On q-Raabe integral and some extensions

Abstract

The Raabe integral $$\int _{0}^1 \log \Gamma _q(x) \,dx$$ for q-gamma function where $$0<q<1$$ was obtained independently by Mahmoud and Agarwal as well as El Bachraoui. We obtain a third method for the computation of this integral. We extend further the q-Raabe integral by evaluating $$\int _{0}^1 x^{k}\log \Gamma _q(x) \,dx$$ for $$k=0,1,\ldots $$ providing q-analogues for $$\int _{0}^1 x^{k}\log \Gamma (x) \,dx$$.