Abstract

In the paper, in terms of the Stirling numbers of the first and second kinds, by three approaches, the author derives simple, meaningful, and significant forms for coefficients in two families of nonlinear ordinary differential equations.

1. Motivation and Main Results

In [ ⁶, Theorems 2.1 and 2.2], it was established inductively and recursively that the function

(1)

satisfies the families of nonlinear ordinary differential functions

(2)

and

(3)

for , where , , , ,

(4)

and

(5)

for 0 < k < n.

In this paper, since

(1) the original proofs of [ ⁶, Theorems 2.1 and 2.2] are long and tedious,

(2) the expressions in (4) and (5) are too complex to be remembered, understood, and computed easily,

we will provide three simple and standard proofs for [ ⁶, Theorems 2.1 and 2.2] and, more importantly, derive simple, meaningful, and significant expressions for the quantities and .

Our main results can be stated as the following theorem.

Theorem 1. For , the function defined by (1) satisfies

(6)

and

(7)

where and stand for the Stirling numbers of the first and second kinds.

2. Proofs of Theorem 1

In this section, we provide three proofs for Theorem 1 as follows.

First proof. It is well known [ ¹, Theorem 11.4] and [ ², p. 139, Theorem C] that the Faàdi Bruno formula can be described in terms of the Bell polynomials of the second kind by

(8)

The identities

(9)

and

(10)

for and can be found in [ ¹,p. 412] and [ ², p. 135]. Applying (8), (9), and (10) in sequence and denoting yield

Combining this with the identity

(11)

in [ ²⁴, Theorem 2.2] results in (6).

The identity (7) follows from applying [ ², p. 213, eq. (5c)] or [ ³⁴, p. 171, Theorem 12.1], which reads that

for a collection of constants and independent of , to (6). The first proof of Theorem 1 is complete.

Second proof. A sequence of polynomials of order are called ^{5, 9, 22} the derivative polynomials of a function if and only if for . In [ ³⁸, Theorem 1.1], it was obtained that the derivative polynomials of the function can be computed by

(12)

for and . In [ ³⁸, Theorem 1.3], it was obtained that the nonlinear differential equations

(13)

have a common solution for and . Letting in (12) and (13) gives

and

The second proof of Theorem 1 is thus complete.

Third proof. By virtue of Theorem 2.1 in ³, Theorems 3.1 and 3.2 in ³⁷, and Lemma 2.1 in ³⁸, it follows that

(14)

and

(15)

where are real constants, either and or and . See also ^{4, 35, 36}.

Taking and in (14) and (15) leads to

and

These two identities can be further rearranged as

and

The third proof of Theorem 1 is thus complete.

3. Remarks

Finally, we list several remarks on our main results and closely related things.

Remark 1. Comparing (2) and (3) with (6) and (7) figures out that

and

which are simpler, more meaningful, and more significant than the expressions in (4) and (5).

Remark 2. In [ ³⁴, p. 118, Eq. (9.18)], it is listed that

This identity is different from (11) and

(16)

for in [ ²⁴, Theorem 2.2].

Remark 3. Any one among three proofs is simpler and shorter than the one in the paper ⁶.

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. This paper is a slightly modified version of the preprint ¹³.

References