Abstract
In the paper, by virtue of the Faà di Bruno formula, with the aid of some properties of the Bell polynomials of the second kind, and by means of a general formula for derivatives of the ratio between two differentiable functions, the authors establish explicit, determinantal, and recurrent formulas for generalized Eulerian polynomials.
Highlights
Eulerian polynomials An (t) for n ≥ 0 can be generated ([1], p. 2) by10.3390/axioms10010037 ∞ 1−t un = A ( t ), n ∑ n!
In [5], among other things, it was proven that Eulerian polynomials An (t) and higher(α) order Eulerian polynomials Ak (t) satisfy n n!
The theory of polynomials is important in mathematics and mathematical sciences
Summary
(α) and higher-order Eulerian polynomials An (t) can be generated ([2], p. 206) by Accepted: 16 March 2021. (α) and higher-order Eulerian polynomials An (t) can be generated In [3], among other things, Eulerian polynomials An (t) and higher-order published maps and institutional affil-. K =0 where S(n, k) for n ≥ k ≥ 0 denotes the Stirling numbers of the second kind. In [5], among other things, it was proven that Eulerian polynomials An (t) and higher(α) order Eulerian polynomials Ak (t) satisfy n n!. Lemma 1 below), with the help of two properties of the Bell polynomials of the second kind (see Lemmas 2 and 3 below), and by means of a general formula for derivatives of the ratio between two differentiable functions (see Lemma 4 below), we establish explicit, determinantal, and recurrent formulas for generalized Eulerian polynomials Tn (t, a, d)
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