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OrthogonalSeries

 GetInfo
 return information about hypergeometric orthogonal polynomials

 Calling Sequence GetInfo(P, subject, optional_arg)

Parameters

 P - hypergeometric polynomial subject - literal name; one of recurrence, structural, hypergeom, derivative, and derivative_representation optional_arg - (optional) equation of the form root=val where val is an expression

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{OrthogonalSeries}\right):$
 > $\mathrm{GetInfo}\left(\mathrm{LaguerreL}\left(n,1,x\right),\mathrm{derivative_representation}\right)$
 ${\mathrm{LaguerreL}}{}\left({n}{,}{1}{,}{x}\right){=}{\mathrm{LaguerreL}}{}\left({n}{,}{2}{,}{x}\right){-}{\mathrm{LaguerreL}}{}\left({n}{-}{1}{,}{2}{,}{x}\right)$ (1)
 > $\mathrm{GetInfo}\left(\mathrm{LaguerreL}\left(n,1,x\right),\mathrm{hypergeom}\right)$
 ${x}{}\left(\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}{}{\mathrm{LaguerreL}}{}\left({n}{,}{1}{,}{x}\right)\right){+}\left({-}{x}{+}{2}\right){}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{LaguerreL}}{}\left({n}{,}{1}{,}{x}\right)\right){+}{n}{}{\mathrm{LaguerreL}}{}\left({n}{,}{1}{,}{x}\right){=}{0}{,}{\mathrm{_B}}{}\left({n}\right){=}\frac{{1}}{{n}{!}}$ (2)
 > $\mathrm{GetInfo}\left(\mathrm{LaguerreL}\left(n,1,x\right),\mathrm{recurrence}\right)$
 ${x}{}{\mathrm{LaguerreL}}{}\left({n}{,}{1}{,}{x}\right){=}\left({2}{+}{2}{}{n}\right){}{\mathrm{LaguerreL}}{}\left({n}{,}{1}{,}{x}\right){+}\left({-}{1}{-}{n}\right){}{\mathrm{LaguerreL}}{}\left({n}{-}{1}{,}{1}{,}{x}\right){+}\left({-}{1}{-}{n}\right){}{\mathrm{LaguerreL}}{}\left({n}{+}{1}{,}{1}{,}{x}\right)$ (3)
 > $\mathrm{GetInfo}\left(\mathrm{LaguerreL}\left(n,1,x\right),\mathrm{structural}\right)$
 ${x}{}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{LaguerreL}}{}\left({n}{,}{1}{,}{x}\right)\right){=}{n}{}{\mathrm{LaguerreL}}{}\left({n}{,}{1}{,}{x}\right){+}\left({-}{1}{-}{n}\right){}{\mathrm{LaguerreL}}{}\left({n}{-}{1}{,}{1}{,}{x}\right)$ (4)
 > $\mathrm{GetInfo}\left(\mathrm{JacobiP}\left(n,\mathrm{α},\mathrm{β},x\right),\mathrm{structural},\mathrm{root}=-1\right)$
 $\left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{JacobiP}}{}\left({n}{,}{\mathrm{α}}{,}{\mathrm{β}}{,}{x}\right)\right){=}\frac{{2}{}\left({\mathrm{α}}{+}{\mathrm{β}}{+}{1}{+}{n}\right){}{n}{}\left({\mathrm{α}}{-}{\mathrm{β}}\right){}{\mathrm{JacobiP}}{}\left({n}{,}{\mathrm{α}}{,}{\mathrm{β}}{,}{x}\right)}{\left({2}{}{n}{+}{\mathrm{α}}{+}{\mathrm{β}}{+}{2}\right){}\left({2}{}{n}{+}{\mathrm{α}}{+}{\mathrm{β}}\right)}{+}\frac{{2}{}\left({\mathrm{α}}{+}{\mathrm{β}}{+}{1}{+}{n}\right){}\left({n}{+}{\mathrm{β}}\right){}\left({n}{+}{\mathrm{α}}\right){}{\mathrm{JacobiP}}{}\left({n}{-}{1}{,}{\mathrm{α}}{,}{\mathrm{β}}{,}{x}\right)}{\left({2}{}{n}{+}{\mathrm{α}}{+}{\mathrm{β}}\right){}\left({2}{}{n}{+}{1}{+}{\mathrm{α}}{+}{\mathrm{β}}\right)}{-}\frac{{2}{}\left({\mathrm{α}}{+}{\mathrm{β}}{+}{1}{+}{n}\right){}\left({n}{+}{1}\right){}{n}{}{\mathrm{JacobiP}}{}\left({n}{+}{1}{,}{\mathrm{α}}{,}{\mathrm{β}}{,}{x}\right)}{\left({2}{}{n}{+}{\mathrm{α}}{+}{\mathrm{β}}{+}{2}\right){}\left({2}{}{n}{+}{1}{+}{\mathrm{α}}{+}{\mathrm{β}}\right)}$ (5)
 > $\mathrm{GetInfo}\left(\mathrm{HermiteH}\left(n,x\right),\mathrm{hypergeom}\right)$
 $\frac{{{\partial }}^{{2}}}{{\partial }{{x}}^{{2}}}{}{\mathrm{HermiteH}}{}\left({n}{,}{x}\right){-}{2}{}{x}{}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{HermiteH}}{}\left({n}{,}{x}\right)\right){+}{2}{}{n}{}{\mathrm{HermiteH}}{}\left({n}{,}{x}\right){=}{0}{,}{\mathrm{_B}}{}\left({n}\right){=}{\left({-}{1}\right)}^{{n}}$ (6)
 > $\mathrm{GetInfo}\left(\mathrm{HermiteH}\left(n,x\right),\mathrm{structural}\right)$
 $\frac{{\partial }}{{\partial }{x}}{}{\mathrm{HermiteH}}{}\left({n}{,}{x}\right){=}{2}{}{n}{}{\mathrm{HermiteH}}{}\left({n}{-}{1}{,}{x}\right)$ (7)
 > $\mathrm{GetInfo}\left(\mathrm{HermiteH}\left(n,x\right),\mathrm{derivative}\right)$
 $\frac{{\partial }}{{\partial }{x}}{}{\mathrm{HermiteH}}{}\left({n}{,}{x}\right){=}{2}{}{n}{}{\mathrm{HermiteH}}{}\left({n}{-}{1}{,}{x}\right)$ (8)
 > $\mathrm{GetInfo}\left(\mathrm{ChebyshevT}\left(n,x\right),\mathrm{structural}\right)$
 $\left({-}{{x}}^{{2}}{+}{1}\right){}\left(\frac{{\partial }}{{\partial }{x}}{}{\mathrm{ChebyshevT}}{}\left({n}{,}{x}\right)\right){=}\frac{{1}}{{2}}{}{n}{}{\mathrm{ChebyshevT}}{}\left({n}{-}{1}{,}{x}\right){-}\frac{{1}}{{2}}{}{n}{}{\mathrm{ChebyshevT}}{}\left({n}{+}{1}{,}{x}\right)$ (9)
 > $\mathrm{GetInfo}\left(\mathrm{ChebyshevT}\left(n,x\right),\mathrm{derivative}\right)$
 $\frac{{\partial }}{{\partial }{x}}{}{\mathrm{ChebyshevT}}{}\left({n}{,}{x}\right){=}{n}{}{\mathrm{ChebyshevU}}{}\left({n}{-}{1}{,}{x}\right)$ (10)