Continuous Hahn polynomials

<h2 id="orthogonality">Orthogonality</h2>
The continuous Hahn polynomials pn(x;a,b,c,d) are orthogonal with respect to the weight function

w
        (
        x
        )
        =
        Γ
        (
        a
        +
        i
        x
        )
        
        Γ
        (
        b
        +
        i
        x
        )
        
        Γ
        (
        c
        −
        i
        x
        )
        
        Γ
        (
        d
        −
        i
        x
        )
        .
      
    
    {\displaystyle w(x)=\Gamma (a+ix)\,\Gamma (b+ix)\,\Gamma (c-ix)\,\Gamma (d-ix).}

In particular, they satisfy the orthogonality relation<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a><a class="footnote-ref" id="fnref:2" href="#fn:2">2</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

1
                    
                      2
                      π
                    
                  
                
                
                  ∫
                  
                    −
                    ∞
                  
                  
                    ∞
                  
                
                Γ
                (
                a
                +
                i
                x
                )
                
                Γ
                (
                b
                +
                i
                x
                )
                
                Γ
                (
                c
                −
                i
                x
                )
                
                Γ
                (
                d
                −
                i
                x
                )
                
                
                  p
                  
                    m
                  
                
                (
                x
                ;
                a
                ,
                b
                ,
                c
                ,
                d
                )
                
                
                  p
                  
                    n
                  
                
                (
                x
                ;
                a
                ,
                b
                ,
                c
                ,
                d
                )
                
                d
                x
              
            
            
              
              
                
                
                
                =
                
                  
                    
                      Γ
                      (
                      n
                      +
                      a
                      +
                      c
                      )
                      
                      Γ
                      (
                      n
                      +
                      a
                      +
                      d
                      )
                      
                      Γ
                      (
                      n
                      +
                      b
                      +
                      c
                      )
                      
                      Γ
                      (
                      n
                      +
                      b
                      +
                      d
                      )
                    
                    
                      n
                      !
                      (
                      2
                      n
                      +
                      a
                      +
                      b
                      +
                      c
                      +
                      d
                      −
                      1
                      )
                      
                      Γ
                      (
                      n
                      +
                      a
                      +
                      b
                      +
                      c
                      +
                      d
                      −
                      1
                      )
                    
                  
                
                
                
                  δ
                  
                    n
                    m
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}&{\frac {1}{2\pi }}\int _{-\infty }^{\infty }\Gamma (a+ix)\,\Gamma (b+ix)\,\Gamma (c-ix)\,\Gamma (d-ix)\,p_{m}(x;a,b,c,d)\,p_{n}(x;a,b,c,d)\,dx\\&\qquad \qquad ={\frac {\Gamma (n+a+c)\,\Gamma (n+a+d)\,\Gamma (n+b+c)\,\Gamma (n+b+d)}{n!(2n+a+b+c+d-1)\,\Gamma (n+a+b+c+d-1)}}\,\delta _{nm}\end{aligned}}}

for 
 
 
 
 ℜ
 (
 a
 )
 >
 0
 
 
 {\displaystyle \Re (a)>0}
 
, 
 
 
 
 ℜ
 (
 b
 )
 >
 0
 
 
 {\displaystyle \Re (b)>0}
 
, 
 
 
 
 ℜ
 (
 c
 )
 >
 0
 
 
 {\displaystyle \Re (c)>0}
 
, 
 
 
 
 ℜ
 (
 d
 )
 >
 0
 
 
 {\displaystyle \Re (d)>0}
 
, 
 
 
 
 c
 =
 
 
 a
 ¯
 
 
 
 
 {\displaystyle c={\overline {a}}}
 
, 
 
 
 
 d
 =
 
 
 b
 ¯
 
 
 
 
 {\displaystyle d={\overline {b}}}
 
.

<h2 id="recurrence-and-difference-relations">Recurrence and difference relations</h2>
The sequence of continuous Hahn polynomials satisfies the recurrence relation<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

x
        
          p
          
            n
          
        
        (
        x
        )
        =
        
          p
          
            n
            +
            1
          
        
        (
        x
        )
        +
        i
        (
        
          A
          
            n
          
        
        +
        
          C
          
            n
          
        
        )
        
          p
          
            n
          
        
        (
        x
        )
        −
        
          A
          
            n
            −
            1
          
        
        
          C
          
            n
          
        
        
          p
          
            n
            −
            1
          
        
        (
        x
        )
        ,
      
    
    {\displaystyle xp_{n}(x)=p_{n+1}(x)+i(A_{n}+C_{n})p_{n}(x)-A_{n-1}C_{n}p_{n-1}(x),}

where
                
                
              
              
                
                  p
                  
                    n
                  
                
                (
                x
                )
                =
                
                  
                    
                      n
                      !
                      (
                      n
                      +
                      a
                      +
                      b
                      +
                      c
                      +
                      d
                      −
                      1
                      )
                      !
                    
                    
                      (
                      2
                      n
                      +
                      a
                      +
                      b
                      +
                      c
                      +
                      d
                      −
                      1
                      )
                      !
                    
                  
                
                
                  p
                  
                    n
                  
                
                (
                x
                ;
                a
                ,
                b
                ,
                c
                ,
                d
                )
                ,
              
            
            
              
              
                
                  A
                  
                    n
                  
                
                =
                −
                
                  
                    
                      (
                      n
                      +
                      a
                      +
                      b
                      +
                      c
                      +
                      d
                      −
                      1
                      )
                      (
                      n
                      +
                      a
                      +
                      c
                      )
                      (
                      n
                      +
                      a
                      +
                      d
                      )
                    
                    
                      (
                      2
                      n
                      +
                      a
                      +
                      b
                      +
                      c
                      +
                      d
                      −
                      1
                      )
                      (
                      2
                      n
                      +
                      a
                      +
                      b
                      +
                      c
                      +
                      d
                      )
                    
                  
                
                ,
              
            
            
              
                
                  and
                
                
              
              
                
                  C
                  
                    n
                  
                
                =
                
                  
                    
                      n
                      (
                      n
                      +
                      b
                      +
                      c
                      −
                      1
                      )
                      (
                      n
                      +
                      b
                      +
                      d
                      −
                      1
                      )
                    
                    
                      (
                      2
                      n
                      +
                      a
                      +
                      b
                      +
                      c
                      +
                      d
                      −
                      2
                      )
                      (
                      2
                      n
                      +
                      a
                      +
                      b
                      +
                      c
                      +
                      d
                      −
                      1
                      )
                    
                  
                
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}{\text{where}}\quad &p_{n}(x)={\frac {n!(n+a+b+c+d-1)!}{(2n+a+b+c+d-1)!}}p_{n}(x;a,b,c,d),\\&A_{n}=-{\frac {(n+a+b+c+d-1)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}},\\{\text{and}}\quad &C_{n}={\frac {n(n+b+c-1)(n+b+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}}.\end{aligned}}}

<h2 id="rodrigues-formula">Rodrigues formula</h2>
The continuous Hahn polynomials are given by the Rodrigues-like formula<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

Γ
                (
                a
                +
                i
                x
                )
                
                Γ
                (
                b
                +
                i
                x
                )
                
                Γ
                (
                c
                −
                i
                x
                )
                
                Γ
                (
                d
                −
                i
                x
                )
                
                
                  p
                  
                    n
                  
                
                (
                x
                ;
                a
                ,
                b
                ,
                c
                ,
                d
                )
              
            
            
              
              
                
                
                =
                
                  
                    
                      (
                      −
                      1
                      
                        )
                        
                          n
                        
                      
                    
                    
                      n
                      !
                    
                  
                
                
                  
                    
                      d
                      
                        n
                      
                    
                    
                      d
                      
                        x
                        
                          n
                        
                      
                    
                  
                
                
                  (
                  
                    Γ
                    
                      (
                      
                        a
                        +
                        
                          
                            n
                            2
                          
                        
                        +
                        i
                        x
                      
                      )
                    
                    
                    Γ
                    
                      (
                      
                        b
                        +
                        
                          
                            n
                            2
                          
                        
                        +
                        i
                        x
                      
                      )
                    
                    
                    Γ
                    
                      (
                      
                        c
                        +
                        
                          
                            n
                            2
                          
                        
                        −
                        i
                        x
                      
                      )
                    
                    
                    Γ
                    
                      (
                      
                        d
                        +
                        
                          
                            n
                            2
                          
                        
                        −
                        i
                        x
                      
                      )
                    
                  
                  )
                
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}&\Gamma (a+ix)\,\Gamma (b+ix)\,\Gamma (c-ix)\,\Gamma (d-ix)\,p_{n}(x;a,b,c,d)\\&\qquad ={\frac {(-1)^{n}}{n!}}{\frac {d^{n}}{dx^{n}}}\left(\Gamma \left(a+{\frac {n}{2}}+ix\right)\,\Gamma \left(b+{\frac {n}{2}}+ix\right)\,\Gamma \left(c+{\frac {n}{2}}-ix\right)\,\Gamma \left(d+{\frac {n}{2}}-ix\right)\right).\end{aligned}}}

<h2 id="generating-functions">Generating functions</h2>
The continuous Hahn polynomials have the following generating function:<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

∑
                  
                    n
                    =
                    0
                  
                  
                    ∞
                  
                
                
                  
                    
                      Γ
                      (
                      n
                      +
                      a
                      +
                      b
                      +
                      c
                      +
                      d
                      )
                      
                      Γ
                      (
                      a
                      +
                      c
                      +
                      1
                      )
                      
                      Γ
                      (
                      a
                      +
                      d
                      +
                      1
                      )
                    
                    
                      Γ
                      (
                      a
                      +
                      b
                      +
                      c
                      +
                      d
                      )
                      
                      Γ
                      (
                      n
                      +
                      a
                      +
                      c
                      +
                      1
                      )
                      
                      Γ
                      (
                      n
                      +
                      a
                      +
                      d
                      +
                      1
                      )
                    
                  
                
                (
                −
                i
                t
                
                  )
                  
                    n
                  
                
                
                  p
                  
                    n
                  
                
                (
                x
                ;
                a
                ,
                b
                ,
                c
                ,
                d
                )
              
            
            
              
              
                
                
                =
                (
                1
                −
                t
                
                  )
                  
                    1
                    −
                    a
                    −
                    b
                    −
                    c
                    −
                    d

3
                  
                
                
                  F
                  
                    2
                  
                
                
                  (
                  
                    
                      
                        
                          
                            
                              
                                1
                                2
                              
                            
                            (
                            a
                            +
                            b
                            +
                            c
                            +
                            d
                            −
                            1
                            )
                            ,
                            
                              
                                1
                                2
                              
                            
                            (
                            a
                            +
                            b
                            +
                            c
                            +
                            d
                            )
                            ,
                            a
                            +
                            i
                            x
                          
                        
                        
                          
                            a
                            +
                            c
                            ,
                            a
                            +
                            d
                          
                        
                      
                    
                    ;
                    −
                    
                      
                        
                          4
                          t
                        
                        
                          (
                          1
                          −
                          t
                          
                            )
                            
                              2
                            
                          
                        
                      
                    
                  
                  )
                
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}&\sum _{n=0}^{\infty }{\frac {\Gamma (n+a+b+c+d)\,\Gamma (a+c+1)\,\Gamma (a+d+1)}{\Gamma (a+b+c+d)\,\Gamma (n+a+c+1)\,\Gamma (n+a+d+1)}}(-it)^{n}p_{n}(x;a,b,c,d)\\&\qquad =(1-t)^{1-a-b-c-d}{}_{3}F_{2}\left({\begin{array}{c}{\frac {1}{2}}(a+b+c+d-1),{\frac {1}{2}}(a+b+c+d),a+ix\\a+c,a+d\end{array}};-{\frac {4t}{(1-t)^{2}}}\right).\end{aligned}}}

A second, distinct generating function is given by

∑
          
            n
            =
            0
          
          
            ∞
          
        
        
          
            
              Γ
              (
              a
              +
              c
              +
              1
              )
              
              Γ
              (
              b
              +
              d
              +
              1
              )
            
            
              Γ
              (
              n
              +
              a
              +
              c
              +
              1
              )
              
              Γ
              (
              n
              +
              b
              +
              d
              +
              1
              )
            
          
        
        
          t
          
            n
          
        
        
          p
          
            n
          
        
        (
        x
        ;
        a
        ,
        b
        ,
        c
        ,
        d
        )
        =
        
          
          
            1
          
        
        
          F
          
            1
          
        
        
          (
          
            
              
                
                  
                    a
                    +
                    i
                    x
                  
                
                
                  
                    a
                    +
                    c
                  
                
              
            
            ;
            −
            i
            t
          
          )
        
        
          
          
            1
          
        
        
          F
          
            1
          
        
        
          (
          
            
              
                
                  
                    d
                    −
                    i
                    x
                  
                
                
                  
                    b
                    +
                    d
                  
                
              
            
            ;
            i
            t
          
          )
        
        .
      
    
    {\displaystyle \sum _{n=0}^{\infty }{\frac {\Gamma (a+c+1)\,\Gamma (b+d+1)}{\Gamma (n+a+c+1)\,\Gamma (n+b+d+1)}}t^{n}p_{n}(x;a,b,c,d)=\,_{1}F_{1}\left({\begin{array}{c}a+ix\\a+c\end{array}};-it\right)\,_{1}F_{1}\left({\begin{array}{c}d-ix\\b+d\end{array}};it\right).}

<h2 id="relation-to-other-polynomials">Relation to other polynomials</h2>
<ul><li>The <a href="/facts/Wilson_polynomials/F1MasRSf">Wilson polynomials</a> are a generalization of the continuous Hahn polynomials.</li>
<li>The <a href="/facts/Bateman_polynomials/Usu03KwH">Bateman polynomials</a> Fn(x) are related to the special case a=b=c=d=1/2 of the continuous Hahn polynomials by</li></ul>

p
          
            n
          
        
        
          (
          
            x
            ;
            
              
                
                  1
                  2
                
              
            
            ,
            
              
                
                  1
                  2
                
              
            
            ,
            
              
                
                  1
                  2
                
              
            
            ,
            
              
                
                  1
                  2
                
              
            
          
          )
        
        =
        
          i
          
            n
          
        
        n
        !
        
          F
          
            n
          
        
        
          (
          
            2
            i
            x
          
          )
        
        .
      
    
    {\displaystyle p_{n}\left(x;{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {1}{2}}\right)=i^{n}n!F_{n}\left(2ix\right).}

<ul><li>The <a href="/facts/Jacobi_polynomials/5oKJKB1W">Jacobi polynomials</a> Pn(α,β)(x) can be obtained as a limiting case of the continuous Hahn polynomials:<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a></li></ul>

P
          
            n
          
          
            (
            α
            ,
            β
            )
          
        
        =
        
          lim
          
            t
            →
            ∞
          
        
        
          t
          
            −
            n
          
        
        
          p
          
            n
          
        
        
          (
          
            
              
                
                  1
                  2
                
              
            
            x
            t
            ;
            
              
                
                  1
                  2
                
              
            
            (
            α
            +
            1
            −
            i
            t
            )
            ,
            
              
                
                  1
                  2
                
              
            
            (
            β
            +
            1
            +
            i
            t
            )
            ,
            
              
                
                  1
                  2
                
              
            
            (
            α
            +
            1
            +
            i
            t
            )
            ,
            
              
                
                  1
                  2
                
              
            
            (
            β
            +
            1
            −
            i
            t
            )
          
          )
        
        .
      
    
    {\displaystyle P_{n}^{(\alpha ,\beta )}=\lim _{t\to \infty }t^{-n}p_{n}\left({\tfrac {1}{2}}xt;{\tfrac {1}{2}}(\alpha +1-it),{\tfrac {1}{2}}(\beta +1+it),{\tfrac {1}{2}}(\alpha +1+it),{\tfrac {1}{2}}(\beta +1-it)\right).}

<ul><li>Hahn, Wolfgang (1949), "Über Orthogonalpolynome, die q-Differenzengleichungen genügen", Mathematische Nachrichten, 2: 4–34, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1002%2Fmana.19490020103">10.1002/mana.19490020103</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0025-584X">0025-584X</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0030647">0030647</a></li>
<li>Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-642-05014-5">10.1007/978-3-642-05014-5</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-642-05013-8, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2656096">2656096</a></li>
<li>Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), <a href="http://dlmf.nist.gov/18.19">"Hahn Class: Definitions"</a>, in <a href="/facts/Frank_W._J._Olver/nQYQVRHc">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <a href="/facts/Digital_Library_of_Mathematical_Functions/ymNQBbgK">NIST Handbook of Mathematical Functions</a>, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-19225-5, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a>.</li>
<li>Andrews, George E.; Askey, Richard; Roy, Ranjan (1999), Special functions, Encyclopedia of Mathematics and its Applications 71, Cambridge: <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-62321-6</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Koekoek, Lesky, & Swarttouw (2010), p. 200. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Askey, R. (1985), "Continuous Hahn polynomials", J. Phys. A: Math. Gen. 18: pp. L1017-L1019. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Andrews, Askey, & Roy (1999), p. 333. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Koekoek, Lesky, & Swarttouw (2010), p. 201. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Koekoek, Lesky, & Swarttouw (2010), p. 202. <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Koekoek, Lesky, & Swarttouw (2010), p. 202. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Koekoek, Lesky, & Swarttouw (2010), p. 203. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Continuous Hahn polynomials open-in-new

Continuous Hahn polynomials