Neumann polynomial

In mathematics, the Neumann polynomials, introduced by <a href="/facts/Carl_Neumann/1MfDa0mo">Carl Neumann</a> for the special case 
 
 
 
 α
 =
 0
 
 
 {\displaystyle \alpha =0}
 
, are a sequence of polynomials in 
 
 
 
 1
 
 /
 
 t
 
 
 {\displaystyle 1/t}
 
 used to expand functions in term of <a href="/facts/Bessel_function/hSyvFPqC">Bessel functions</a>.
The first few polynomials are

O
          
            0
          
          
            (
            α
            )
          
        
        (
        t
        )
        =
        
          
            1
            t
          
        
        ,
      
    
    {\displaystyle O_{0}^{(\alpha )}(t)={\frac {1}{t}},}

O
          
            1
          
          
            (
            α
            )
          
        
        (
        t
        )
        =
        2
        
          
            
              α
              +
              1
            
            
              t
              
                2
              
            
          
        
        ,
      
    
    {\displaystyle O_{1}^{(\alpha )}(t)=2{\frac {\alpha +1}{t^{2}}},}

O
          
            2
          
          
            (
            α
            )
          
        
        (
        t
        )
        =
        
          
            
              2
              +
              α
            
            t
          
        
        +
        4
        
          
            
              (
              2
              +
              α
              )
              (
              1
              +
              α
              )
            
            
              t
              
                3
              
            
          
        
        ,
      
    
    {\displaystyle O_{2}^{(\alpha )}(t)={\frac {2+\alpha }{t}}+4{\frac {(2+\alpha )(1+\alpha )}{t^{3}}},}

O
          
            3
          
          
            (
            α
            )
          
        
        (
        t
        )
        =
        2
        
          
            
              (
              1
              +
              α
              )
              (
              3
              +
              α
              )
            
            
              t
              
                2
              
            
          
        
        +
        8
        
          
            
              (
              1
              +
              α
              )
              (
              2
              +
              α
              )
              (
              3
              +
              α
              )
            
            
              t
              
                4
              
            
          
        
        ,
      
    
    {\displaystyle O_{3}^{(\alpha )}(t)=2{\frac {(1+\alpha )(3+\alpha )}{t^{2}}}+8{\frac {(1+\alpha )(2+\alpha )(3+\alpha )}{t^{4}}},}

O
          
            4
          
          
            (
            α
            )
          
        
        (
        t
        )
        =
        
          
            
              (
              1
              +
              α
              )
              (
              4
              +
              α
              )
            
            
              2
              t
            
          
        
        +
        4
        
          
            
              (
              1
              +
              α
              )
              (
              2
              +
              α
              )
              (
              4
              +
              α
              )
            
            
              t
              
                3
              
            
          
        
        +
        16
        
          
            
              (
              1
              +
              α
              )
              (
              2
              +
              α
              )
              (
              3
              +
              α
              )
              (
              4
              +
              α
              )
            
            
              t
              
                5
              
            
          
        
        .
      
    
    {\displaystyle O_{4}^{(\alpha )}(t)={\frac {(1+\alpha )(4+\alpha )}{2t}}+4{\frac {(1+\alpha )(2+\alpha )(4+\alpha )}{t^{3}}}+16{\frac {(1+\alpha )(2+\alpha )(3+\alpha )(4+\alpha )}{t^{5}}}.}

A general form for the polynomial is

O
          
            n
          
          
            (
            α
            )
          
        
        (
        t
        )
        =
        
          
            
              α
              +
              n
            
            
              2
              α
            
          
        
        
          ∑
          
            k
            =
            0
          
          
            ⌊
            n
            
              /
            
            2
            ⌋
          
        
        (
        −
        1
        
          )
          
            n
            −
            k
          
        
        
          
            
              (
              n
              −
              k
              )
              !
            
            
              k
              !
            
          
        
        
          
            
              (
            
            
              
                −
                α
              
              
                n
                −
                k
              
            
            
              )
            
          
        
        
          
            (
            
              
                2
                t
              
            
            )
          
          
            n
            +
            1
            −
            2
            k
          
        
        ,
      
    
    {\displaystyle O_{n}^{(\alpha )}(t)={\frac {\alpha +n}{2\alpha }}\sum _{k=0}^{\lfloor n/2\rfloor }(-1)^{n-k}{\frac {(n-k)!}{k!}}{-\alpha  \choose n-k}\left({\frac {2}{t}}\right)^{n+1-2k},}

and they have the "generating function"

(
                
                  
                    z
                    2
                  
                
                )
              
              
                α
              
            
            
              Γ
              (
              α
              +
              1
              )
            
          
        
        
          
            1
            
              t
              −
              z
            
          
        
        =
        
          ∑
          
            n
            =
            0
          
        
        
          O
          
            n
          
          
            (
            α
            )
          
        
        (
        t
        )
        
          J
          
            α
            +
            n
          
        
        (
        z
        )
        ,
      
    
    {\displaystyle {\frac {\left({\frac {z}{2}}\right)^{\alpha }}{\Gamma (\alpha +1)}}{\frac {1}{t-z}}=\sum _{n=0}O_{n}^{(\alpha )}(t)J_{\alpha +n}(z),}

where J are <a href="/facts/Bessel_function/hSyvFPqC">Bessel functions</a>.
To expand a function f in the form

f
        (
        z
        )
        =
        
          
            (
            
              
                2
                z
              
            
            )
          
          
            α
          
        
        
          ∑
          
            n
            =
            0
          
        
        
          a
          
            n
          
        
        
          J
          
            α
            +
            n
          
        
        (
        z
        )
        
      
    
    {\displaystyle f(z)=\left({\frac {2}{z}}\right)^{\alpha }\sum _{n=0}a_{n}J_{\alpha +n}(z)\,}

for 
 
 
 
 
 |
 
 t
 
 |
 
 <
 c
 
 
 {\displaystyle |t|<c}
 
, compute

a
          
            n
          
        
        =
        
          
            
              Γ
              (
              α
              +
              1
              )
            
            
              2
              π
              i
            
          
        
        
          ∮
          
            
              |
            
            t
            
              |
            
            =
            
              c
              ′
            
          
        
        f
        (
        t
        )
        
          O
          
            n
          
          
            (
            α
            )
          
        
        (
        t
        )
        
        d
        t
        ,
      
    
    {\displaystyle a_{n}={\frac {\Gamma (\alpha +1)}{2\pi i}}\oint _{|t|=c'}f(t)O_{n}^{(\alpha )}(t)\,dt,}

where 
 
 
 
 
 c
 ′
 
 <
 c
 
 
 {\displaystyle c'<c}
 
 and c is the distance of the nearest singularity of f(z) from 
 
 
 
 z
 =
 0
 
 
 {\displaystyle z=0}
 
.

Neumann polynomial open-in-new

Neumann polynomial