5.17 Barnes’ \mathop{G\/}\nolimits-Function (Double Gamma Function)5.19 Mathematical Applications

§5.18 q-Gamma and q-Beta Functions

Contents

§5.18(ii) q-Gamma Function

When 0<q<1,

5.18.4 \mathop{\Gamma _{{q}}\/}\nolimits\!\left(z\right)=\left(q;q\right)_{{\infty}}(1-q)^{{1-z}}/\left(q^{z};q\right)_{{\infty}},
5.18.5 \mathop{\Gamma _{{q}}\/}\nolimits\!\left(1\right)=\mathop{\Gamma _{{q}}\/}\nolimits\!\left(2\right)=1,
5.18.6 n!_{q}=\mathop{\Gamma _{{q}}\/}\nolimits\!\left(n+1\right),
5.18.7 \mathop{\Gamma _{{q}}\/}\nolimits\!\left(z+1\right)=\frac{1-q^{z}}{1-q}\mathop{\Gamma _{{q}}\/}\nolimits\!\left(z\right).

Also, \mathop{\ln\/}\nolimits\mathop{\Gamma _{{q}}\/}\nolimits\!\left(x\right) is convex for x>0, and the analog of the Bohr-Mollerup theorem (§5.5(iv)) holds.

For generalized asymptotic expansions of \mathop{\ln\/}\nolimits\mathop{\Gamma _{{q}}\/}\nolimits\!\left(z\right) as |z|\to\infty see Olde Daalhuis (1994) and Moak (1984).

§5.18(iii) q-Beta Function