title: Q-Racah polynomials
text: # Q-Racah polynomials
  
  In mathematics, the q-Racah polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Askey & Wilson (1979). Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.
  
  ## Definition
  
  The polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by
  
      $p_{n}(q^{-x}+q^{x+1}cd;a,b,c,d;q)=_{4}\phi _{3}\left[{\begin{matrix}q^{-n}&abq^{n+1}&q^{-x}&q^{x+1}cd\\aq&bdq&cq\\\end{matrix}};q;q\right]$
  
  They are sometimes given with changes of variables as
  
      $W_{n}(x;a,b,c,N;q)=_{4}\phi _{3}\left[{\begin{matrix}q^{-n}&abq^{n+1}&q^{-x}&cq^{x-n}\\aq&bcq&q^{-N}\\\end{matrix}};q;q\right]$
  
  ## Relation to other polynomials
  
  q-Racah polynomials→Racah polynomials
  
  