Quantum q-Krawtchouk polynomials

<h2 id="definition">Definition</h2>
The polynomials are given in terms of <a href="/facts/Basic_hypergeometric_function/LuW9vAIs">basic hypergeometric functions</a> by

K
          
            n
          
          
            q
            t
            m
          
        
        (
        
          q
          
            −
            x
          
        
        ;
        p
        ,
        N
        ;
        q
        )
        =

2
          
        
        
          ϕ
          
            1
          
        
        
          [
          
            
              
                
                  
                    
                      q
                      
                        −
                        n
                      
                    
                    ,
                    
                      q
                      
                        −
                        x
                      
                    
                  
                
                
                  
                    
                      q
                      
                        −
                        N
                      
                    
                  
                
              
            
            ;
            q
            ;
            p
            
              q
              
                n
                +
                1
              
            
          
          ]
        
        
        n
        =
        0
        ,
        1
        ,
        2
        ,
        .
        .
        .
        ,
        N
        .
      
    
    {\displaystyle K_{n}^{qtm}(q^{-x};p,N;q)={}_{2}\phi _{1}\left[{\begin{matrix}q^{-n},q^{-x}\\q^{-N}\end{matrix}};q;pq^{n+1}\right]\qquad n=0,1,2,...,N.}

<ul><li>Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, 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), Hypergeometric orthogonal polynomials and their q-analogues, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/9602214">math/9602214</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>Koekoek, Roelof; Swarttouw, René F. (1996), The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/9602214">math/9602214</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1996math......2214K">1996math......2214K</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.), <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></ul>

Quantum q-Krawtchouk polynomials open-in-new

Quantum q-Krawtchouk polynomials