Some generating functions for \(q\)-polynomials (Q2337883)

Summary: Demonstrating the striking symmetry between calculus and \(q\)-calculus, we obtain \(q\)-analogues of the Bateman, Pasternack, Sylvester, and Cesàro polynomials. Using these, we also obtain \(q\)-analogues for some of their generating functions. Our \(q\)-generating functions are given in terms of the basic hypergeometric series \(_4\phi_5\), \(_5 \phi_5\), \(_4 \phi_3\), \(_3 \phi_2\), \(_2 \phi_1\), and \(q\)-Pochhammer symbols. Starting with our \(q\)-generating functions, we are also able to find some new classical generating functions for the Pasternack and Bateman polynomials.