The Shifted Legendre Polynomials

    The "shifted Legendre polynomials
    
        [Graphics:Images/LegendrePolyMod_gr_346.gif]  are orthogonal on [Graphics:Images/LegendrePolyMod_gr_347.gif],  
    
    where [Graphics:Images/LegendrePolyMod_gr_348.gif] are the Legendre polynomials on [Graphics:Images/LegendrePolyMod_gr_349.gif].   

Exploration.

[Graphics:../Images/LegendrePolyMod_gr_350.gif]

[Graphics:../Images/LegendrePolyMod_gr_351.gif]

 

 

[Graphics:../Images/LegendrePolyMod_gr_352.gif]

[Graphics:../Images/LegendrePolyMod_gr_353.gif]

 

 

For  [Graphics:../Images/LegendrePolyMod_gr_354.gif] we have.  

[Graphics:../Images/LegendrePolyMod_gr_355.gif]

[Graphics:../Images/LegendrePolyMod_gr_356.gif]

[Graphics:../Images/LegendrePolyMod_gr_357.gif]

[Graphics:../Images/LegendrePolyMod_gr_358.gif]

[Graphics:../Images/LegendrePolyMod_gr_359.gif]

 

Therefore, the shifted Legendre polynomials  [Graphics:../Images/LegendrePolyMod_gr_360.gif]  are orthogonal on [Graphics:../Images/LegendrePolyMod_gr_361.gif].   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2005