title: Q-difference polynomial
text: # Q-difference polynomial
  
  In combinatorial mathematics, the q-difference polynomials or q-harmonic polynomials are a polynomial sequence defined in terms of the q-derivative. They are a generalized type of Brenke polynomial, and generalize the Appell polynomials. See also Sheffer sequence.
  
  ## Definition
  
  The q-difference polynomials satisfy the relation
  
      $\left({\frac {d}{dz}}\right)_{q}p_{n}(z)={\frac {p_{n}(qz)-p_{n}(z)}{qz-z}}={\frac {q^{n}-1}{q-1}}p_{n-1}(z)=[n]_{q}p_{n-1}(z)$
  
  where the derivative symbol on the left is the q-derivative. In the limit of $q\to 1$, this becomes the definition of the Appell polynomials:
  
      ${\frac {d}{dz}}p_{n}(z)=np_{n-1}(z).$
  
  ## Generating function
  
  The generalized generating function for these polynomials is of the type of generating function for Brenke polynomials, namely
  
      $A(w)e_{q}(zw)=\sum _{n=0}^{\infty }{\frac {p_{n}(z)}{[n]_{q}!}}w^{n}$
  
  where $e_{q}(t)$ is the q-exponential:
  
      $e_{q}(t)=\sum _{n=0}^{\infty }{\frac {t^{n}}{[n]_{q}!}}=\sum _{n=0}^{\infty }{\frac {t^{n}(1-q)^{n}}{(q;q)_{n}}}.$
  
  Here, $[n]_{q}!$ is the q-factorial and
  
      $(q;q)_{n}=(1-q^{n})(1-q^{n-1})\cdots (1-q)$
  
  is the q-Pochhammer symbol. The function $A(w)$ is arbitrary but assumed to have an expansion
  
      $A(w)=\sum _{n=0}^{\infty }a_{n}w^{n}{\mbox{ with }}a_{0}\neq 0.$
  
  Any such $A(w)$ gives a sequence of q-difference polynomials.
  
  