The 
-binomial
coefficient is a two-parameter generalization of the
q-binomial coefficient. For nonnegative
integers 
and 
with 
,
it is defined by
![[n; k]_(p,q)=product_(i=1)^k(p^(n-i+1)-q^(n-i+1))/(p^i-q^i)=([n]_(p,q)!)/([k]_(p,q)![n-k]_(p,q)!),](/images/equations/pq-BinomialCoefficient/NumberedEquation1.svg) |
where 
,
![[j]_(p,q)=(p^j-q^j)/(p-q)](/images/equations/pq-BinomialCoefficient/Inline6.svg)
,
and the corresponding factorial is ![[j]_(p,q)!=[j]_(p,q)[j-1]_(p,q)...[1]_(p,q)](/images/equations/pq-BinomialCoefficient/Inline7.svg)
. The coefficient
is defined to be zero for 
(Corcino 2008).
It is symmetric under interchange of 
and 
, reduces to the q-binomial
coefficient when 
, and reduces to the binomial
coefficient in the limit 
. It satisfies the recurrence
equation
![[n+1; k]_(p,q)=p^k[n; k]_(p,q)+q^(n-k+1)[n; k-1]_(p,q).](/images/equations/pq-BinomialCoefficient/NumberedEquation2.svg) |
-Binomial Coefficients." Integers 8, A29,
1-16, 2008.