Christoffel-Darboux Identity

Author

Eric W. Weisstein
April 18, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/Christoffel-DarbouxIdentity.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Christoffel-DarbouxIdentity.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Check

l=Partition[{
ChebyshevT,Pi 2^(DiscreteDelta[#]-1)&,
ChebyshevU,Pi/2&,
HermiteH,2^# #! Sqrt[Pi]&,
LaguerreL,1&,
LegendreP,2/(2#+1)&
},2];

Transpose[Simplify[Table[Function[{ϕ,γ},Subtract@@{Sum[((ϕ[k,x]ϕ[k,y])/(γ[k])),{k,0,m}],((ϕ[m+1,x]ϕ[m,y]-ϕ[m,x]ϕ[m+1,y])/(((Coefficient[ϕ[m+1,x],x,m+1])/(Coefficient[ϕ[m,x],x,m]))γ[m](x-y)))}]@@@l,{m,0,5}]]]//TableForm