VIEWING SOME ORDINARY DIFFERENTIAL EQUATIONS FROM THE ANGLE OF DERIVATIVE POLYNOMIALS

In the paper, the authors view some ordinary differential equations and their solutions from the angle of (the generalized) derivative polynomials and simplify some known identities for the Bernoulli numbers and polynomials, the Frobenius-Euler polynomials, the Euler numbers and polynomials, in terms of the Stirling numbers of the first and second kinds.


Introduction
We recall some results obtained in two different and independent directions.We also recall below a concept in combinatorics about higher derivatives of functions.
1.1.The first direction.In [13, p. 1127 and 1131], it was obtained that 1 In the preprint [17] and its formally published version [6], the above two identities were generalized inductively and recursively by the following eight identities: , , , , where 1 ≤ m ≤ k, the quantities and S(i + 1, i + 1) S(i + 2, i + 1) • • • S(i + j − 1, i + 1) for j ∈ N. In [6, Remark 5.1], it was pointed out that the four functions 1 − e −t , and 1 e t − 1 are respectively the unique solutions to the linear ordinary equations for all k ∈ N and n = 1, 2, where In [6,Remark 5.2], it was concluded that All of the above results in [6,17] were established by induction.
In [25, Theorems 3.1 and 3.2], by the Faà di Bruno formula and combinatorial techniques, the above identities in [6,17] were generalized and unified as follows.For real numbers α, λ ∈ R, ; where s(n, k) for n ≥ k ≥ 1 denote the Stirling numbers of the first kind.
In [4, Theorem 2.1], the above identities in [6,17,25] were inductively proved once again and were rewritten as follows.For real constants λ = 0 and α = 0 and for k ∈ N, when λ > 0 and t = − ln λ α or when λ < 0 and t ∈ R, we have 1.2.The second direction.In [9,10], it was obtained inductively and recursively that (1) the function F (t) = 1 u−e t is a solution of the nonlinear differential equation (2) for m ∈ N and n ≥ 0, we have , where H n (u) are called the nth Frobenius-Euler numbers which can be generated by (3) for m ∈ N and n ≥ 0, we have (4) for m ∈ N and n ≥ 0, we have where H n (x|u) are called the Frobenius-Euler polynomials which can be generated by Influenced by the papers [9,10], the authors in [7] derived inductively and recursively several formulas for the Bernoulli polynomials of the rth order n (0).In [22,Theorem 2.1], it was procured inductively and recursively that the function F (t) = 1 e t +1 is a solution of the nonlinear differential equation By virtue of this, as did in [7], the authors of [22] presented several formulas for the Euler numbers and polynomials of the rth order E (r) in terms of the so-called Euler numbers n (0) which are different from the classical and traditional Euler numbers In [8], it was acquired inductively and recursively that the function is a solution of the nonlinear differential equation for n ∈ N, where Thereafter, the authors in [8] established an identity involving the higher order Bernoulli numbers of the second kind b In [12], it was procured inductively and recursively that (1) the function for n ∈ N, where (2) the function for n ∈ N. Hereafter, the authors in [12] gave some new identities involving degenerate Euler numbers and polynomials.
For x ∈ R, let denotes the falling factorial of x.In [3, Theorem 2.1], it was presented inductively and recursively among other things that the function F (t) = (1+t) x 2+t is a solution of the linear differential equations for n ≥ 0, where a 0 (n, x) = (−1) n n! and In [11], it was demonstrated inductively and recursively that (1) the function is a solution of the nonlinear differential equation where (1.11) Hereafter, the authors gave two identities for the Changhee polynomials of the rth order Ch (r) n (x) defined by t n n! .
1.3.Derivative polynomials.Suppose f is a function whose derivative is a polynomial in f , that is, f (x) = P (f (x)) for some polynomial P .Then all the higher order derivatives of f are also polynomials in f , so we have a sequence of polynomials P n defined by f (n) (x) = P n (f (x)) for n ≥ 0. As usual, we call P n (u) the derivative polynomials of f .See [2,24] and closely related references therein.Now we more accurately introduce a new notion below.If there exists a sequence of polynomials P n (u) = n k=0 a k u k for a k ∈ C and a sequence of functions h n,k (x) for n, k ≥ 0 such that f (n) (x) = n+q k=0 a k h n,k (x)f k (x) for n, q ≥ 0, then we call P n+q (u) the generalized derivative polynomials of f (x) with respect to h n,k (x).

Alternative viewpoints and derivative polynomials
Now we are in a position to discuss the above conclusions from alternative viewpoints and (the generalized) derivative polynomials.
2.1.Almost all the above linear or nonlinear ordinary differential equations and their solutions can be alternatively regarded as problems of (the generalized) derivative polynomials.
2.2.Taking λ = −1 and α = 1 in (1.1) and simplifying yield Comparing this with (1.10) reveals that the coefficients a n,k in (1.11) is just equal to (k − 1)!S(n + 1, k).In other words, We can also say that the generalized derivative polynomials of the function 1 e t +1 is with respect to h k,m (t) ≡ 1 for k ≥ 1 and 1 ≤ m ≤ k +1.It is much straightforward and simple to see that the derivative polynomials of the function 1 t+c is P n (x) = (−1) n n!x n+1 .Consequently, we find an alternative viewpoint to examine the results in [11].
2.3.By Leibnlz's theorem for differentiation of a product, we obtain This implies that the coefficients a j (n, x) in (1.9) can be simplified explicitly as Moreover, it follows that In other words, the generalized derivative polynomials of the function (1+t) x 2+t is P 1+0 (u) = u with respect to 2.4.In combinatorial analysis, the Faà di Bruno formula plays an important role and can be described in terms of the Bell polynomials of the second kind for n ≥ 0, see [1,p. 139,Theorem C].Replacing f (u) and u = h(t) in (2.1) respectively by 1 u±1 and (1 + λt) 1/λ yields where we used in the above line the formula listed in [1, p. 135].
In [1, p. 133], it was listed that for k ≥ 0. Therefore, we obtain that Accordingly, it follows that In a word, we derive that for n ≥ 0. This means that the functions 1 (1+λt) 1/λ ±1 have the same generalized derivative polynomials By the way, we note that the form of the equation (2.3) is different from the one in (1.6) and (1.8).The coefficients in the brackets on the right-hand side of the equation (2.3) are more nice, more explicit, easier to compute than the coefficients a i (n, λ) defined by (1.7) in (1.6) and (1.8).
2.5.In [14,18], it was obtained that where a n,2 = (n − 1)! and Comparing this with (1.5) and rearranging lead to This connects (1.5) with the Stirling numbers of the first kind s(n, i) and the generalized derivative polynomials: the generalized derivative polynomials of with respect to h m,i ≡ 1 (1+t) m for m ≥ 0. Hence, we are viewing the paper [8] from a different angle., n ∈ N. (2.5) As a result, Theorem 2.1 in [22] can be simplified as (2.4) and has something to do with the Stirling numbers of the first kind s(n, k).

Setting
for k ∈ N. Comparing this with (1.3) recovers the formula (2.5) which is an alternative expression for the Stirling numbers of the first kind s(n, k).Therefore, all the above-mentioned results in [9,10] can be restated simply in terms of the Stirling numbers of the first kind s(n, k).
2.8.In [4,6,25], some formulas for computing the Bernoulli numbers, the Euler polynomials, the Apostol-Bernoulli numbers, the Eulerian polynomials, and the Fubini numbers in terms of the Stirling numbers of the second kind were established.These formulas are more concise, simpler, more meaningful than those in [7,8,9,10,11,22], as showed above.Due to limitation on the length of the paper, we will not elaborate in further detail.2.9.In [5,15,20,21,23,24], there are more information and new conclusions about (the generalized) derivative polynomials, explicit formulas for the Bernoulli numbers and polynomials, for the Euler numbers and polynomials, for higher derivatives of some elementary functions, properties of the functions ±1 e ±t −1 , and the like.Due to limitation on the length of the paper, we will not elaborate in further detail yet.
2.10.It should be common knowledge that mathematicians should try to represent, explain, or interpret a new mathematical quantities in terms of some known and popular quantities in mathematics.Once a new mathematical quantity were established connections or relations with some famous or important quantities, it would be more meaningful in mathematics.