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In this paper, the asymptotic formulas for Eulerian numbers, refined Euleriannumbers and the coefficients of descent polynomials are obtained directly fromthe spline interpretations of these numbers. Having related these numbersdirectly to B-splines [15], we can take advantage of many powerful splinetechniques to derive various results of these numbers. The asymptotic formulasfor the Eulerian numbers Ad;k agree with the previously known results whichwere given by L. Carlitz et al.(1972)[2] and S.Tanny (1973) [18], but theconvergence order is much better. We also give the asymptotic representationsof refined Eulerian numbers which is in terms of the Hermite polynomials.

The asymptotic properties of Eulerian numbers and refined Eulerian numbers: A Spline perspective