Legendre rational functions

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Legendre rational functions are a sequence of <a href="/facts/Orthogonal_functions/2MmN826Q">orthogonal functions</a> on [0, ∞). They are obtained by composing the <a href="/facts/Cayley_transform/2jbbGdA9">Cayley transform</a> with <a href="/facts/Legendre_polynomials/nDvV4em0">Legendre polynomials</a>.
A rational Legendre function of degree n is defined as:

R
          
            n
          
        
        (
        x
        )
        =
        
          
            
              2
            
            
              x
              +
              1
            
          
        
        
        
          P
          
            n
          
        
        
          (
          
            
              
                x
                −
                1
              
              
                x
                +
                1
              
            
          
          )
        
      
    
    {\displaystyle R_{n}(x)={\frac {\sqrt {2}}{x+1}}\,P_{n}\left({\frac {x-1}{x+1}}\right)}

where 
 
 
 
 
 P
 
 n
 
 
 (
 x
 )
 
 
 {\displaystyle P_{n}(x)}
 
 is a Legendre polynomial. These functions are <a href="/facts/Eigenfunction/62EJS0XB">eigenfunctions</a> of the singular <a href="/facts/Sturm%25E2%2580%2593Liouville_problem/xwvX8d9d">Sturm–Liouville problem</a>:

(
        x
        +
        1
        )
        
          
            d
            
              d
              x
            
          
        
        
          (
          
            x
            
              
                d
                
                  d
                  x
                
              
            
            
              [
              
                
                  (
                  
                    x
                    +
                    1
                  
                  )
                
                v
                (
                x
                )
              
              ]
            
          
          )
        
        +
        λ
        v
        (
        x
        )
        =
        0
      
    
    {\displaystyle (x+1){\frac {d}{dx}}\left(x{\frac {d}{dx}}\left[\left(x+1\right)v(x)\right]\right)+\lambda v(x)=0}

with eigenvalues

λ
          
            n
          
        
        =
        n
        (
        n
        +
        1
        )
        
      
    
    {\displaystyle \lambda _{n}=n(n+1)\,}

Legendre rational functions open-in-new

Legendre rational functions