path integral takes the form
P_{n}(x)={\frac {n!}{2\pi i}}c_{n}\oint _{C}{\frac {G(x,u)}{u^{n+1}}}\,du
G(x,u)={\frac {w(t){\frac {dt}{du}}}{w(x)B(t)}}
where now the closed path C encircles the origin. In the equation for
is an implicit function of
in the power series given earlier gives
{\displaystyle {\frac {1}{2\pi i}}\oint _{C}{\frac {G(x,u)}{u^{n+1}}}\,du={\frac {1}{2\pi i}}\oint _{C}{\frac {\sum _{m=0}^{\infty }u^{m}P_{m}(x)}{u^{n+1}}}\,du=P_{n}(x)}
term has a nonzero residue, which is
{\displaystyle n!\,c_{n}}
coefficient was dropped since normalizations are conventions which can be inserted afterwards as discussed earlier.
By expressing t in terms of u in the general formula just given for
may be found. As a simple example, let
(Hermite polynomials) so that
{\displaystyle w(x)=\exp \left(-{\frac {x^{2}}{2}}\right)}
{\displaystyle w(t)=\exp \left(-{\frac {(u+x)^{2}}{2}}\right)}
{\displaystyle G(x,u)=\exp \left(-xu-{\frac {u^{2}}{2}}\right)}
{\displaystyle (1-x^{2})P_{n}''(x)-2xP_{n}'(x)+n(n+1)P_{n}(x)=0}
For Legendre polynomials, the generating function is defined as
G(x,u)=\sum _{n=0}^{\infty }u^{n}P_{n}(x)
The contour integral gives the Schläfli integral for Legendre polynomials:
{\displaystyle P_{n}(x)={\frac {1}{2\pi i2^{n}}}\oint _{C}{\frac {(t^{2}-1)^{n}}{(t-x)^{n+1}}}dt}
{\displaystyle 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}
{\displaystyle t_{\pm }={\frac {1}{u}}(1\pm {\sqrt {1-2ux+u^{2}}})}
{\displaystyle t_{-}\approx x,t_{+}\to \infty }
, which heuristically suggests that the integral should be the residue around
{\displaystyle G(x,u)={\frac {1}{\sqrt {1-2ux+u^{2}}}}}
{\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {d^{n}}{dx^{n}}}\!\left[e^{-x^{2}}\right]=\left(2x-{\frac {d}{dx}}\right)^{n}\cdot 1.}
{\displaystyle H_{n}''-2xH_{n}'+2nH_{n}=0}
The generating function is defined as
{\displaystyle G(x,u)=\sum _{n=0}^{\infty }{\frac {H_{n}(x)}{n!}}\,u^{n}.}
The contour integral gives
{\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac {n!}{2\pi i}}\oint _{C}{\frac {e^{-t^{2}}}{(t-x)^{n+1}}}\,dt.}
{\displaystyle {\begin{aligned}G(x,u)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}e^{x^{2}}}{n!}}{\frac {n!}{2\pi i}}\,u^{n}\oint _{C}{\frac {e^{-t^{2}}}{(t-x)^{n+1}}}\,dt\\&=e^{x^{2}}{\frac {1}{2\pi i}}\oint _{C}e^{-t^{2}}\left(\sum _{n=0}^{\infty }{\frac {(-1)^{n}u^{n}}{(t-x)^{n+1}}}\right)dt\\&=e^{x^{2}}{\frac {1}{2\pi i}}\oint _{C}e^{-t^{2}}{\frac {1}{t-x+u}}\\&=e^{x^{2}}\,e^{-(x-u)^{2}}\\&=e^{2xu-u^{2}}\end{aligned}}}
{\displaystyle L_{n}^{(\alpha )}(x)={x^{-\alpha }e^{x} \over n!}{d^{n} \over dx^{n}}\left(e^{-x}x^{n+\alpha }\right)={\frac {x^{-\alpha }}{n!}}\left({\frac {d}{dx}}-1\right)^{n}x^{n+\alpha }.}
{\displaystyle xL_{n}^{(\alpha )}(x)''+(\alpha +1-x)L_{n}^{(\alpha )}(x)'+nL_{n}^{(\alpha )}(x)=0~.}
The generating function is defined as
{\displaystyle G(x,u):=\sum _{n=0}^{\infty }u^{n}L_{n}^{(\alpha )}(x)}
By the same method, we have
{\displaystyle G(x,u)={\frac {1}{(1-u)^{\alpha +1}}}e^{-{\frac {ux}{1-u}}}}
{\displaystyle P_{n}^{(\alpha ,\beta )}(x)={\frac {(-1)^{n}}{2^{n}n!}}(1-x)^{-\alpha }(1+x)^{-\beta }{\frac {d^{n}}{dx^{n}}}\left\{(1-x)^{\alpha }(1+x)^{\beta }\left(1-x^{2}\right)^{n}\right\}.}
{\displaystyle \left(1-x^{2}\right)P_{n}^{(\alpha ,\beta )}{}''+(\beta -\alpha -(\alpha +\beta +2)x)P_{n}^{(\alpha ,\beta )}{}'+n(n+\alpha +\beta +1)P_{n}^{(\alpha ,\beta )}=0.}
{\displaystyle \sum _{n=0}^{\infty }P_{n}^{(\alpha ,\beta )}(x)u^{n}=2^{\alpha +\beta }R^{-1}(1-u+R)^{-\alpha }(1+u+R)^{-\beta },}
{\textstyle R={\sqrt {1-2ux+u^{2}}}}
, and the branch of square root is chosen so that
