Bessel polynomials

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Bessel polynomials are an <a href="/facts/Orthogonal_polynomials/9cyRbDYU">orthogonal</a> sequence of <a href="/facts/Polynomial/Lzak8VVx">polynomials</a>. There are a number of different but closely related definitions. The definition favored by mathematicians is given by the series: 101

y
          
            n
          
        
        (
        x
        )
        =
        
          ∑
          
            k
            =
            0
          
          
            n
          
        
        
          
            
              (
              n
              +
              k
              )
              !
            
            
              (
              n
              −
              k
              )
              !
              k
              !
            
          
        
        
        
          
            (
            
              
                x
                2
              
            
            )
          
          
            k
          
        
        .
      
    
    {\displaystyle y_{n}(x)=\sum _{k=0}^{n}{\frac {(n+k)!}{(n-k)!k!}}\,\left({\frac {x}{2}}\right)^{k}.}

Another definition, favored by electrical engineers, is sometimes known as the reverse Bessel polynomials: 8 : 15

θ
          
            n
          
        
        (
        x
        )
        =
        
          x
          
            n
          
        
        
        
          y
          
            n
          
        
        (
        1
        
          /
        
        x
        )
        =
        
          ∑
          
            k
            =
            0
          
          
            n
          
        
        
          
            
              (
              n
              +
              k
              )
              !
            
            
              (
              n
              −
              k
              )
              !
              k
              !
            
          
        
        
        
          
            
              x
              
                n
                −
                k
              
            
            
              2
              
                k
              
            
          
        
        .
      
    
    {\displaystyle \theta _{n}(x)=x^{n}\,y_{n}(1/x)=\sum _{k=0}^{n}{\frac {(n+k)!}{(n-k)!k!}}\,{\frac {x^{n-k}}{2^{k}}}.}

The coefficients of the second definition are the same as the first but in reverse order. For example, the third-degree Bessel polynomial is

y
          
            3
          
        
        (
        x
        )
        =
        15
        
          x
          
            3
          
        
        +
        15
        
          x
          
            2
          
        
        +
        6
        x
        +
        1
      
    
    {\displaystyle y_{3}(x)=15x^{3}+15x^{2}+6x+1}

while the third-degree reverse Bessel polynomial is

θ
          
            3
          
        
        (
        x
        )
        =
        
          x
          
            3
          
        
        +
        6
        
          x
          
            2
          
        
        +
        15
        x
        +
        15.
      
    
    {\displaystyle \theta _{3}(x)=x^{3}+6x^{2}+15x+15.}

The reverse Bessel polynomial is used in the design of <a href="/facts/Bessel_filter/1DYEo6ds">Bessel electronic filters</a>.

Bessel polynomials open-in-new

Bessel polynomials