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Some generating functions for \(q\)-polynomials. (English) Zbl 1425.33009
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.
MSC:
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
33C20 Generalized hypergeometric series, \({}_pF_q\)
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