| Summary: | Demonstrating the striking symmetry between calculus and <i>q</i>-calculus, we obtain <i>q</i>-analogues of the Bateman, Pasternack, Sylvester, and Cesàro polynomials. Using these, we also obtain <i>q</i>-analogues for some of their generating functions. Our <i>q</i>-generating functions are given in terms of the basic hypergeometric series <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mrow></mrow> <mn>4</mn> </msub> <msub> <mi>ϕ</mi> <mn>5</mn> </msub> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mrow></mrow> <mn>5</mn> </msub> <msub> <mi>ϕ</mi> <mn>5</mn> </msub> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mrow></mrow> <mn>4</mn> </msub> <msub> <mi>ϕ</mi> <mn>3</mn> </msub> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mrow></mrow> <mn>3</mn> </msub> <msub> <mi>ϕ</mi> <mn>2</mn> </msub> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mrow></mrow> <mn>2</mn> </msub> <msub> <mi>ϕ</mi> <mn>1</mn> </msub> </mrow> </semantics> </math> </inline-formula>, and <i>q</i>-Pochhammer symbols. Starting with our <i>q</i>-generating functions, we are also able to find some new classical generating functions for the Pasternack and Bateman polynomials.
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