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.