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5 Gamma FunctionProperties

§5.18 q-Gamma and q-Beta Functions


§5.18(i) q-Factorials

5.18.1 (a;q)n=k=0n-1(1-aqk),
5.18.2 n!q=1(1+q)(1+q++qn-1)=(q;q)n(1-q)-n.

When |q|<1,

5.18.3 (a;q)=k=0(1-aqk).

See also §17.2(i).

§5.18(ii) q-Gamma Function

When 0<q<1,

5.18.4 Γq(z)=(q;q)(1-q)1-z/(qz;q),
5.18.5 Γq(1)=Γq(2)=1,
5.18.6 n!q=Γq(n+1),
5.18.7 Γq(z+1)=1-qz1-qΓq(z).

Also, lnΓq(x) is convex for x>0, and the analog of the Bohr-Mollerup theorem (§5.5(iv)) holds.

If 0<q<r<1, then

5.18.8 Γq(x)<Γr(x),

when 0<x<1 or when x>2, and

5.18.9 Γq(x)>Γr(x),

when 1<x<2.

5.18.10 limq1-Γq(z)=Γ(z).

For generalized asymptotic expansions of lnΓq(z) as |z| see Olde Daalhuis (1994) and Moak (1984). For the q-digamma or q-psi function ψq(z)=Γq(z)/Γq(z) see Salem (2013).

§5.18(iii) q-Beta Function

5.18.11 Bq(a,b) =Γq(a)Γq(b)Γq(a+b).
5.18.12 Bq(a,b) =01ta-1(tq;q)(tqb;q)dqt,
0<q<1, a>0, b>0.

For q-integrals see §17.2(v).