Some identities related to Eulerian polynomials and involving the Stirling numbers

Abstract

In the paper, the authors establish two identities, which can be regarded as nonlinear differential equations, for the generating function of Eulerian polynomials, find two identities for the Stirling numbers of the second kind, present two identities for Eulerian polynomials and higher order Eulerian polynomials, and pose two open problems about summability of two finite sums involving the Stirling numbers of the second kind. Some of these conclusions meaningfully and significantly simplify several known results.

Explicit Expressions Related to Degenerate Cauchy Numbers and Their Generating Function

“…Remark 5. The motivations in the papers [5,6,9,10,11,12,13,14,15,16,20,21,24,25,26,27,30,31,32,38,35,36,37,40,41,42] are same as the one in this paper.…”

Determinantal expressions and recurrence relations for Fubini and Eulerian polynomials

“…The equations (5.12) and (5.13) were also established in [19,Theorem 1] by three alternative approaches. By similar methods to those in [23,19,28,29,31] and closely related references therein, we can easily deduce…”

“…As an inversion formula of (5.10), Theorem 4 in [29] states that 11) where s(n, k), which can be generated by…”

“…Remark 6.3. On 24 July 2018, when commenting on the paper [29] on ResearchGate, Professor Dr. Boyadzhiev (Ohio Northern University) recommended the website http://www.luschny.de/ math/euler/EulerianPolynomials.html and two papers [3,9]. He said that, when writing about Eulerian polynomials, it is good to mention that they originate from Euler and their theory is very classical-generating function and other formulas have been known for very long time.…”

Simplification of Coefficients in Two Families of Nonlinear Ordinary Differential Equations

“…Remark 4. The motivations in the papers [3,4,7,8,10,11,12,[14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]38] are same as the one in this paper. Remark 5.…”