Editing Classical orthogonal polynomials (section)

== Differential equation ==

The classical orthogonal polynomials arise from a differential equation of the form

:<math> Q(x) \, f'' +  L(x)\,f' + \lambda  f = 0 </math>

where ''Q'' is a given quadratic (at most) polynomial, and ''L'' is a given linear polynomial.  The function ''f'', and the constant ''λ'', are to be found.

:(Note that it makes sense for such an equation to have a polynomial solution.
:Each term in the equation is a polynomial, and the degrees are consistent.)

This is a [[Sturm–Liouville theory|Sturm–Liouville]] type of equation.  Such equations generally have singularities in their solution functions f except for particular values of ''λ''.  They can be thought of as [[eigenvalue|eigenvector/eigenvalue]] problems:  Letting ''D'' be the [[differential operator]], <math>D(f) = Q f'' + L f'</math>, and changing the sign of ''λ'', the problem is to find the eigenvectors (eigenfunctions) f, and the
corresponding eigenvalues ''λ'', such that f does not have singularities and ''D''(''f'') = ''λf''.

The solutions of this differential equation have singularities unless ''λ'' takes on
specific values.  There is a series of numbers ''&lambda;''<sub>0</sub>, ''&lambda;''<sub>1</sub>, ''&lambda;''<sub>2</sub>, ... that led to a series of polynomial solutions ''P''<sub>0</sub>, ''P''<sub>1</sub>, ''P''<sub>2</sub>, ... if one of the following sets of conditions are met:

# ''Q'' is actually quadratic, ''L'' is linear, ''Q'' has two distinct real roots, the root of ''L'' lies strictly between the roots of ''Q'', and the leading terms of ''Q'' and ''L'' have the same sign.
# ''Q'' is not actually quadratic, but is linear, ''L'' is linear, the roots of ''Q'' and ''L'' are different, and the leading terms of ''Q'' and ''L'' have the same sign if the root of ''L'' is less than the root of ''Q'', or vice versa.
# ''Q'' is just a nonzero constant, ''L'' is linear, and the leading term of ''L'' has the opposite sign of ''Q''.

These three cases lead to the '''Jacobi-like''', '''Laguerre-like''', and '''Hermite-like''' polynomials, respectively.

In each of these three cases, we have the following:

* The solutions are a series of polynomials ''P''<sub>0</sub>, ''P''<sub>1</sub>, ''P''<sub>2</sub>, ..., each ''P''<sub>''n''</sub> having degree ''n'', and corresponding to a number &lambda;<sub>''n''</sub>.
* The interval of orthogonality is bounded by whatever roots ''Q'' has.
* The root of ''L'' is inside the interval of orthogonality.
* Letting <math>R(x) = e^{\int \frac{L(x)}{Q(x)}\,dx}</math>, the polynomials are orthogonal under the weight function <math>W(x) =\frac{R(x)}{Q(x)}</math>
* ''W''(''x'') has no zeros or infinities inside the interval, though it may have zeros or infinities at the end points.
* ''W''(''x'') gives a finite inner product to any polynomials.
* ''W''(''x'') can be made to be greater than 0 in the interval.  (Negate the entire differential equation if necessary so that ''Q''(''x'') > 0 inside the interval.)

Because of the constant of integration, the quantity ''R''(''x'') is determined only up to an arbitrary positive multiplicative constant.  It will be used only in homogeneous differential equations
(where this doesn't matter) and in the definition of the weight function (which can also be
indeterminate.)  The tables below will give the "official" values of ''R''(''x'') and ''W''(''x'').

=== 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>

=== Orthogonality ===

The differential equation for a particular ''λ'' may be written (omitting explicit dependence on x)

:<math>Q\ddot{f}_n+L\dot{f}_n+\lambda_nf_n=0</math>

multiplying by <math>(R/Q)f_m</math> yields

:<math>Rf_m\ddot{f}_n+\frac{R}{Q}Lf_m\dot{f}_n+\frac{R}{Q}\lambda_nf_mf_n=0</math>

and reversing the subscripts yields

:<math>Rf_n\ddot{f}_m+\frac{R}{Q}Lf_n\dot{f}_m+\frac{R}{Q}\lambda_mf_nf_m=0</math>

subtracting and integrating:

:<math>
\int_a^b \left[R(f_m\ddot{f}_n-f_n\ddot{f}_m)+
\frac{R}{Q}L(f_m\dot{f}_n-f_n\dot{f}_m)\right] \, dx
+(\lambda_n-\lambda_m)\int_a^b \frac{R}{Q}f_mf_n \, dx = 0
</math>

but it can be seen that

:<math>
\frac{d}{dx}\left[R(f_m\dot{f}_n-f_n\dot{f}_m)\right]=
R(f_m\ddot{f}_n-f_n\ddot{f}_m)\,\,+\,\,R\frac{L}{Q}(f_m\dot{f}_n-f_n\dot{f}_m)
</math>

so that:

:<math>\left[R(f_m\dot{f}_n-f_n\dot{f}_m)\right]_a^b\,\,+\,\,(\lambda_n-\lambda_m)\int_a^b \frac{R}{Q}f_mf_n \, dx=0</math>

If the polynomials ''f'' are such that the term on the left is zero, and <math>\lambda_m \ne \lambda_n</math> for <math>m \ne n</math>, then the orthogonality relationship will hold:

:<math>\int_a^b \frac{R}{Q}f_mf_n \, dx=0</math>

for <math>m \ne n</math>.