Charlier polynomials

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, Charlier polynomials (also called Poisson–Charlier polynomials) are a family of <a href="/facts/Orthogonal_polynomials/9cyRbDYU">orthogonal polynomials</a> introduced by <a href="/facts/Carl_Charlier/pPfP9nzx">Carl Charlier</a>.
They are given in terms of the <a href="/facts/Generalized_hypergeometric_function/fqXXDHdR">generalized hypergeometric function</a> by

C
          
            n
          
        
        (
        x
        ;
        μ
        )
        =

2
          
        
        
          F
          
            0
          
        
        (
        −
        n
        ,
        −
        x
        ;
        −
        ;
        −
        1
        
          /
        
        μ
        )
        =
        (
        −
        1
        
          )
          
            n
          
        
        n
        !
        
          L
          
            n
          
          
            (
            −
            1
            −
            x
            )
          
        
        
          (
          
            −
            
              
                1
                μ
              
            
          
          )
        
        ,
      
    
    {\displaystyle C_{n}(x;\mu )={}_{2}F_{0}(-n,-x;-;-1/\mu )=(-1)^{n}n!L_{n}^{(-1-x)}\left(-{\frac {1}{\mu }}\right),}

where 
 
 
 
 L
 
 
 {\displaystyle L}
 
 are <a href="/facts/Laguerre_polynomials/7RM3ypWl">generalized Laguerre polynomials</a>. They satisfy the <a href="/facts/Orthogonality_relation/WMp5Q6YE">orthogonality relation</a>

∑
          
            x
            =
            0
          
          
            ∞
          
        
        
          
            
              μ
              
                x
              
            
            
              x
              !
            
          
        
        
          C
          
            n
          
        
        (
        x
        ;
        μ
        )
        
          C
          
            m
          
        
        (
        x
        ;
        μ
        )
        =
        
          μ
          
            −
            n
          
        
        
          e
          
            μ
          
        
        n
        !
        
          δ
          
            n
            m
          
        
        ,
        
        μ
        >
        0.
      
    
    {\displaystyle \sum _{x=0}^{\infty }{\frac {\mu ^{x}}{x!}}C_{n}(x;\mu )C_{m}(x;\mu )=\mu ^{-n}e^{\mu }n!\delta _{nm},\quad \mu >0.}

They form a <a href="/facts/Sheffer_sequence/CpjBBWcM">Sheffer sequence</a> related to the <a href="/facts/Poisson_process/8uphjAUc">Poisson process</a>, similar to how <a href="/facts/Hermite_polynomials/MLQNY0ez">Hermite polynomials</a> relate to the <a href="/facts/Brownian_motion/Th0g0wkr">Brownian motion</a>.

Charlier polynomials open-in-new

Charlier polynomials