Editing Classical orthogonal polynomials (section)

=== Rodrigues' formula  ===

{{main article|Rodrigues' formula}}
Under the assumptions of the preceding section,
''P''<sub>''n''</sub>(''x'') is proportional to <math>\frac{1}{W(x)} \  \frac{d^n}{dx^n}\left(W(x)[Q(x)]^n\right).</math>

This is known as [[Rodrigues' formula]], after [[Olinde Rodrigues]].  It is often written

:<math>P_n(x) = \frac{1}{{e_n}W(x)} \  \frac{d^n}{dx^n}\left(W(x)[Q(x)]^n\right)</math>

where the numbers ''e''<sub>''n''</sub> depend on the standardization.  The standard values of ''e''<sub>''n''</sub> will be given in the tables below.

===The numbers ''λ''<sub>''n''</sub>===
Under the assumptions of the preceding section, we have

:<math>\lambda_n = - n \left( \frac{n-1}{2} Q'' + L' \right).</math>

(Since ''Q'' is quadratic and ''L'' is linear, <math>Q''</math> and <math>L'</math> are constants, so these are just numbers.)

=== Second form for the differential equation ===
Let

:<math>R(x) = e^{\int \frac{L(x)}{Q(x)}\,dx}.</math>

Then

:<math>(Ry')' = R\,y'' + R'\,y' = R\,y'' + \frac{R\,L}{Q}\,y'.</math>

Now multiply the differential equation

:<math>Q\,y'' + L\,y' + \lambda y = 0</math>

by ''R''/''Q'', getting

:<math>R\,y'' + \frac{R\,L}{Q}\,y' + \frac{R\,\lambda}{Q}\,y = 0</math>

or

:<math>(Ry')' + \frac{R\,\lambda}{Q}\,y = 0.</math>

This is the standard Sturm–Liouville form for the equation.

===Third form for the differential equation===
Let <math>S(x) = \sqrt{R(x)} = e^{\int \frac{L(x)}{2\,Q(x)}\,dx}.</math>

Then

:<math>S' = \frac{S\,L}{2\,Q}.</math>

Now multiply the differential equation

:<math>Q\,y'' + L \,y' + \lambda y = 0</math>

by ''S''/''Q'', getting

:<math>S\,y'' + \frac{S\,L} Q \,y' + \frac{S\,\lambda} Q \,y = 0</math>

or

:<math>S\,y'' + 2\,S'\,y' + \frac{S\,\lambda} Q \,y = 0</math>

But <math>(S\,y)'' = S\,y'' + 2\,S'\,y' + S''\,y</math>, so

:<math>(S\,y)'' + \left(\frac{S\,\lambda} Q - S''\right)\,y = 0,</math>

or, letting ''u'' = ''Sy'',

:<math>u'' + \left(\frac \lambda Q - \frac{S''} S \right)\,u = 0.</math>

=== Formulas involving derivatives ===
Under the assumptions of the preceding section, let ''P''{{su|b=''n''|p=[''r'']}} denote the ''r''-th derivative of ''P''<sub>''n''</sub>.
(We put the "r" in brackets to avoid confusion with an exponent.)
''P''{{su|b=''n''|p=[''r'']}} is a polynomial of degree ''n''&nbsp;&minus;&nbsp;''r''.  Then we have the following:

* (orthogonality) For fixed r, the polynomial sequence ''P''{{su|b=''r''|p=[''r'']}}, ''P''{{su|b=''r'' + 1|p=[''r'']}}, ''P''{{su|b=''r'' + 2|p=[''r'']}}, ... are orthogonal, weighted by <math>WQ^r</math>.
* (generalized [[Olinde Rodrigues|Rodrigues']] formula) ''P''{{su|b=''n''|p=[''r'']}} is proportional to <math>\frac{1}{W(x)[Q(x)]^r} \  \frac{d^{n-r}}{dx^{n-r}}\left(W(x)[Q(x)]^n\right).</math>
* (differential equation) ''P''{{su|b=''n''|p=[''r'']}} is a solution of <math>{Q}\,y'' + (rQ'+L)\,y' + [\lambda_n-\lambda_r]\,y = 0</math>, where &lambda;<sub>''r''</sub> is the same function as &lambda;<sub>''n''</sub>, that is, <math>\lambda_r = - r \left( \frac{r-1}{2} Q'' + L' \right)</math>
* (differential equation, second form) ''P''{{su|b=''n''|p=[''r'']}} is a solution of <math>(RQ^{r}y')' + [\lambda_n-\lambda_r]RQ^{r-1}\,y = 0</math>

There are also some mixed recurrences.  In each of these, the numbers ''a'', ''b'', and ''c'' depend on ''n''
and ''r'', and are unrelated in the various formulas.

* <math>P_n^{[r]} = aP_{n+1}^{[r+1]} + bP_n^{[r+1]} + cP_{n-1}^{[r+1]}</math>
* <math>P_n^{[r]} = (ax+b)P_n^{[r+1]} + cP_{n-1}^{[r+1]}</math>
* <math>QP_n^{[r+1]} = (ax+b)P_n^{[r]} + cP_{n-1}^{[r]}</math>

There are an enormous number of other formulas involving orthogonal polynomials
in various ways.  Here is a tiny sample of them, relating to the Chebyshev,
associated Laguerre, and Hermite polynomials:

* <math>2\,T_{m}(x)\,T_{n}(x) = T_{m+n}(x) + T_{m-n}(x)</math>
* <math>H_{2n}(x) = (-4)^{n}\,n!\,L_{n}^{(-1/2)}(x^2)</math>
* <math>H_{2n+1}(x) = 2(-4)^{n}\,n!\,x\,L_{n}^{(1/2)}(x^2)</math>