Continuous dual Hahn polynomials

In mathematics, the continuous dual Hahn polynomials are a family of <a href="/facts/Orthogonal_polynomials/9cyRbDYU">orthogonal polynomials</a> in the <a href="/facts/Askey_scheme/cmELnDGN">Askey scheme</a> of hypergeometric orthogonal polynomials. They are defined in terms of <a href="/facts/Generalized_hypergeometric_function/fqXXDHdR">generalized hypergeometric functions</a> by

S
          
            n
          
        
        (
        
          x
          
            2
          
        
        ;
        a
        ,
        b
        ,
        c
        )
        =

3
          
        
        
          F
          
            2
          
        
        (
        −
        n
        ,
        a
        +
        i
        x
        ,
        a
        −
        i
        x
        ;
        a
        +
        b
        ,
        a
        +
        c
        ;
        1
        )
        .
         
      
    
    {\displaystyle S_{n}(x^{2};a,b,c)={}_{3}F_{2}(-n,a+ix,a-ix;a+b,a+c;1).\ }

Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.
Closely related polynomials include the <a href="/facts/Dual_Hahn_polynomials/HFbftPIm">dual Hahn polynomials</a> Rn(x;γ,δ,N), the <a href="/facts/Continuous_Hahn_polynomials/FU60SwfE">continuous Hahn polynomials</a> pn(x,a,b, a, b), and the <a href="/facts/Hahn_polynomials/M0cpTwkm">Hahn polynomials</a>. These polynomials all have q-analogs with an extra parameter q, such as the <a href="/facts/Q-Hahn_polynomials/JuvbD4Kk">q-Hahn polynomials</a> Qn(x;α,β, N;q), and so on.

Continuous dual Hahn polynomials open-in-new

Continuous dual Hahn polynomials