q-Beta Function

A q-Analog of the Beta Function

$$B(a,b)=\int_0^1 t^{a-1}(1-t)^{q-1}\,dt={\Gamma(a)\Gamma(b)\over\Gamma(a+b)},$$

where $\Gamma(z)$ is a Gamma Function, is given by

$$B_q(a,b)\equiv \int_0^1 t^{b-1}(qt;q)_{a-1}\,d(a,t)={\Gamma_q(b)\Gamma_q(a)\over\Gamma_q(a+b)},$$

where $\Gamma_q(a)$ is a q-Gamma Function and $(a;q)_n$ is a q-Series coefficient (Andrews 1986, pp. 11-12).

See also q-Factorial, q-Gamma Function

References

Andrews, G. E. $q$-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., 1986.