Dual q-Krawtchouk 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> by 
</p>

K
          
            n
          
        
        (
        λ
        (
        x
        )
        ;
        c
        ,
        N
        
          |
        
        q
        )
        =

3
          
        
        
          ϕ
          
            2
          
        
        (
        
          q
          
            −
            n
          
        
        ,
        
          q
          
            −
            x
          
        
        ,
        c
        
          q
          
            x
            −
            N
          
        
        ;
        
          q
          
            −
            N
          
        
        ,
        0
        
          |
        
        q
        ;
        q
        )
        ,
        
        n
        =
        0
        ,
        1
        ,
        2
        ,
        .
        .
        .
        ,
        N
        ,
      
    
    {\displaystyle K_{n}(\lambda (x);c,N|q)={}_{3}\phi _{2}(q^{-n},q^{-x},cq^{x-N};q^{-N},0|q;q),\quad n=0,1,2,...,N,}

where 
  
    
      
        λ
        (
        x
        )
        =
        
          q
          
            −
            x
          
        
        +
        c
        
          q
          
            x
            −
            N
          
        
        .
      
    
    {\displaystyle \lambda (x)=q^{-x}+cq^{x-N}.}

<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>

Dual q-Krawtchouk polynomials open-in-new

Dual q-Krawtchouk polynomials