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\(q\)-Eulerian polynomials and polynomials with only real zeros. (English) Zbl 1180.05009

Summary: Let \(f\) and \(F\) be two polynomials satisfying \(F(x)=u(x)f(x)+v(x)f^{\prime}(x)\). We characterize the relation between the location and multiplicity of the real zeros of \(f\) and \(F\), which generalizes and unifies many known results, including the results of F. Brenti [“Unimodal, log-concave and Pólya frequency sequences in combinatorics”, Mem. Am. Math. Soc. 413 (1989; Zbl 0697.05011)] and P. Brändén [“On linear transformations preserving the Pólya frequency property”, Trans. Am. Math. Soc. 358, No.8, 3697–3716 (2006; Zbl 1086.05007)] about the \(q\)-Eulerian polynomials.

MSC:

05A15 Exact enumeration problems, generating functions
26C10 Real polynomials: location of zeros
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