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Β§5.18 q-Gamma and q-Beta Functions

Contents
  1. Β§5.18(i) q-Factorials
  2. Β§5.18(ii) q-Gamma Function
  3. Β§5.18(iii) q-Beta Function

Β§5.18(i) q-Factorials

5.18.1 (a;q)n=∏k=0nβˆ’1(1βˆ’a⁒qk),
n=0,1,2,…,
5.18.2 n!q=1⁒(1+q)⁒⋯⁒(1+q+β‹―+qnβˆ’1)=(q;q)n⁒(1βˆ’q)βˆ’n.

When |q|<1,

5.18.3 (a;q)∞=∏k=0∞(1βˆ’a⁒qk).

See also Β§17.2(i).

Β§5.18(ii) q-Gamma Function

When 0<q<1,

5.18.4 Ξ“q⁑(z)=(q;q)∞⁒(1βˆ’q)1βˆ’z/(qz;q)∞,
5.18.5 Ξ“q⁑(1)=Ξ“q⁑(2)=1,
5.18.6 n!q=Ξ“q⁑(n+1),
5.18.7 Ξ“q⁑(z+1)=1βˆ’qz1βˆ’q⁒Γq⁑(z).

Also, ln⁑Γq⁑(x) is convex for x>0, and the analog of the Bohr–Mollerup theorem (Β§5.5(iv)) holds.

If 0<q<r<1, then

5.18.8 Ξ“q⁑(x)<Ξ“r⁑(x),

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

5.18.9 Ξ“q⁑(x)>Ξ“r⁑(x),

when 1<x<2.

5.18.10 limqβ†’1βˆ’β‘Ξ“q⁑(z)=Γ⁑(z).

For generalized asymptotic expansions of ln⁑Γq⁑(z) as |z|β†’βˆž see Olde Daalhuis (1994) and Moak (1984). For the q-digamma or q-psi function ψq⁒(z)=Ξ“q′⁑(z)/Ξ“q⁑(z) see Salem (2013).

Β§5.18(iii) q-Beta Function

5.18.11 Bq⁑(a,b) =Ξ“q⁑(a)⁒Γq⁑(b)Ξ“q⁑(a+b).
5.18.12 Bq⁑(a,b) =∫01taβˆ’1⁒(t⁒q;q)∞(t⁒qb;q)∞⁒dqt,
0<q<1, β„œβ‘a>0, β„œβ‘b>0.

For q-integrals see Β§17.2(v).