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q-Beta Function

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A q-analog of the beta function

B(a,b)=int_0^1t^(a-1)(1-t)^(b-1)dt
(1)
=(Gamma(a)Gamma(b))/(Gamma(a+b)),
(2)

where Gamma(z) is a gamma function, is given by

B_q(a,b)=int_0^1t^(b-1)(qt;q)_(a-1)d(q,t)
(3)
=(Gamma_q(a)Gamma_q(b))/(Gamma_q(a+b)),
(4)

where Gamma_q(a) is a q-gamma function, (a;q)_n is a q-Pochhammer symbol coefficient, and

int_0^1f(x)d(q,x)=(1-q)sum_(i=0)^inftyf(q^i)q^i
(5)

is a q-integral (Andrews 1986, pp. 11-12).

See also

q-Factorial, q-Gamma Function, q-Pochhammer Symbol

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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.

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q-Beta Function

Cite this as:

Weisstein, Eric W. "q-Beta Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/q-BetaFunction.html

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