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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.
Highlights
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
It is clear that the above formulas [4] and [5] for ai(n, t) cannot be computed either by hand or by computer software
It is easier to compute the quantities in brackets of the nonlinear ordinary differential equations [8] and [9] than to compute the quantity ai(n, t) in [5]
Summary
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. T = 1 n=0 and that higher order Eulerian polynomials A(nα)(t) for integers n ≥ 0 and real numbers α > 0 can be generated by 1−t ex(t−1) − t α Identity, generating function, Eulerian polynomial, higher order Eulerian polynomial, Stirling number, open problem. In [7, Theorem 1], Kims established inductively and recurrently that the generating function
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