Editing Rodrigues' formula (section)

== 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
 
<math display="”block”">G(x,u)=\sum_{n=0}^\infty u^nP_n(x)</math> 

The <math>P_n(x)</math> functions here may not have the standard normalizations. But we can write this equivalently as
 
<math display="”block”">G(x,u)=\sum_{n=0}^\infty \frac{u^n}{N_n}N_nP_n(x)</math> 

where the <math>N_n</math> 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

<math display="”block”">G(x,\alpha u)=\sum_{n=0}^\infty \frac{\alpha^n u^n}{N_n}N_nP_n(x)</math> 

This gives an alternate form of the generating function.

By [[Cauchy's integral formula]], Rodrigues’ formula is equivalent to<math display="block">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</math>where the integral is along a counterclockwise closed loop around <math>x</math>. Let

<math display=”block”>u=\frac{t-x}{B(t)}</math>

Then the complex path integral takes the form

<math display=”block”>P_n(x)=\frac{n!}{2\pi i}c_n\oint_C \frac{G(x,u)}{u^{n+1}}\,du</math>

<math display=”block”>G(x,u)=\frac{w(t)\frac{dt}{du}}{w(x)B(t)}</math>

where now the closed path C encircles the origin.  In the equation for <math>G(x,u)</math>, <math>t</math> is an implicit function of <math>u</math>. Expanding <math>G(x,u)</math> in the power series given earlier gives

<math>\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^mP_m(x)}{u^{n+1}}\,du=P_n(x)</math>

Only the <math>m=n</math> term has a nonzero residue, which is <math>P_n(x)</math>.  The <math>n!\,c_n</math> 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 <math>G(x,u)</math>, explicit formulas for <math>G(x,u)</math> may be found.  As a simple example, let <math>B(x)=1</math> and <math>A(x)=-x</math> (Hermite polynomials) so that <math>w(x)=\exp\left(-\frac{x^2}{2}\right)</math>, <math>t=u+x</math>, <math>w(t)=\exp\left(-\frac{(u+x)^2}{2}\right)</math> and so <math>G(x,u)=\exp\left(-xu-\frac{u^2}{2}\right)</math>.