Continuous q-Hahn polynomials

<h2 id="definition">Definition</h2>
<p>The  polynomials are given in terms of <a href="/facts/Basic_hypergeometric_function/LuW9vAIs">basic hypergeometric functions</a> and the <a href="/facts/Q-Pochhammer_symbol/keASLfpe">q-Pochhammer symbol</a> by <a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>

p
          
            n
          
        
        (
        x
        ;
        a
        ,
        b
        ,
        c
        ,
        d
        
          |
        
        q
        )
        =
        
          a
          
            −
            n
          
        
        
          e
          
            −
            i
            n
            u
          
        
        (
        a
        b
        
          e
          
            2
            i
            u
          
        
        ,
        a
        c
        ,
        a
        d
        ;
        q
        
          )
          
            n

4
          
        
        
          ϕ
          
            3
          
        
        (
        
          q
          
            −
            n
          
        
        ,
        a
        b
        c
        d
        
          q
          
            n
            −
            1
          
        
        ,
        a
        
          e
          
            i
            
              (
              t
              +
              2
              u
              )
            
          
        
        ,
        a
        
          e
          
            −
            i
            t
          
        
        ;
        a
        b
        
          e
          
            2
            i
            u
          
        
        ,
        a
        c
        ,
        a
        d
        ;
        q
        ;
        q
        )
      
    
    {\displaystyle p_{n}(x;a,b,c,d|q)=a^{-n}e^{-inu}(abe^{2iu},ac,ad;q)_{n}{}_{4}\phi _{3}(q^{-n},abcdq^{n-1},ae^{i{(t+2u)}},ae^{-it};abe^{2iu},ac,ad;q;q)}

<p>
  
    
      
        x
        =
        cos
        ⁡
        (
        t
        +
        u
        )
      
    
    {\displaystyle x=\cos(t+u)}

</p>
<h2 id="gallery">Gallery</h2>
<table><tbody><tr><td></td><td></td><td></td></tr></tbody></table>
<table><tbody><tr><td></td><td></td><td></td></tr></tbody></table>

<ul><li>Gasper, George; Rahman, Mizan (2004), <i>Basic hypergeometric series</i>, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-83357-8, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2128719">2128719</a></li>
<li>Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), <i>Hypergeometric orthogonal polynomials and their q-analogues</i>, 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">"Chapter 18: Orthogonal Polynomials"</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.), <i><a href="/facts/Digital_Library_of_Mathematical_Functions/ymNQBbgK">NIST Handbook of Mathematical Functions</a></i>, 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></ul>

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

<ol>
<li id="fn:1"><p>Roelof p433, Springer 2010 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Continuous q-Hahn polynomials open-in-new

Continuous q-Hahn polynomials