Let $$S_k(t)=(1-x)(1-qx)\dots(1-q^k x)\sum_{n\geq0}{(1+q+\dots+q^n)^k x^n}$$ be a q-analog of Carlitz's identity. I look for references where the properties of these Eulerian polynomials are studied.
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$\begingroup$ This seems different from the two $q$-analogues, given arxiv.org/pdf/1201.4941.pdf and mathoverflow.net/questions/322848/… $\endgroup$– T. AmdeberhanCommented Mar 8, 2021 at 18:21
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$\begingroup$ There are some lines missing from the question. I don't know whether this is a problem with my computer or with MO. $\endgroup$– Richard StanleyCommented Mar 8, 2021 at 20:03
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$\begingroup$ @T.Amdeberhan: still not good. $\endgroup$– Richard StanleyCommented Mar 8, 2021 at 20:31
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$\begingroup$ @T.Amdeberhan: for some reason it is okay now. $\endgroup$– Richard StanleyCommented Mar 9, 2021 at 1:50
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$\begingroup$ This is the joint distribution of DES and MAJ on the symmetric group $\mathfrak{S}_n$. I don't know, however, where your formula first appeared. $\endgroup$– Richard StanleyCommented Mar 9, 2021 at 2:04
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