A $q$-Analogue of the Euler Gamma Integral

We discuss $q$-analogues of the Euler reflection formula and the Euler gamma integral. The central role here is played by the Ramanujan $q$-extension of the Euler integral representation for the gamma function, which allows deriving the Mellin integral transformations for the $q$-polynomials of Stieltjes–Wigert, Rogers–Szegö, Laguerre, and Wall, for the alternative $q$-polynomials of Charlier, and for the little $q$-polynomials of Jacobi.