(p,q)-Binomial Coefficient

The -binomial coefficient is a two-parameter generalization of the q-binomial coefficient. For nonnegative integers and with , it is defined by

where, initially for , , and the corresponding factorial is . After cancellation, the product and factorial quotient in the definition give the same polynomial in and . At parameter values for which a displayed denominator vanishes, the coefficient is defined by the value of this polynomial continuation. 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