The GetInfo(P, subject) command returns information about the hypergeometric polynomial P that depends on the value of subject.

hypergeom: hypergeometric functional equation satisfied by P and the normalization coefficient of the Rodrigues formula.

recurrence: three-term recurrence for P.

derivative: derivative of P.

structural: structural relation(s). If the optional equation root=val is specified, GetInfo returns the partial structural relation with respect to val. This is available for only continuous hypergeometric polynomials.

derivative_representation: derivative representation for P. If the optional equation root=val is specified, GetInfo returns the partial derivative representation with respect to val. This is available for only continuous hypergeometric polynomials.

with(OrthogonalSeries) :

GetInfo(LaguerreL(n,1,x),derivative_representation);

$\mathrm{LaguerreL}\left(n\,1\,x\right)=\mathrm{LaguerreL}\left(n\,2\,x\right)-\mathrm{LaguerreL}\left(n-1\,2\,x\right)$

GetInfo(LaguerreL(n,1,x),hypergeom);

$x\left(\frac{{\ⅆ}^2}{\ⅆx^2}\mathrm{LaguerreL}\left(n\,1\,x\right)\right)+\left(-x+2\right)\left(\frac{\ⅆ}{\ⅆx}\mathrm{LaguerreL}\left(n\,1\,x\right)\right)+n\mathrm{LaguerreL}\left(n\,1\,x\right)=0,\mathrm{\_B}\left(n\right)=\frac{1}{n!}$

GetInfo(LaguerreL(n,1,x),recurrence);

$x\mathrm{LaguerreL}\left(n\,1\,x\right)=\left(2+2n\right)\mathrm{LaguerreL}\left(n\,1\,x\right)+\left(-1-n\right)\mathrm{LaguerreL}\left(n-1\,1\,x\right)+\left(-1-n\right)\mathrm{LaguerreL}\left(n+1\,1\,x\right)$

GetInfo(LaguerreL(n,1,x),structural);

$x\left(\frac{\ⅆ}{\ⅆx}\mathrm{LaguerreL}\left(n\,1\,x\right)\right)=n\mathrm{LaguerreL}\left(n\,1\,x\right)+\left(-1-n\right)\mathrm{LaguerreL}\left(n-1\,1\,x\right)$

GetInfo(JacobiP(n,alpha,beta,x),structural,root=-1);

GetInfo(HermiteH(n,x),hypergeom);

$\frac{{\ⅆ}^2}{\ⅆx^2}\mathrm{HermiteH}\left(n\,x\right)-2x\left(\frac{\ⅆ}{\ⅆx}\mathrm{HermiteH}\left(n\,x\right)\right)+2n\mathrm{HermiteH}\left(n\,x\right)=0,\mathrm{\_B}\left(n\right)={\left(\mathrm{-1}\right)}^n$

GetInfo(HermiteH(n,x),structural);

$\frac{\ⅆ}{\ⅆx}\mathrm{HermiteH}\left(n\,x\right)=2n\mathrm{HermiteH}\left(n-1\,x\right)$

GetInfo(HermiteH(n,x),derivative);

$\frac{\ⅆ}{\ⅆx}\mathrm{HermiteH}\left(n\,x\right)=2n\mathrm{HermiteH}\left(n-1\,x\right)$

GetInfo(ChebyshevT(n,x),structural);

GetInfo(ChebyshevT(n,x),derivative);

$\frac{\ⅆ}{\ⅆx}\mathrm{ChebyshevT}\left(n\,x\right)=n\mathrm{ChebyshevU}\left(n-1\,x\right)$