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Carlitz and Scoville introduced the polynomials $A_n(x,y|{\alpha},{\beta})$,which we refer to as the $(\alpha, \beta)$-Eulerian polynomials. Thesepolynomials count permutations based on Eulerian-Stirling statistics, includingdescents, ascents, left-to-right maxima, and right-to-left maxima. Carlitz andScoville obtained the generating function of $A_n(x,y|{\alpha},{\beta})$. Inthis paper, we introduce a new family of polynomials,$P_n(u_1,u_2,u_3,u_4|{\alpha},{\beta})$, defined on permutations, incorporatingdescent-Stirling statistics including valleys, exterior peaks, right doubledescents, left double ascents, left-to-right maxima, and right-to-left maxima.By employing the grammatical calculus introduced by Chen, we establish theconnection between the generating function of$P_n(u_1,u_2,u_3,u_4|{\alpha},{\beta})$ and the generating function of the$(\alpha,\beta)$-Eulerian polynomials $A_n(x,y|{\alpha},{\beta})$ introduced byCarlitz and Scoville. Using this connection, we derive the generating functionof $P_n(u_1,u_2,u_3,u_4|{\alpha},{\beta})$, which can be specialized to obtainthe $(\alpha,\beta)$-extensions of generating functions for peaks, left peaks,double ascents, right double ascents and left-right double ascents given byDavid-Barton, Elizalde and Noy, Entringer, Gessel, Kitaev and Zhuang. Moreover,we establish two relations between $P_n(u_1,u_2,u_3,u_4|{\alpha},{\beta})$ and$A_n(x,y|{\alpha},{\beta})$, which enable us to derive$(\alpha,\beta)$-extensions of results of Stembridge, Petersen, Br\"and\'en,and Zhuang. Specializing $(\alpha,\beta)$-extensions of Stembridge's formulaand the left peak version of Stembridge's formula allows us to derive the$(\alpha,\beta)$-extensions of the tangent and secant numbers.

The $(\alpha,\beta)$-Eulerian Polynomials and Descent-Stirling Statistics on Permutations